https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Myh2910&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T05:07:01ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=Schur%27s_Inequality&diff=156251Schur's Inequality2021-06-17T22:19:52Z<p>Myh2910: </p>
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<div>'''Schur's Inequality''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai Schur.<br />
<br />
== Theorem ==<br />
Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>0</math>:<br />
<br />
<cmath>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0</cmath><br />
<br />
The four [[equality condition | equality cases]] occur when <math>a=b=c</math> or when two of <math>a,b,c</math> are equal and the third is <math>{0}</math>.<br />
<br />
=== Common Cases ===<br />
The <math>r=1</math> case yields the well-known inequality:<br />
<cmath>a^3+b^3+c^3+3abc \ge a^2 b+a^2 c+b^2 a+b^2 c+c^2 a+c^2 b</cmath><br />
<br />
When <math>r=2</math>, an equivalent form is:<br />
<cmath>a^4+b^4+c^4+abc(a+b+c) \ge a^3 b+a^3 c+b^3 a+b^3 c+c^3 a+c^3 b</cmath><br />
<br />
=== Proof ===<br />
<br />
Without loss of Generality, let <math>{a\ge b\ge c}</math>. Note that <math>a^r(a-b)(a-c)+b^r(b-a)(b-c)</math> <math>= a^r(a-b)(a-c)-b^r(a-b)(b-c) = (a-b)(a^r(a-c)-b^r(b-c))</math>. Clearly, <math>a^r\ge b^r \ge 0</math>, and <math>a-c \geq b-c \geq 0</math>. Thus, <math>(a-b)(a^r(a-c)-b^r(b-c)) \geq 0 \implies a^r(a-b)(a-c)+b^r(b-a)(b-c) \geq 0</math>. However, <math>c^r(c-a)(c-b) \geq 0</math>, and thus the proof is complete.<br />
<br />
=== Generalized Form ===<br />
It has been shown by [[Valentin Vornicu]] that a more general form of Schur's Inequality exists. Consider <math>a,b,c,x,y,z \in \mathbb{R}</math>, where <math>{a \geq b \geq c}</math>, and either <math>x \geq y \geq z</math> or <math>z \geq y \geq x</math>. Let <math>k \in \mathbb{Z}^{+}</math>, and let <math>f:\mathbb{R} \rightarrow \mathbb{R}_{0}^{+}</math> be either convex or monotonic. Then,<br />
<cmath>f(x)(a-b)^k(a-c)^k+f(y)(b-a)^k(b-c)^k+f(z)(c-a)^k(c-b)^k \geq 0.</cmath><br />
<br />
The standard form of Schur's is the case of this inequality where <math>x=a,\ y=b,\ z=c,\ k=1,\ f(m)=m^r</math>.<br />
<br />
== References ==<br />
* Mildorf, Thomas; ''Olympiad Inequalities''; January 20, 2006; <http://artofproblemsolving.com/articles/files/MildorfInequalities.pdf><br />
<br />
* Vornicu, Valentin; ''Olimpiada de Matematica... de la provocare la experienta''; GIL Publishing House; Zalau, Romania.<br />
<br />
==See Also==<br />
* [[Olympiad Mathematics]]<br />
* [[Inequalities]]<br />
* [[Number Theory]]<br />
<br />
[[Category:Inequality]]<br />
[[Category:Theorems]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=Chicken_McNugget_Theorem&diff=150387Chicken McNugget Theorem2021-03-27T03:08:11Z<p>Myh2910: /* Olympiad */</p>
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<div>The '''Chicken McNugget Theorem''' (or '''Postage Stamp Problem''' or '''Frobenius Coin Problem''') states that for any two [[relatively prime]] [[positive integer]]s <math>m,n</math>, the greatest integer that cannot be written in the form <math>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>.<br />
<br />
A consequence of the theorem is that there are exactly <math>\frac{(m - 1)(n - 1)}{2}</math> positive integers which cannot be expressed in the form <math>am + bn</math>. The proof is based on the fact that in each pair of the form <math>(k, mn-m-n-k)</math>, exactly one element is expressible.<br />
<br />
== Origins ==<br />
There are many stories surrounding the origin of the Chicken McNugget theorem. However, the most popular by far remains that of the Chicken McNugget. Originally, McDonald's sold its nuggets in packs of 9 and 20. Math enthusiasts were curious to find the largest number of nuggets that could not have been bought with these packs, thus creating the Chicken McNugget Theorem (the answer worked out to be 151 nuggets). The Chicken McNugget Theorem has also been called the Frobenius Coin Problem or the Frobenius Problem, after German mathematician Ferdinand Frobenius inquired about the largest amount of currency that could not have been made with certain types of coins.<br />
<br />
<br />
<br />
<br />
<br />
==Proof Without Words==<br />
<math>\begin{array}{ccccccc}<br />
0\mod{m}&1\mod{m}&2\mod{m}&...&...&...&(m-1)\mod{m}\\<br />
\hline<br />
\cancel{0n}&1&2&&...&&m-1\\<br />
\cancel{0n+m}&...&&\vdots&&...&\\<br />
\cancel{0n+2m}&...&&\cancel{1n}&&...&\\<br />
\cancel{0n+3m}&&&\cancel{1n+m}&&\vdots&\\<br />
\cancel{0n+4m}&&&\cancel{1n+2m}&&\cancel{2n}&\\<br />
\cancel{0n+5m}&&&\cancel{1n+3m}&&\cancel{2n+m}&\\<br />
\vdots&&&\vdots&&\vdots&\\<br />
\cancel{\qquad}&\cancel{\qquad}&\cancel{ \qquad}&\cancel{ \qquad}&\mathbf{(m-1)n-m}&\cancel{\qquad }&\cancel{\qquad }\\<br />
\cancel{\qquad}&\cancel{\qquad}&\cancel{ \qquad}&\cancel{ \qquad}&\cancel{(m-1)n}&\cancel{\qquad }&\cancel{\qquad }<br />
\end{array}</math><br />
<br />
==Proof 1==<br />
<b>Definition</b>. An integer <math>N \in \mathbb{Z}</math> will be called <i>purchasable</i> if there exist nonnegative integers <math>a,b</math> such that <math>am+bn = N</math>.<br />
<br />
We would like to prove that <math>mn-m-n</math> is the largest non-purchasable integer. We are required to show that: <br />
<br />
(1) <math>mn-m-n</math> is non-purchasable<br />
<br />
(2) Every <math>N > mn-m-n</math> is purchasable<br />
<br />
Note that all purchasable integers are nonnegative, thus the set of non-purchasable integers is nonempty.<br />
<br />
<b>Lemma</b>. Let <math>A_{N} \subset \mathbb{Z} \times \mathbb{Z}</math> be the set of solutions <math>(x,y)</math> to <math>xm+yn = N</math>. Then <math>A_{N} = \{(x+kn,y-km) \;:\; k \in \mathbb{Z}\}</math> for any <math>(x,y) \in A_{N}</math>.<br />
<br />
<i>Proof</i>: By [[Bezout's Lemma]], there exist integers <math>x',y'</math> such that <math>x'm+y'n = 1</math>. Then <math>(Nx')m+(Ny')n = N</math>. Hence <math>A_{N}</math> is nonempty. It is easy to check that <math>(Nx'+kn,Ny'-km) \in A_{N}</math> for all <math>k \in \mathbb{Z}</math>. We now prove that there are no others. Suppose <math>(x_{1},y_{1})</math> and <math>(x_{2},y_{2})</math> are solutions to <math>xm+yn=N</math>. Then <math>x_{1}m+y_{1}n = x_{2}m+y_{2}n</math> implies <math>m(x_{1}-x_{2}) = n(y_{2}-y_{1})</math>. Since <math>m</math> and <math>n</math> are coprime and <math>m</math> divides <math>n(y_{2}-y_{1})</math>, <math>m</math> divides <math>y_{2}-y_{1}</math> and <math>y_{2} \equiv y_{1} \pmod{m}</math>. Similarly <math>x_{2} \equiv x_{1} \pmod{n}</math>. Let <math>k_{1},k_{2}</math> be integers such that <math>x_{2}-x_{1} = k_{1}n</math> and <math>y_{2}-y_{1} = k_{2}m</math>. Then <math>m(-k_{1}n) = n(k_{2}m)</math> implies <math>k_{1} = -k_{2}.</math> We have the desired result. <math>\square</math><br />
<br />
<b>Lemma</b>. For any integer <math>N</math>, there exists unique <math>(a_{N},b_{N}) \in \mathbb{Z} \times \{0,1,\ldots,m-1\}</math> such that <math>a_{N}m + b_{N}n = N</math>.<br />
<br />
<i>Proof</i>: By the division algorithm, there exists one and only one <math>k</math> such that <math>0 \le y-km \le m-1</math>. <math>\square</math><br />
<br />
<b>Lemma</b>. <math>N</math> is purchasable if and only if <math>a_{N} \ge 0</math>.<br />
<br />
<i>Proof</i>: If <math>a_{N} \ge 0</math>, then we may simply pick <math>(a,b) = (a_{N},b_{N})</math> so <math>N</math> is purchasable. If <math>a_{N} < 0</math>, then <math>a_{N}+kn < 0</math> if <math>k \le 0</math> and <math>b_{N}-km < 0</math> if <math>k > 0</math>, hence at least one coordinate of <math>(a_{N}+kn,b_{N}-km)</math> is negative for all <math>k \in \mathbb{Z}</math>. Thus <math>N</math> is not purchasable. <math>\square</math><br />
<br />
Thus the set of non-purchasable integers is <math>\{xm+yn \;:\; x<0,0 \le y \le m-1\}</math>. We would like to find the maximum of this set. <br />
Since both <math>m,n</math> are positive, the maximum is achieved when <math>x = -1</math> and <math>y = m-1</math> so that <math>xm+yn = (-1)m+(m-1)n = mn-m-n</math>.<br />
<br />
==Proof 2==<br />
We start with this statement taken from [[Fermat%27s_Little_Theorem#Proof_2_.28Inverses.29|Proof 2 of Fermat's Little Theorem]]:<br />
<br />
"Let <math>S = \{1,2,3,\cdots, p-1\}</math>. Then, we claim that the set <math>a \cdot S</math>, consisting of the product of the elements of <math>S</math> with <math>a</math>, taken modulo <math>p</math>, is simply a permutation of <math>S</math>. In other words, <br />
<br />
<center><cmath>S \equiv \{1a, 2a, \cdots, (p-1)a\} \pmod{p}.</cmath></center><br><br />
<br />
Clearly none of the <math>ia</math> for <math>1 \le i \le p-1</math> are divisible by <math>p</math>, so it suffices to show that all of the elements in <math>a \cdot S</math> are distinct. Suppose that <math>ai \equiv aj \pmod{p}</math> for <math>i \neq j</math>. Since <math>\text{gcd}\, (a,p) = 1</math>, by the cancellation rule, that reduces to <math>i \equiv j \pmod{p}</math>, which is a contradiction."<br />
<br />
Because <math>m</math> and <math>n</math> are coprime, we know that multiplying the residues of <math>m</math> by <math>n</math> simply permutes these residues. Each of these permuted residues is purchasable (using the definition from Proof 1), because, in the form <math>am+bn</math>, <math>a</math> is <math>0</math> and <math>b</math> is the original residue. We now prove the following lemma.<br />
<br />
<b>Lemma</b>: For any nonnegative integer <math>c < m</math>, <math>cn</math> is the least purchasable number <math>\equiv cn \bmod m</math>.<br />
<br />
<i>Proof</i>: Any number that is less than <math>cn</math> and congruent to it <math>\bmod m</math> can be represented in the form <math>cn-dm</math>, where <math>d</math> is a positive integer. If this is purchasable, we can say <math>cn-dm=am+bn</math> for some nonnegative integers <math>a, b</math>. This can be rearranged into <math>(a+d)m=(c-b)n</math>, which implies that <math>(a+d)</math> is a multiple of <math>n</math> (since <math>\gcd(m, n)=1</math>). We can say that <math>(a+d)=gn</math> for some positive integer <math>g</math>, and substitute to get <math>gmn=(c-b)n</math>. Because <math>c < m</math>, <math>(c-b)n < mn</math>, and <math>gmn < mn</math>. We divide by <math>mn</math> to get <math>g<1</math>. However, we defined <math>g</math> to be a positive integer, and all positive integers are greater than or equal to <math>1</math>. Therefore, we have a contradiction, and <math>cn</math> is the least purchasable number congruent to <math>cn \bmod m</math>. <math>\square</math><br />
<br />
This means that because <math>cn</math> is purchasable, every number that is greater than <math>cn</math> and congruent to it <math>\bmod m</math> is also purchasable (because these numbers are in the form <math>am+bn</math> where <math>b=c</math>). Another result of this Lemma is that <math>cn-m</math> is the greatest number <math>\equiv cn \bmod m</math> that is not purchasable. <math>c \leq m-1</math>, so <math>cn-m \leq (m-1)n-m=mn-m-n</math>, which shows that <math>mn-m-n</math> is the greatest number in the form <math>cn-m</math>. Any number greater than this and congruent to some <math>cn \bmod m</math> is purchasable, because that number is greater than <math>cn</math>. All numbers are congruent to some <math>cn</math>, and thus all numbers greater than <math>mn-m-n</math> are purchasable.<br />
<br />
Putting it all together, we can say that for any coprime <math>m</math> and <math>n</math>, <math>mn-m-n</math> is the greatest number not representable in the form <math>am + bn</math> for nonnegative integers <math>a, b</math>. <math>\square</math><br />
<br />
==Corollary==<br />
This corollary is based off of Proof 2, so it is necessary to read that proof before this corollary. We prove the following lemma.<br />
<br />
<b>Lemma:</b> For any integer <math>k</math>, exactly one of the integers <math>k</math>, <math>mn-m-n-k</math> is not purchasable.<br />
<br />
<i>Proof</i>: Because every number is congruent to some residue of <math>m</math> permuted by <math>n</math>, we can set <math>k \equiv cn \bmod m</math> for some <math>c</math>. We can break this into two cases.<br />
<br />
<i>Case 1</i>: <math>k \leq cn-m</math>. This implies that <math>k</math> is not purchasable, and that <math>mn-m-n-k \geq mn-m-n-(cn-m) = n(m-1-c)</math>. <math>n(m-1-c)</math> is a permuted residue, and a result of the lemma in Proof 2 was that a permuted residue is the least number congruent to itself <math>\bmod m</math> that is purchasable. Therefore, <math>mn-m-n-k \equiv n(m-1-c) \bmod m</math> and <math>mn-m-n-k \geq n(m-1-c)</math>, so <math>mn-m-n-k</math> is purchasable.<br />
<br />
<i>Case 2</i>: <math>k > cn-m</math>. This implies that <math>k</math> is purchasable, and that <math>mn-m-n-k < mn-m-n-(cn-m) = n(m-1-c)</math>. Again, because <math>n(m-1-c)</math> is the least number congruent to itself <math>\bmod m</math> that is purchasable, and because <math>mn-m-n-k \equiv n(m-1-c) \bmod m</math> and <math>mn-m-n-k < n(m-1-c)</math>, <math>mn-m-n-k</math> is not purchasable.<br />
<br />
We now limit the values of <math>k</math> to all integers <math>0 \leq k \leq \frac{mn-m-n}{2}</math>, which limits the values of <math>mn-m-n-k</math> to <math>mn-m-n \geq mn-m-n-k \geq \frac{mn-m-n}{2}</math>. Because <math>m</math> and <math>n</math> are coprime, only one of them can be a multiple of <math>2</math>. Therefore, <math>mn-m-n \equiv (0)(1)-0-1 \equiv -1 \equiv 1 \bmod 2</math>, showing that <math>\frac{mn-m-n}{2}</math> is not an integer and that <math>\frac{mn-m-n-1}{2}</math> and <math>\frac{mn-m-n+1}{2}</math> are integers. We can now set limits that are equivalent to the previous on the values of <math>k</math> and <math>mn-m-n-k</math> so that they cover all integers form <math>0</math> to <math>mn-m-n</math> without overlap: <math>0 \leq k \leq \frac{mn-m-n-1}{2}</math> and <math>\frac{mn-m-n+1}{2} \leq mn-m-n-k \leq mn-m-n</math>. There are <math>\frac{mn-m-n-1}{2}+1=\frac{(m-1)(n-1)}{2}</math> values of <math>k</math>, and each is paired with a value of <math>mn-m-n-k</math>, so we can make <math>\frac{(m-1)(n-1)}{2}</math> different ordered pairs of the form <math>(k, mn-m-n-k)</math>. The coordinates of these ordered pairs cover all integers from <math>0</math> to <math>mn-m-n</math> inclusive, and each contains exactly one not-purchasable integer, so that means that there are <math>\frac{(m-1)(n-1)}{2}</math> different not-purchasable integers from <math>0</math> to <math>mn-m-n</math>. All integers greater than <math>mn-m-n</math> are purchasable, so that means there are a total of <math>\frac{(m-1)(n-1)}{2}</math> integers <math>\geq 0</math> that are not purchasable.<br />
<br />
In other words, for every pair of coprime integers <math>m, n</math>, there are exactly <math>\frac{(m-1)(n-1)}{2}</math> nonnegative integers that cannot be represented in the form <math>am + bn</math> for nonnegative integers <math>a, b</math>. <math>\square</math><br />
<br />
==Generalization==<br />
If <math>m</math> and <math>n</math> are not relatively prime, then we can simply rearrange <math>am+bn</math> into the form<br />
<cmath>\gcd(m,n) \left( a\frac{m}{\gcd(m,n)}+b\frac{n}{\gcd(m,n)} \right)</cmath><br />
<math>\frac{m}{\gcd(m,n)}</math> and <math>\frac{n}{\gcd(m,n)}</math> are relatively prime, so we apply Chicken McNugget to find a bound<br />
<cmath>\frac{mn}{\gcd(m,n)^{2}}-\frac{m}{\gcd(m,n)}-\frac{n}{\gcd(m,n)}</cmath><br />
We can simply multiply <math>\gcd(m,n)</math> back into the bound to get<br />
<cmath>\frac{mn}{\gcd(m,n)}-m-n=\textrm{lcm}(m, n)-m-n</cmath><br />
Therefore, all multiples of <math>\gcd(m, n)</math> greater than <math>\textrm{lcm}(m, n)-m-n</math> are representable in the form <math>am+bn</math> for some positive integers <math>a, b</math>.<br />
<br />
=Problems=<br />
<br />
===Introductory===<br />
*Marcy buys paint jars in containers of <math>2</math> and <math>7</math>. What's the largest number of paint jars that Marcy can't obtain? <br />
<br />
Answer: <math>5</math> containers<br />
<br />
*Bay Area Rapid food sells chicken nuggets. You can buy packages of <math>11</math> or <math>7</math>. What is the largest integer <math>n</math> such that there is no way to buy exactly <math>n</math> nuggets? Can you Generalize ?(ACOPS) <br />
<br />
Answer: <math>n=59</math> <br />
<br />
*If a game of American Football has only scores of field goals (<math>3</math> points) and touchdowns with the extra point (<math>7</math> points), then what is the greatest score that cannot be the score of a team in this football game (ignoring time constraints)?<br />
<br />
Answer: <math>11</math> points<br />
<br />
*The town of Hamlet has <math>3</math> people for each horse, <math>4</math> sheep for each cow, and <math>3</math> ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?<br />
<br />
<math>\textbf{(A) }41\qquad\textbf{(B) }47\qquad\textbf{(C) }59\qquad\textbf{(D) }61\qquad\textbf{(E) }66</math> [[2015 AMC 10B Problems/Problem 15|AMC 10B 2015 Problem 15]]<br />
<br />
Answer: <math>47\qquad\textbf{(B) }</math><br />
<br />
===Intermediate===<br />
*Ninety-four bricks, each measuring <math>4''\times10''\times19'',</math> are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes <math>4''\,</math> or <math>10''\,</math> or <math>19''\,</math> to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks? [[1994 AIME Problems/Problem 11|AIME]]<br />
<br />
*Find the sum of all positive integers <math>n</math> such that, given an unlimited supply of stamps of denominations <math>5,n,</math> and <math>n+1</math> cents, <math>91</math> cents is the greatest postage that cannot be formed. [[2019 AIME II Problems/Problem 14|AIME II 2019 Problem 14]]<br />
<br />
===Olympiad===<br />
*On the real number line, paint red all points that correspond to integers of the form <math>81x+100y</math>, where <math>x</math> and <math>y</math> are positive integers. Paint the remaining integer points blue. Find a point <math>P</math> on the line such that, for every integer point <math>T</math>, the reflection of <math>T</math> with respect to <math>P</math> is an integer point of a different colour than <math>T</math>. (India TST)<br />
<br />
*Let <math>S</math> be a set of integers (not necessarily positive) such that<br />
<br />
:(a) there exist <math>a,b \in S</math> with <math>\gcd(a,b)=\gcd(a-2,b-2)=1</math>;<br />
<br />
:(b) if <math>x</math> and <math>y</math> are elements of <math>S</math> (possibly equal), then <math>x^2-y</math> also belongs to <math>S</math>. <br />
<br />
:Prove that <math>S</math> is the set of all integers. (USAMO 2001 Problem 5)<br />
<br />
==See Also==<br />
*[[Theorem]]<br />
*[[Prime]]<br />
<br />
[[Category:Theorems]]<br />
[[Category:Number theory]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=Chicken_McNugget_Theorem&diff=150386Chicken McNugget Theorem2021-03-27T03:01:41Z<p>Myh2910: /* Corollary */</p>
<hr />
<div>The '''Chicken McNugget Theorem''' (or '''Postage Stamp Problem''' or '''Frobenius Coin Problem''') states that for any two [[relatively prime]] [[positive integer]]s <math>m,n</math>, the greatest integer that cannot be written in the form <math>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>.<br />
<br />
A consequence of the theorem is that there are exactly <math>\frac{(m - 1)(n - 1)}{2}</math> positive integers which cannot be expressed in the form <math>am + bn</math>. The proof is based on the fact that in each pair of the form <math>(k, mn-m-n-k)</math>, exactly one element is expressible.<br />
<br />
== Origins ==<br />
There are many stories surrounding the origin of the Chicken McNugget theorem. However, the most popular by far remains that of the Chicken McNugget. Originally, McDonald's sold its nuggets in packs of 9 and 20. Math enthusiasts were curious to find the largest number of nuggets that could not have been bought with these packs, thus creating the Chicken McNugget Theorem (the answer worked out to be 151 nuggets). The Chicken McNugget Theorem has also been called the Frobenius Coin Problem or the Frobenius Problem, after German mathematician Ferdinand Frobenius inquired about the largest amount of currency that could not have been made with certain types of coins.<br />
<br />
<br />
<br />
<br />
<br />
==Proof Without Words==<br />
<math>\begin{array}{ccccccc}<br />
0\mod{m}&1\mod{m}&2\mod{m}&...&...&...&(m-1)\mod{m}\\<br />
\hline<br />
\cancel{0n}&1&2&&...&&m-1\\<br />
\cancel{0n+m}&...&&\vdots&&...&\\<br />
\cancel{0n+2m}&...&&\cancel{1n}&&...&\\<br />
\cancel{0n+3m}&&&\cancel{1n+m}&&\vdots&\\<br />
\cancel{0n+4m}&&&\cancel{1n+2m}&&\cancel{2n}&\\<br />
\cancel{0n+5m}&&&\cancel{1n+3m}&&\cancel{2n+m}&\\<br />
\vdots&&&\vdots&&\vdots&\\<br />
\cancel{\qquad}&\cancel{\qquad}&\cancel{ \qquad}&\cancel{ \qquad}&\mathbf{(m-1)n-m}&\cancel{\qquad }&\cancel{\qquad }\\<br />
\cancel{\qquad}&\cancel{\qquad}&\cancel{ \qquad}&\cancel{ \qquad}&\cancel{(m-1)n}&\cancel{\qquad }&\cancel{\qquad }<br />
\end{array}</math><br />
<br />
==Proof 1==<br />
<b>Definition</b>. An integer <math>N \in \mathbb{Z}</math> will be called <i>purchasable</i> if there exist nonnegative integers <math>a,b</math> such that <math>am+bn = N</math>.<br />
<br />
We would like to prove that <math>mn-m-n</math> is the largest non-purchasable integer. We are required to show that: <br />
<br />
(1) <math>mn-m-n</math> is non-purchasable<br />
<br />
(2) Every <math>N > mn-m-n</math> is purchasable<br />
<br />
Note that all purchasable integers are nonnegative, thus the set of non-purchasable integers is nonempty.<br />
<br />
<b>Lemma</b>. Let <math>A_{N} \subset \mathbb{Z} \times \mathbb{Z}</math> be the set of solutions <math>(x,y)</math> to <math>xm+yn = N</math>. Then <math>A_{N} = \{(x+kn,y-km) \;:\; k \in \mathbb{Z}\}</math> for any <math>(x,y) \in A_{N}</math>.<br />
<br />
<i>Proof</i>: By [[Bezout's Lemma]], there exist integers <math>x',y'</math> such that <math>x'm+y'n = 1</math>. Then <math>(Nx')m+(Ny')n = N</math>. Hence <math>A_{N}</math> is nonempty. It is easy to check that <math>(Nx'+kn,Ny'-km) \in A_{N}</math> for all <math>k \in \mathbb{Z}</math>. We now prove that there are no others. Suppose <math>(x_{1},y_{1})</math> and <math>(x_{2},y_{2})</math> are solutions to <math>xm+yn=N</math>. Then <math>x_{1}m+y_{1}n = x_{2}m+y_{2}n</math> implies <math>m(x_{1}-x_{2}) = n(y_{2}-y_{1})</math>. Since <math>m</math> and <math>n</math> are coprime and <math>m</math> divides <math>n(y_{2}-y_{1})</math>, <math>m</math> divides <math>y_{2}-y_{1}</math> and <math>y_{2} \equiv y_{1} \pmod{m}</math>. Similarly <math>x_{2} \equiv x_{1} \pmod{n}</math>. Let <math>k_{1},k_{2}</math> be integers such that <math>x_{2}-x_{1} = k_{1}n</math> and <math>y_{2}-y_{1} = k_{2}m</math>. Then <math>m(-k_{1}n) = n(k_{2}m)</math> implies <math>k_{1} = -k_{2}.</math> We have the desired result. <math>\square</math><br />
<br />
<b>Lemma</b>. For any integer <math>N</math>, there exists unique <math>(a_{N},b_{N}) \in \mathbb{Z} \times \{0,1,\ldots,m-1\}</math> such that <math>a_{N}m + b_{N}n = N</math>.<br />
<br />
<i>Proof</i>: By the division algorithm, there exists one and only one <math>k</math> such that <math>0 \le y-km \le m-1</math>. <math>\square</math><br />
<br />
<b>Lemma</b>. <math>N</math> is purchasable if and only if <math>a_{N} \ge 0</math>.<br />
<br />
<i>Proof</i>: If <math>a_{N} \ge 0</math>, then we may simply pick <math>(a,b) = (a_{N},b_{N})</math> so <math>N</math> is purchasable. If <math>a_{N} < 0</math>, then <math>a_{N}+kn < 0</math> if <math>k \le 0</math> and <math>b_{N}-km < 0</math> if <math>k > 0</math>, hence at least one coordinate of <math>(a_{N}+kn,b_{N}-km)</math> is negative for all <math>k \in \mathbb{Z}</math>. Thus <math>N</math> is not purchasable. <math>\square</math><br />
<br />
Thus the set of non-purchasable integers is <math>\{xm+yn \;:\; x<0,0 \le y \le m-1\}</math>. We would like to find the maximum of this set. <br />
Since both <math>m,n</math> are positive, the maximum is achieved when <math>x = -1</math> and <math>y = m-1</math> so that <math>xm+yn = (-1)m+(m-1)n = mn-m-n</math>.<br />
<br />
==Proof 2==<br />
We start with this statement taken from [[Fermat%27s_Little_Theorem#Proof_2_.28Inverses.29|Proof 2 of Fermat's Little Theorem]]:<br />
<br />
"Let <math>S = \{1,2,3,\cdots, p-1\}</math>. Then, we claim that the set <math>a \cdot S</math>, consisting of the product of the elements of <math>S</math> with <math>a</math>, taken modulo <math>p</math>, is simply a permutation of <math>S</math>. In other words, <br />
<br />
<center><cmath>S \equiv \{1a, 2a, \cdots, (p-1)a\} \pmod{p}.</cmath></center><br><br />
<br />
Clearly none of the <math>ia</math> for <math>1 \le i \le p-1</math> are divisible by <math>p</math>, so it suffices to show that all of the elements in <math>a \cdot S</math> are distinct. Suppose that <math>ai \equiv aj \pmod{p}</math> for <math>i \neq j</math>. Since <math>\text{gcd}\, (a,p) = 1</math>, by the cancellation rule, that reduces to <math>i \equiv j \pmod{p}</math>, which is a contradiction."<br />
<br />
Because <math>m</math> and <math>n</math> are coprime, we know that multiplying the residues of <math>m</math> by <math>n</math> simply permutes these residues. Each of these permuted residues is purchasable (using the definition from Proof 1), because, in the form <math>am+bn</math>, <math>a</math> is <math>0</math> and <math>b</math> is the original residue. We now prove the following lemma.<br />
<br />
<b>Lemma</b>: For any nonnegative integer <math>c < m</math>, <math>cn</math> is the least purchasable number <math>\equiv cn \bmod m</math>.<br />
<br />
<i>Proof</i>: Any number that is less than <math>cn</math> and congruent to it <math>\bmod m</math> can be represented in the form <math>cn-dm</math>, where <math>d</math> is a positive integer. If this is purchasable, we can say <math>cn-dm=am+bn</math> for some nonnegative integers <math>a, b</math>. This can be rearranged into <math>(a+d)m=(c-b)n</math>, which implies that <math>(a+d)</math> is a multiple of <math>n</math> (since <math>\gcd(m, n)=1</math>). We can say that <math>(a+d)=gn</math> for some positive integer <math>g</math>, and substitute to get <math>gmn=(c-b)n</math>. Because <math>c < m</math>, <math>(c-b)n < mn</math>, and <math>gmn < mn</math>. We divide by <math>mn</math> to get <math>g<1</math>. However, we defined <math>g</math> to be a positive integer, and all positive integers are greater than or equal to <math>1</math>. Therefore, we have a contradiction, and <math>cn</math> is the least purchasable number congruent to <math>cn \bmod m</math>. <math>\square</math><br />
<br />
This means that because <math>cn</math> is purchasable, every number that is greater than <math>cn</math> and congruent to it <math>\bmod m</math> is also purchasable (because these numbers are in the form <math>am+bn</math> where <math>b=c</math>). Another result of this Lemma is that <math>cn-m</math> is the greatest number <math>\equiv cn \bmod m</math> that is not purchasable. <math>c \leq m-1</math>, so <math>cn-m \leq (m-1)n-m=mn-m-n</math>, which shows that <math>mn-m-n</math> is the greatest number in the form <math>cn-m</math>. Any number greater than this and congruent to some <math>cn \bmod m</math> is purchasable, because that number is greater than <math>cn</math>. All numbers are congruent to some <math>cn</math>, and thus all numbers greater than <math>mn-m-n</math> are purchasable.<br />
<br />
Putting it all together, we can say that for any coprime <math>m</math> and <math>n</math>, <math>mn-m-n</math> is the greatest number not representable in the form <math>am + bn</math> for nonnegative integers <math>a, b</math>. <math>\square</math><br />
<br />
==Corollary==<br />
This corollary is based off of Proof 2, so it is necessary to read that proof before this corollary. We prove the following lemma.<br />
<br />
<b>Lemma:</b> For any integer <math>k</math>, exactly one of the integers <math>k</math>, <math>mn-m-n-k</math> is not purchasable.<br />
<br />
<i>Proof</i>: Because every number is congruent to some residue of <math>m</math> permuted by <math>n</math>, we can set <math>k \equiv cn \bmod m</math> for some <math>c</math>. We can break this into two cases.<br />
<br />
<i>Case 1</i>: <math>k \leq cn-m</math>. This implies that <math>k</math> is not purchasable, and that <math>mn-m-n-k \geq mn-m-n-(cn-m) = n(m-1-c)</math>. <math>n(m-1-c)</math> is a permuted residue, and a result of the lemma in Proof 2 was that a permuted residue is the least number congruent to itself <math>\bmod m</math> that is purchasable. Therefore, <math>mn-m-n-k \equiv n(m-1-c) \bmod m</math> and <math>mn-m-n-k \geq n(m-1-c)</math>, so <math>mn-m-n-k</math> is purchasable.<br />
<br />
<i>Case 2</i>: <math>k > cn-m</math>. This implies that <math>k</math> is purchasable, and that <math>mn-m-n-k < mn-m-n-(cn-m) = n(m-1-c)</math>. Again, because <math>n(m-1-c)</math> is the least number congruent to itself <math>\bmod m</math> that is purchasable, and because <math>mn-m-n-k \equiv n(m-1-c) \bmod m</math> and <math>mn-m-n-k < n(m-1-c)</math>, <math>mn-m-n-k</math> is not purchasable.<br />
<br />
We now limit the values of <math>k</math> to all integers <math>0 \leq k \leq \frac{mn-m-n}{2}</math>, which limits the values of <math>mn-m-n-k</math> to <math>mn-m-n \geq mn-m-n-k \geq \frac{mn-m-n}{2}</math>. Because <math>m</math> and <math>n</math> are coprime, only one of them can be a multiple of <math>2</math>. Therefore, <math>mn-m-n \equiv (0)(1)-0-1 \equiv -1 \equiv 1 \bmod 2</math>, showing that <math>\frac{mn-m-n}{2}</math> is not an integer and that <math>\frac{mn-m-n-1}{2}</math> and <math>\frac{mn-m-n+1}{2}</math> are integers. We can now set limits that are equivalent to the previous on the values of <math>k</math> and <math>mn-m-n-k</math> so that they cover all integers form <math>0</math> to <math>mn-m-n</math> without overlap: <math>0 \leq k \leq \frac{mn-m-n-1}{2}</math> and <math>\frac{mn-m-n+1}{2} \leq mn-m-n-k \leq mn-m-n</math>. There are <math>\frac{mn-m-n-1}{2}+1=\frac{(m-1)(n-1)}{2}</math> values of <math>k</math>, and each is paired with a value of <math>mn-m-n-k</math>, so we can make <math>\frac{(m-1)(n-1)}{2}</math> different ordered pairs of the form <math>(k, mn-m-n-k)</math>. The coordinates of these ordered pairs cover all integers from <math>0</math> to <math>mn-m-n</math> inclusive, and each contains exactly one not-purchasable integer, so that means that there are <math>\frac{(m-1)(n-1)}{2}</math> different not-purchasable integers from <math>0</math> to <math>mn-m-n</math>. All integers greater than <math>mn-m-n</math> are purchasable, so that means there are a total of <math>\frac{(m-1)(n-1)}{2}</math> integers <math>\geq 0</math> that are not purchasable.<br />
<br />
In other words, for every pair of coprime integers <math>m, n</math>, there are exactly <math>\frac{(m-1)(n-1)}{2}</math> nonnegative integers that cannot be represented in the form <math>am + bn</math> for nonnegative integers <math>a, b</math>. <math>\square</math><br />
<br />
==Generalization==<br />
If <math>m</math> and <math>n</math> are not relatively prime, then we can simply rearrange <math>am+bn</math> into the form<br />
<cmath>\gcd(m,n) \left( a\frac{m}{\gcd(m,n)}+b\frac{n}{\gcd(m,n)} \right)</cmath><br />
<math>\frac{m}{\gcd(m,n)}</math> and <math>\frac{n}{\gcd(m,n)}</math> are relatively prime, so we apply Chicken McNugget to find a bound<br />
<cmath>\frac{mn}{\gcd(m,n)^{2}}-\frac{m}{\gcd(m,n)}-\frac{n}{\gcd(m,n)}</cmath><br />
We can simply multiply <math>\gcd(m,n)</math> back into the bound to get<br />
<cmath>\frac{mn}{\gcd(m,n)}-m-n=\textrm{lcm}(m, n)-m-n</cmath><br />
Therefore, all multiples of <math>\gcd(m, n)</math> greater than <math>\textrm{lcm}(m, n)-m-n</math> are representable in the form <math>am+bn</math> for some positive integers <math>a, b</math>.<br />
<br />
=Problems=<br />
<br />
===Introductory===<br />
*Marcy buys paint jars in containers of <math>2</math> and <math>7</math>. What's the largest number of paint jars that Marcy can't obtain? <br />
<br />
Answer: <math>5</math> containers<br />
<br />
*Bay Area Rapid food sells chicken nuggets. You can buy packages of <math>11</math> or <math>7</math>. What is the largest integer <math>n</math> such that there is no way to buy exactly <math>n</math> nuggets? Can you Generalize ?(ACOPS) <br />
<br />
Answer: <math>n=59</math> <br />
<br />
*If a game of American Football has only scores of field goals (<math>3</math> points) and touchdowns with the extra point (<math>7</math> points), then what is the greatest score that cannot be the score of a team in this football game (ignoring time constraints)?<br />
<br />
Answer: <math>11</math> points<br />
<br />
*The town of Hamlet has <math>3</math> people for each horse, <math>4</math> sheep for each cow, and <math>3</math> ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?<br />
<br />
<math>\textbf{(A) }41\qquad\textbf{(B) }47\qquad\textbf{(C) }59\qquad\textbf{(D) }61\qquad\textbf{(E) }66</math> [[2015 AMC 10B Problems/Problem 15|AMC 10B 2015 Problem 15]]<br />
<br />
Answer: <math>47\qquad\textbf{(B) }</math><br />
<br />
===Intermediate===<br />
*Ninety-four bricks, each measuring <math>4''\times10''\times19'',</math> are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes <math>4''\,</math> or <math>10''\,</math> or <math>19''\,</math> to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks? [[1994 AIME Problems/Problem 11|AIME]]<br />
<br />
*Find the sum of all positive integers <math>n</math> such that, given an unlimited supply of stamps of denominations <math>5,n,</math> and <math>n+1</math> cents, <math>91</math> cents is the greatest postage that cannot be formed. [[2019 AIME II Problems/Problem 14|AIME II 2019 Problem 14]]<br />
<br />
===Olympiad===<br />
*On the real number line, paint red all points that correspond to integers of the form <math>81x+100y</math>, where <math>x</math> and <math>y</math> are positive integers. Paint the remaining integer points blue. Find a point <math>P</math> on the line such that, for every integer point <math>T</math>, the reflection of <math>T</math> with respect to <math>P</math> is an integer point of a different colour than <math>T</math>. (India TST)<br />
<br />
*Let <math>S</math> be a set of integers (not necessarily positive) such that<br />
<br />
(a) there exist <math>a,b \in S</math> with <math>\gcd(a,b)=\gcd(a-2,b-2)=1</math>;<br />
<br />
(b) if <math>x</math> and <math>y</math> are elements of <math>S</math> (possibly equal), then <math>x^2-y</math> also belongs to <math>S</math>. <br />
<br />
Prove that <math>S</math> is the set of all integers. (USAMO)<br />
<br />
==See Also==<br />
*[[Theorem]]<br />
*[[Prime]]<br />
<br />
[[Category:Theorems]]<br />
[[Category:Number theory]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=Peru_mathematics_competitions&diff=148119Peru mathematics competitions2021-03-01T18:35:35Z<p>Myh2910: /* ONEM */</p>
<hr />
<div>== ONEM ==<br />
ONEM (Olimpiada Nacional Escolar de Matemática) is a mathematical competition that has been held annually since 2004, organized by the Ministry of Education of Peru (MINEDU) and the Olympiad Commission of the Peruvian Mathematical Society, and is aimed at students from secondary education throughout the country.<br />
<br />
In addition, the three students who obtain the highest scores for each level (regardless of their category) in the fourth phase, represent Peru in the Rioplatense Mathematical Olympiad, which takes place every year in Buenos Aires, Argentina in December.<br />
<br />
== External Links ==<br />
* [https://onemperu.wordpress.com Jorge Tipe's unofficial blog]<br />
* [http://selectivos-peru.blogspot.com Selectivos Perú]<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=Peru_mathematics_competitions&diff=147990Peru mathematics competitions2021-02-26T15:22:21Z<p>Myh2910: /* ONEM */</p>
<hr />
<div>== ONEM ==<br />
ONEM (Olimpiada Nacional Escolar de Matemática) is a mathematical competition that has been held annually since 2004, organized by the Ministry of Education of Peru (MINEDU) and the Olympiad Commission of the Peruvian Mathematical Society, and is aimed at students from secondary education throughout the country.<br />
<br />
In addition, the three students who obtain the highest scores for each level (regardless of their category) in the fourth phase, represent Peru in the Rioplatense Mathematical Olympiad, which takes place every year in Buenos Aires, Argentina in the month from December.<br />
<br />
== External Links ==<br />
* [https://onemperu.wordpress.com Jorge Tipe's unofficial blog]<br />
* [http://selectivos-peru.blogspot.com Selectivos Perú]<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=List_of_mathematics_competitions&diff=147989List of mathematics competitions2021-02-26T15:20:30Z<p>Myh2910: </p>
<hr />
<div>This list is intended to be global. If other international or contests from other nations or regions are documented elsewhere, they should be added here as well.<br />
<br />
This is a directory of internal links to more helpful pages about mathematics competitions. This is not the place to list individual competitions.<br />
<br />
<br />
<br />
== International mathematics competitions ==<br />
<br />
[[List of international mathematics competitions]].<br />
<br />
== Regional mathematics competitions ==<br />
<br />
[[List of regional mathematics competitions]].<br />
<br />
== National mathematics competitions ==<br />
=== Argentina ===<br />
<br />
[[Argentina mathematics competitions]].<br />
<br />
=== Australia ===<br />
<br />
[[Australia mathematics competitions]].<br />
<br />
=== Austria ===<br />
<br />
[[Austria mathematics competitions]].<br />
<br />
=== Bangladesh ===<br />
<br />
[[Bangladesh mathematics competition]]<br />
=== Belgium ===<br />
<br />
[[Belgium mathematics competitions]].<br />
<br />
=== Brazil ===<br />
Brazilian mathematics competitions are categorized in two, one for public schools and other for particular schools.<br />
*[[Brazil mathematics competitions]].<br />
<br />
There's also some exams with the part of mathematics being as hard as an olympiad, but in the last years some of the questions in these exams are even harder than some olympiads. They're the military exams, such as [[Military Institute of Engineering]] ([[Military Institute of Engineering|IME]] or [[Military Institute of Engineering|Instituto Militar de Engenharia]]) and [[Technological Institute of Aeronautics]] ([[Technological Institute of Aeronautics|ITA]] or [[Technological Institute of Aeronautics|Instituto Tecnológico de Aeronáutica]]).<br />
<br />
=== Bulgaria ===<br />
<br />
[[Bulgaria mathematics competitions]].<br />
<br />
=== Canada ===<br />
<br />
[[List of Canada mathematics competitions]].<br />
<br />
=== China ===<br />
<br />
[[List of China mathematics competitions]].<br />
<br />
=== Cyprus ===<br />
<br />
[[Cyprus mathematics competitions]].<br />
<br />
=== Denmark ===<br />
<br />
[[Denmark mathematics competitions]].<br />
<br />
=== Germany ===<br />
<br />
[[List of Germany mathematics competitions]].<br />
<br />
=== Hungary ===<br />
<br />
[[Hungary mathematics competitions]].<br />
<br />
=== Greece ===<br />
<br />
[[Greece mathematics competitions]].<br />
<br />
===India===<br />
<br />
[[India mathematics competitions]].<br />
<br />
===Indonesia===<br />
<br />
[[Indonesia mathematics competitions]].<br />
<br />
===Ireland===<br />
<br />
[[Ireland mathematics competitions]].<br />
<br />
===Israel===<br />
<br />
[[Israeli mathematics competitions]].<br />
<br />
=== Mexico ===<br />
<br />
[[Mexico mathematics competitions]].<br />
<br />
=== Netherlands ===<br />
<br />
[[Netherlands mathematics competitions]].<br />
<br />
=== Norway ===<br />
<br />
[[Norway mathematics competitions]].<br />
<br />
=== Peru ===<br />
<br />
[[Peru mathematics competitions]].<br />
<br />
=== Philippines ===<br />
<br />
[[Philippines mathematics competitions]].<br />
<br />
=== Poland ===<br />
<br />
[[Poland mathematics competitions]].<br />
<br />
=== Portugal ===<br />
<br />
[[Portugal mathematics competitions]].<br />
<br />
=== Romania ===<br />
<br />
[[Romania mathematics competitions]].<br />
<br />
=== Singapore ===<br />
<br />
[[Singapore mathematics competitions]].<br />
<br />
=== South Korea ===<br />
<br />
[[Korean mathematics competitions]].<br />
<br />
=== Slovakia ===<br />
<br />
[[Slovaki mathematics competitions]].<br />
<br />
=== South Africa ===<br />
<br />
[[South Africa mathematics competitions]].<br />
<br />
=== Sweden ===<br />
<br />
[[Sweden mathematics competitions]].<br />
<br />
=== Thailand ===<br />
<br />
[[Thailand mathematics competitions]].<br />
<br />
=== United Kingdom ===<br />
<br />
[[United Kingdom mathematics competitions]].<br />
<br />
=== United States ===<br />
Mathematics competitions in the United States are so numerous that we categorize them according to the level of schooling of competing students.<br />
<br />
*[[List of United States elementary school mathematics competitions]].<br />
*[[List of United States middle school mathematics competitions]].<br />
*[[List of United States high school mathematics competitions]].<br />
*[[List of United States college mathematics competitions]].<br />
<br />
=== Uruguay ===<br />
<br />
[[Uruguay mathematics competitions]].<br />
<br />
== See also == <br />
* [http://en.wikipedia.org/wiki/List_of_mathematics_competitions List of mathematics competitions on Wikipedia]<br />
* [[Mathematics competition resources]]<br />
* [[Mathematics scholarships]]<br />
* [[Mathematical olympiads]]<br />
* [[Mathematical problem solving]]<br />
* [[World Federation of National Mathematics Competitions]]<br />
* [[Academic competitions]]<br />
* [[AoPSWiki:Competition ratings]]<br />
<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=Peru_mathematics_competitions&diff=147988Peru mathematics competitions2021-02-26T15:19:35Z<p>Myh2910: Created page with "== ONEM == ONEM (Olimpiada Nacional Escolar de Matemática) is a mathematical competition that has been held annually since 2004, organized by the Ministry of Education of Per..."</p>
<hr />
<div>== ONEM ==<br />
ONEM (Olimpiada Nacional Escolar de Matemática) is a mathematical competition that has been held annually since 2004, organized by the Ministry of Education of Peru (MINEDU) and the Olympics Commission of the Peruvian Mathematical Society, and is aimed at students from secondary education throughout the country.<br />
<br />
In addition, the three students who obtain the highest scores for each level (regardless of their category) in the fourth phase, represent Peru in the Rioplatense Mathematical Olympiad, which takes place every year in Buenos Aires, Argentina in the month from December.<br />
== External Links ==<br />
* [https://onemperu.wordpress.com Jorge Tipe's unofficial blog]<br />
* [http://selectivos-peru.blogspot.com Selectivos Perú]<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Competition_ratings&diff=147987AoPS Wiki:Competition ratings2021-02-26T15:01:09Z<p>Myh2910: </p>
<hr />
<div>This page contains an approximate estimation of the difficulty level of various [[List of mathematics competitions|competitions]]. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.<br />
<br />
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution. <br />
<br />
As you may have guessed with time many competitions got more challenging because many countries got more access to books targeted at olympiad preparation. But especially web site where one can discuss Olympiads such as our very own AoPS!<br />
<br />
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC problems] and 10 is hardest level, e.g. [http://www.mathlinks.ro/resources.php?c=37&cid=47 China IMO Team Selection Test.] When considering problem difficulty '''put more emphasis on problem-solving aspects and less so on technical skill requirements'''.<br />
<br />
= Scale =<br />
All levels are estimated and refer to ''averages''. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this. <br />
# Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, #1-10 AMC 10s, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problems<br />
# For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #11-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc), traditional middle/high school word problems with extremely complex problem solving<br />
# Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, easier #1-5 on AIMEs, etc.)<br />
# Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.<br />
# More difficult AIME problems (#10-12), simple proof-based problems (JBMO), etc<br />
# High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1,4s).<br />
# Tougher Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.<br />
# High-level Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.<br />
# Expert Olympiad-level questions, eg #3,6s on difficult Olympiad contests.<br />
# Super Expert problems, problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).<br />
<br />
= Competitions =<br />
<br />
==Introductory Competitions==<br />
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntroductory+mathematics+competitions here].<br />
<br />
=== [[MOEMS]] ===<br />
*Division E: '''1'''<br />
*: ''The whole number <math>N</math> is divisible by <math>7</math>. <math>N</math> leaves a remainder of <math>1</math> when divided by <math>2,3,4,</math> or <math>5</math>. What is the smallest value that <math>N</math> can be?'' ([http://www.moems.org/sample_files/SampleE.pdf Solution])<br />
*Division M: '''1'''<br />
*: ''The value of a two-digit number is <math>10</math> times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.'' ([http://www.moems.org/sample_files/SampleM.pdf Solution])<br />
<br />
=== [[AMC 8]] ===<br />
<br />
* Problem 1 - Problem 12: '''1''' <br />
*: ''The <math>\emph{harmonic mean}</math> of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?'' ([[2018 AMC 8 Problems/Problem 10|Solution]])<br />
* Problem 13 - Problem 25: '''1.5-2'''<br />
*: ''How many positive factors does <math>23,232</math> have?'' ([[2018 AMC 8 Problems/Problem 18|Solution]])<br />
<br />
=== [[Mathcounts]] ===<br />
<br />
* Countdown: '''1-2.'''<br />
* Sprint: '''1-1.5''' (school/chapter), '''1.5-2''' (State), '''2-2.5''' (National)<br />
* Target: '''1-2''' (school/chapter), '''1.5-2.5''' (State), '''2.5-3.5''' (National)<br />
<br />
=== [[AMC 10]] ===<br />
<br />
* Problem 1 - 10: '''1-2'''<br />
*: ''A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box?'' ([[2016 AMC 10A Problems/Problem 5|Solution]])<br />
* Problem 11 - 20: '''2-3'''<br />
*: ''For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?'' ([[2019 AMC 10A Problems/Problem 18|Solution]])<br />
* Problem 21 - 25: '''3.5-4.5'''<br />
*: ''The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?'' ([[2017 AMC 10B Problems/Problem 24|Solution]])<br />
<br />
===[[CEMC|CEMC Multiple Choice Tests]]===<br />
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.<br />
<br />
* Part A: '''1-1.5'''<br />
*: ''How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number?'' (2015 Gauss 7 Problem 10)<br />
* Part B: '''1-2'''<br />
*: ''Two lines with slopes <math>\tfrac14</math> and <math>\tfrac54</math> intersect at <math>(1,1)</math>. What is the area of the triangle formed by these two lines and the vertical line <math>x = 5</math>?'' (2017 Cayley Problem 19)<br />
* Part C (Gauss/Pascal): '''2-2.5'''<br />
*: ''Suppose that <math>\tfrac{2009}{2014} + \tfrac{2019}{n} = \tfrac{a}{b}</math>, where <math>a</math>, <math>b</math>, and <math>n</math> are positive integers with <math>\tfrac{a}{b}</math> in lowest terms. What is the sum of the digits of the smallest positive integer <math>n</math> for which <math>a</math> is a multiple of 1004?'' (2014 Pascal Problem 25)<br />
* Part C (Cayley/Fermat): '''2.5-3'''<br />
*: ''Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets?'' (2018 Fermat Problem 24)<br />
<br />
===[[CEMC|CEMC Fryer/Galois/Hypatia]]===<br />
<br />
* Problem 1-2: '''1-2'''<br />
* Problem 3-4 (early parts): '''2-3'''<br />
* Problem 3-4 (later parts): '''3-5'''<br />
<br />
===Problem Solving Books for Introductory Students===<br />
<br />
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.<br />
<br />
===[[Prealgebra by AoPS]]===<br />
'''1-2'''<br />
===[[Introduction to Algebra by AoPS]]===<br />
'''1-3.5'''<br />
===[[Introduction to Counting and Probability by AoPS]]===<br />
'''1-3.5'''<br />
===[[Introduction to Number Theory by AoPS]]===<br />
'''1-3'''<br />
===[[Introduction to Geometry by AoPS]]===<br />
'''1-4'''<br />
<br />
==Intermediate Competitions==<br />
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntermediate+mathematics+competitions here].<br />
<br />
=== [[AMC 12]] ===<br />
<br />
* Problem 1-10: '''1.5-2'''<br />
*: ''What is the value of <cmath>\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?</cmath>'' ([[2018 AMC 12B Problems/Problem 7|Solution]])<br />
* Problem 11-20: '''2.5-3.5'''<br />
*: ''An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?'' ([[2006 AMC 12B Problems/Problem 18|Solution]])<br />
* Problem 21-25: '''4.5-6'''<br />
*: ''Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?'' ([[2009 AMC 12A Problems/Problem 23|Solution]])<br />
<br />
=== [[AIME]] ===<br />
<br />
* Problem 1 - 5: '''3-3.5'''<br />
*: ''Consider the integer <cmath>N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>.'' ([[2019 AIME I Problems/Problem 1|Solution]])<br />
* Problem 6 - 9: '''4-4.5''' <br />
*: ''How many positive integers <math>N</math> less than <math>1000</math> are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>?'' ([[2009 AIME I Problems/Problem 6|Solution]])<br />
* Problem 10 - 12: '''5-5.5'''<br />
*: Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by <math>1000</math>.Let <math>S</math> be the sum of all elements in <math>R</math>. Find the remainder when <math>S</math> is divided by <math>1000</math> ([[2011 AIME I Problems/Problem 11|Solution]])<br />
* Problem 13 - 15: '''6-6.5'''<br />
*: ''Let <cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath> Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let <cmath>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</cmath> where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>.'' ([[2003 AIME II Problems/Problem 15|Solution]])<br />
<br />
=== [[ARML]] ===<br />
<br />
* Individuals, Problem 1: '''2'''<br />
<br />
* Individuals, Problems 2, 3, 4, 5, 7, and 9: '''3'''<br />
<br />
* Individuals, Problems 6 and 8: '''4''' <br />
<br />
* Individuals, Problem 10: '''5.5'''<br />
<br />
* Team/power, Problem 1-5: '''3.5''' <br />
<br />
* Team/power, Problem 6-10: '''5'''<br />
<br />
===[[HMMT|HMMT (November)]]===<br />
* Individual Round, Problem 6-8: '''4'''<br />
* Individual Round, Problem 10: '''4.5'''<br />
* Team Round: '''4-5'''<br />
* Guts: '''3.5-5.25'''<br />
<br />
===[[CEMC|CEMC Euclid]]===<br />
<br />
* Problem 1-6: '''1-3'''<br />
* Problem 7-10: '''3-5'''<br />
<br />
===[[Purple Comet! Math Meet|Purple Comet]]===<br />
<br />
* Problems 1-10 (MS): '''1.5-3'''<br />
* Problems 11-20 (MS): '''3-4.5'''<br />
* Problems 1-10 (HS): '''2-3.5'''<br />
* Problems 11-20 (HS): '''3.5'''<br />
* Problems 21-30 (HS): '''4.5-6'''<br />
<br />
=== [[Philippine Mathematical Olympiad Qualifying Round]] ===<br />
<br />
* Problem 1-15: '''2'''<br />
* Problem 16-25: '''3'''<br />
* Problem 26-30: '''4'''<br />
<br />
===[[Lexington Math Tournament|LMT]]===<br />
<br />
* Easy Problems: '''1-2'''<br />
*: ''Let trapezoid <math>ABCD</math> be such that <math>AB||CD</math>. Additionally, <math>AC = AD = 5</math>, <math>CD = 6</math>, and <math>AB = 3</math>. Find <math>BC</math>. ''<br />
* Medium Problems: '''2-4'''<br />
*: ''Let <math>\triangle LMN</math> have side lengths <math>LM = 15</math>, <math>MN = 14</math>, and <math>NL = 13</math>. Let the angle bisector of <math>\angle MLN</math> meet the circumcircle of <math>\triangle LMN</math> at a point <math>T \ne L</math>. Determine the area of <math>\triangle LMT</math>. ''<br />
* Hard Problems: '''5-7'''<br />
*: ''A magic <math>3 \times 5</math> board can toggle its cells between black and white. Define a pattern to be an assignment of black or white to each of the board’s <math>15</math> cells (so there are <math>2^{15}</math> patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than <math>3</math> cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day 1, compute the maximum number of days it can stay alive.''<br />
<br />
==Problem Solving Books for Intermediate Students==<br />
<br />
Remark: As stated above, there are many books for Intermediate students that have not been published by AoPS. Below is a list of intermediate books that AoPS has published and their difficulty. The left-hand number corresponds to the difficulty of the easiest review problems, while the right-hand number corresponds to the difficulty of the hardest challenge problems.<br />
<br />
===[[Intermediate Algebra by AoPS]]===<br />
'''2.5-6.5/7''', may vary across chapters<br />
<br />
===[[Intermediate Counting & Probability by AoPS]]===<br />
'''3.5-7.5/8''', may vary across chapters<br />
<br />
===[[Precalculus by AoPS]]===<br />
'''2-8''', may vary across chapters<br />
<br />
==Beginner Olympiad Competitions==<br />
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3ABeginner+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMTS]] ===<br />
USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:<br />
* Problem 1-2: '''3-4'''<br />
*: ''Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter.'' ([http://usamts.org/Solutions/Solution2_3_16.pdf Solution])<br />
* Problem 3-5: '''4-6'''<br />
*: ''Call a positive real number groovy if it can be written in the form <math>\sqrt{n} + \sqrt{n + 1}</math> for some positive integer <math>n</math>. Show that if <math>x</math> is groovy, then for any positive integer <math>r</math>, the number <math>x^r</math> is groovy as well.'' ([http://usamts.org/Solutions/Solutions_20_1.pdf Solution])<br />
<br />
=== [[Indonesia Mathematical Olympiad|Indonesia MO]] ===<br />
* Problem 1/5: '''3.5'''<br />
*: '' In a drawer, there are at most <math>2009</math> balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is <math>\frac12</math>. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?'' ([https://artofproblemsolving.com/community/c6h294065 Solution])<br />
* Problem 2/6: '''4.5'''<br />
*: ''Find the lowest possible values from the function <cmath>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</cmath> for any real numbers <math>x</math>.'' ([https://artofproblemsolving.com/community/c6h294067 Solution])<br />
* Problem 3/7: '''5'''<br />
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if <cmath>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</cmath> Given 2 positive integers <math>a,b > 1</math> which are relatively prime, prove that there exists a ''good'' pair <math>(m,n)</math> with <math>a\mid m</math> and <math>b\mid n</math>, but <math>a\nmid n</math> and <math>b\nmid m</math>.'' ([https://artofproblemsolving.com/community/c6h294068 Solution])<br />
* Problem 4/8: '''6'''<br />
*: ''Given an acute triangle <math>ABC</math>. The incircle of triangle <math>ABC</math> touches <math>BC,CA,AB</math> respectively at <math>D,E,F</math>. The angle bisector of <math>\angle A</math> cuts <math>DE</math> and <math>DF</math> respectively at <math>K</math> and <math>L</math>. Suppose <math>AA_1</math> is one of the altitudes of triangle <math>ABC</math>, and <math>M</math> be the midpoint of <math>BC</math>.''<br />
<br />
::''(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.''<br />
<br />
::''(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' ([https://artofproblemsolving.com/community/c6h294069 Solution])<br />
<br />
=== [[Central American Olympiad]] ===<br />
* Problem 1: '''4'''<br />
*: ''Find all three-digit numbers <math>abc</math> (with <math>a \neq 0</math>) such that <math>a^{2} + b^{2} + c^{2}</math> is a divisor of 26.'' ([https://artofproblemsolving.com/community/c6h161957p903856 Solution])<br />
* Problem 2,4,5: '''5-6'''<br />
*: ''Show that the equation <math>a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005</math> has no integer solutions.'' ([https://artofproblemsolving.com/community/c6h46028p291301 Solution])<br />
* Problem 3/6: '''6.5''' <br />
*: ''Let <math>ABCD</math> be a convex quadrilateral. <math>I = AC\cap BD</math>, and <math>E</math>, <math>H</math>, <math>F</math> and <math>G</math> are points on <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively, such that <math>EF \cap GH = I</math>. If <math>M = EG \cap AC</math>, <math>N = HF \cap AC</math>, show that <math>\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}</math>.'' ([https://artofproblemsolving.com/community/c6h146421p828841 Solution])<br />
<br />
=== [[JBMO]] ===<br />
<br />
* Problem 1: '''4'''<br />
*: ''Find all real numbers <math>a,b,c,d</math> such that <br />
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''<br />
* Problem 2: '''4.5-5'''<br />
*: ''Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAC=\angle BDC=36^\circ</math>, <math>\angle CBD=18^\circ</math> and <math>\angle BAC=72^\circ</math>. The diagonals intersect at point <math>P</math>. Determine the measure of <math>\angle APD</math>.''<br />
* Problem 3: '''5'''<br />
*: ''Find all prime numbers <math>p,q,r</math>, such that <math>\frac pq-\frac4{r+1}=1</math>.''<br />
* Problem 4: '''6'''<br />
*: ''A <math>4\times4</math> table is divided into <math>16</math> white unit square cells. Two cells are called neighbors if they share a common side. A '''move''' consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly <math>n</math> moves all the <math>16</math> cells were black. Find all possible values of <math>n</math>.''<br />
<br />
==Olympiad Competitions==<br />
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AOlympiad+mathematics+competitions here].<br />
<br />
=== [[USAJMO]] ===<br />
* Problem 1/4: '''5'''<br />
*: ''There are <math>a+b</math> bowls arranged in a row, numbered <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive integers. Initially, each of the first <math>a</math> bowls contains an apple, and each of the last <math>b</math> bowls contains a pear.''<br />
<br />
::''A legal move consists of moving an apple from bowl <math>i</math> to bowl <math>i+1</math> and a pear from bowl <math>j</math> to bowl <math>j-1</math>, provided that the difference <math>i-j</math> is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first <math>b</math> bowls each containing a pear and the last <math>a</math> bowls each containing an apple. Show that this is possible if and only if the product <math>ab</math> is even.'' ([[2019 USAJMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''6-6.5'''<br />
*: ''Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>'' ([[2018 USAJMO Problems/Problem 2|Solution]])<br />
<br />
* Problem 3/6: '''7'''<br />
*: ''Two rational numbers <math>\tfrac{m}{n}</math> and <math>\tfrac{n}{m}</math> are written on a blackboard, where <math>m</math> and <math>n</math> are relatively prime positive integers. At any point, Evan may pick two of the numbers <math>x</math> and <math>y</math> written on the board and write either their arithmetic mean <math>\tfrac{x+y}{2}</math> or their harmonic mean <math>\tfrac{2xy}{x+y}</math> on the board as well. Find all pairs <math>(m,n)</math> such that Evan can write <math>1</math> on the board in finitely many steps.'' ([[2019 USAJMO Problems/Problem 6|Solution]])<br />
<br />
===[[HMMT|HMMT (February)]]===<br />
* Individual Round, Problem 1-5: '''5'''<br />
* Individual Round, Problem 6-10: '''5.5-6'''<br />
* Team Round: '''7.5'''<br />
* HMIC: '''8'''<br />
<br />
=== [[Canadian MO]] ===<br />
<br />
* Problem 1: '''5.5'''<br />
* Problem 2: '''6'''<br />
* Problem 3: '''6.5''' <br />
* Problem 4: '''7-7.5'''<br />
* Problem 5: '''7.5-8'''<br />
<br />
=== Austrian MO ===<br />
<br />
* Regional Competition for Advanced Students, Problems 1-4: '''5''' <br />
* Federal Competition for Advanced Students, Part 1. Problems 1-4: '''6''' <br />
* Federal Competition for Advanced Students, Part 2, Problems 1-6: '''7'''<br />
<br />
=== [[Iberoamerican Math Olympiad]] ===<br />
<br />
* Problem 1/4: '''5.5'''<br />
* Problem 2/5: '''6.5'''<br />
* Problem 3/6: '''7.5'''<br />
<br />
=== [[APMO]] ===<br />
*Problem 1: '''6'''<br />
*Problem 2: '''7'''<br />
*Problem 3: '''7'''<br />
*Problem 4: '''7.5'''<br />
*Problem 5: '''8.5'''<br />
<br />
=== Balkan MO ===<br />
<br />
* Problem 1: '''6'''<br />
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. '' <br />
* Problem 2: '''6.5'''<br />
*: '' Let <math>MN</math> be a line parallel to the side <math>BC</math> of a triangle <math>ABC</math>, with <math>M</math> on the side <math>AB</math> and <math>N</math> on the side <math>AC</math>. The lines <math>BN</math> and <math>CM</math> meet at point <math>P</math>. The circumcircles of triangles <math>BMP</math> and <math>CNP</math> meet at two distinct points <math>P</math> and <math>Q</math>. Prove that <math>\angle BAQ = \angle CAP</math>. ''<br />
* Problem 3: '''7.5'''<br />
*: '' A <math>9 \times 12</math> rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres <math>C_1,C_2...,C_{96}</math> in such way that the following to conditions are both fulfilled''<br />
<br />
::<math>(\rm i)</math> ''the distances <math>C_1C_2,...C_{95}C_{96}, C_{96}C_{1}</math> are all equal to <math>\sqrt {13}</math>''<br />
<br />
::<math>(\rm ii)</math> ''the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry?''<br />
* Problem 4: '''8'''<br />
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that'' <cmath>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2\text{ for all }m,n \in S.</cmath><br />
<br />
==Hard Olympiad Competitions==<br />
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AHard+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMO]] ===<br />
* Problem 1/4: '''6-7'''<br />
*: ''Let <math>\mathcal{P}</math> be a convex polygon with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n - 3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.'' ([[2008 USAMO Problems/Problem 4|Solution]]) <br />
* Problem 2/5: '''7-8'''<br />
*: ''Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x \le y</math>, then erase <math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.'' ([[2008 USAMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''8-9'''<br />
*: ''Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n </math> real roots.'' ([[2002 USAMO Problems/Problem 3|Solution]])<br />
<br />
=== [[USA TST]] ===<br />
<br />
<br />
<br />
* Problem 1/4/7: '''6.5-7'''<br />
* Problem 2/5/8: '''7.5-8'''<br />
* Problem 3/6/9: '''8.5-9'''<br />
<br />
=== [[Putnam]] ===<br />
<br />
* Problem A/B,1-2: '''7'''<br />
*: ''Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola <math>xyz = 1</math> and both branches of the hyperbola <math>xwy = - 1.</math> (A set <math>S</math> in the plane is called ''convex'' if for any two points in <math>S</math> the line segment connecting them is contained in <math>S.</math>)'' ([https://artofproblemsolving.com/community/c7h177227p978383 Solution])<br />
* Problem A/B,3-4: '''8'''<br />
*: ''Let <math>H</math> be an <math>n\times n</math> matrix all of whose entries are <math>\pm1</math> and whose rows are mutually orthogonal. Suppose <math>H</math> has an <math>a\times b</math> submatrix whose entries are all <math>1.</math> Show that <math>ab\le n</math>.'' ([https://artofproblemsolving.com/community/c7h64435p383280 Solution])<br />
* Problem A/B,5-6: '''9'''<br />
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution])<br />
<br />
=== [[China TST]] ===<br />
<br />
* Problem 1/4: '''8-8.5''' <br />
*: ''Given an integer <math>m,</math> prove that there exist odd integers <math>a,b</math> and a positive integer <math>k</math> such that <cmath>2m=a^{19}+b^{99}+k*2^{1000}.</cmath>''<br />
* Problem 2/5: '''9''' <br />
*: ''Given a positive integer <math>n=1</math> and real numbers <math>a_1 < a_2 < \ldots < a_n,</math> such that <math>\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,</math> prove that for any positive real number <math>x,</math> <cmath>\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.</cmath>''<br />
* Problem 3/6: '''9.5-10'''<br />
*: ''Let <math>n>1</math> be an integer and let <math>a_0,a_1,\ldots,a_n</math> be non-negative real numbers. Define <math>S_k=\sum_{i=0}^k \binom{k}{i}a_i</math> for <math>k=0,1,\ldots,n</math>. Prove that<cmath>\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.</cmath>''<br />
<br />
=== [[IMO]] ===<br />
<br />
* Problem 1/4: '''5.5-7'''<br />
*: ''Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.'' ([[2018 IMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''7-8'''<br />
*: ''Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.'' ([[2006 IMO Problems/Problem 5|Solution]])<br />
<br />
* Problem 3/6: '''9-10'''<br />
*: ''Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.'' ([https://artofproblemsolving.com/community/c6h101488p572824 Solution])<br />
<br />
=== [[IMO Shortlist]] ===<br />
<br />
* Problem 1-2: '''5.5-7'''<br />
* Problem 3-4: '''7-8'''<br />
* Problem 5+: '''8-10'''<br />
<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Competition_ratings&diff=147986AoPS Wiki:Competition ratings2021-02-26T14:59:39Z<p>Myh2910: /* Balkan MO */</p>
<hr />
<div>This page contains an approximate estimation of the difficulty level of various [[List of mathematics competitions|competitions]]. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.<br />
<br />
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution. <br />
<br />
As you may have guessed with time many competitions got more challenging because many countries got more access to books targeted at olympiad preparation. But especially web site where one can discuss Olympiads such as our very own AoPS!<br />
<br />
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC problems] and 10 is hardest level, e.g. [http://www.mathlinks.ro/resources.php?c=37&cid=47 China IMO Team Selection Test.] When considering problem difficulty '''put more emphasis on problem-solving aspects and less so on technical skill requirements'''.<br />
<br />
= Scale =<br />
All levels are estimated and refer to ''averages''. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this. <br />
# Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, #1-10 AMC 10s, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problems<br />
# For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #11-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc), traditional middle/high school word problems with extremely complex problem solving<br />
# Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, easier #1-5 on AIMEs, etc.)<br />
# Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.<br />
# More difficult AIME problems (#10-12), simple proof-based problems (JBMO), etc<br />
# High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1,4s).<br />
# Tougher Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.<br />
# High-level Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.<br />
# Expert Olympiad-level questions, eg #3,6s on difficult Olympiad contests.<br />
# Super Expert problems, problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).<br />
<br />
= Competitions =<br />
<br />
==Introductory Competitions==<br />
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntroductory+mathematics+competitions here].<br />
<br />
=== [[MOEMS]] ===<br />
*Division E: '''1'''<br />
*: ''The whole number <math>N</math> is divisible by <math>7</math>. <math>N</math> leaves a remainder of <math>1</math> when divided by <math>2,3,4,</math> or <math>5</math>. What is the smallest value that <math>N</math> can be?'' ([http://www.moems.org/sample_files/SampleE.pdf Solution])<br />
*Division M: '''1'''<br />
*: ''The value of a two-digit number is <math>10</math> times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.'' ([http://www.moems.org/sample_files/SampleM.pdf Solution])<br />
<br />
=== [[AMC 8]] ===<br />
<br />
* Problem 1 - Problem 12: '''1''' <br />
*: ''The <math>\emph{harmonic mean}</math> of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?'' ([[2018 AMC 8 Problems/Problem 10|Solution]])<br />
* Problem 13 - Problem 25: '''1.5-2'''<br />
*: ''How many positive factors does <math>23,232</math> have?'' ([[2018 AMC 8 Problems/Problem 18|Solution]])<br />
<br />
=== [[Mathcounts]] ===<br />
<br />
* Countdown: '''1-2.'''<br />
* Sprint: '''1-1.5''' (school/chapter), '''1.5-2''' (State), '''2-2.5''' (National)<br />
* Target: '''1-2''' (school/chapter), '''1.5-2.5''' (State), '''2.5-3.5''' (National)<br />
<br />
=== [[AMC 10]] ===<br />
<br />
* Problem 1 - 10: '''1-2'''<br />
*: ''A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box?'' ([[2016 AMC 10A Problems/Problem 5|Solution]])<br />
* Problem 11 - 20: '''2-3'''<br />
*: ''For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?'' ([[2019 AMC 10A Problems/Problem 18|Solution]])<br />
* Problem 21 - 25: '''3.5-4.5'''<br />
*: ''The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?'' ([[2017 AMC 10B Problems/Problem 24|Solution]])<br />
<br />
===[[CEMC|CEMC Multiple Choice Tests]]===<br />
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.<br />
<br />
* Part A: '''1-1.5'''<br />
*: ''How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number?'' (2015 Gauss 7 Problem 10)<br />
* Part B: '''1-2'''<br />
*: ''Two lines with slopes <math>\tfrac14</math> and <math>\tfrac54</math> intersect at <math>(1,1)</math>. What is the area of the triangle formed by these two lines and the vertical line <math>x = 5</math>?'' (2017 Cayley Problem 19)<br />
* Part C (Gauss/Pascal): '''2-2.5'''<br />
*: ''Suppose that <math>\tfrac{2009}{2014} + \tfrac{2019}{n} = \tfrac{a}{b}</math>, where <math>a</math>, <math>b</math>, and <math>n</math> are positive integers with <math>\tfrac{a}{b}</math> in lowest terms. What is the sum of the digits of the smallest positive integer <math>n</math> for which <math>a</math> is a multiple of 1004?'' (2014 Pascal Problem 25)<br />
* Part C (Cayley/Fermat): '''2.5-3'''<br />
*: ''Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets?'' (2018 Fermat Problem 24)<br />
<br />
===[[CEMC|CEMC Fryer/Galois/Hypatia]]===<br />
<br />
* Problem 1-2: '''1-2'''<br />
* Problem 3-4 (early parts): '''2-3'''<br />
* Problem 3-4 (later parts): '''3-5'''<br />
<br />
===Problem Solving Books for Introductory Students===<br />
<br />
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.<br />
<br />
===[[Prealgebra by AoPS]]===<br />
1-2<br />
===[[Introduction to Algebra by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Counting and Probability by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Number Theory by AoPS]]===<br />
1-3<br />
===[[Introduction to Geometry by AoPS]]===<br />
1-4<br />
<br />
==Intermediate Competitions==<br />
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntermediate+mathematics+competitions here].<br />
<br />
=== [[AMC 12]] ===<br />
<br />
* Problem 1-10: '''1.5-2'''<br />
*: ''What is the value of <cmath>\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?</cmath>'' ([[2018 AMC 12B Problems/Problem 7|Solution]])<br />
* Problem 11-20: '''2.5-3.5'''<br />
*: ''An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?'' ([[2006 AMC 12B Problems/Problem 18|Solution]])<br />
* Problem 21-25: '''4.5-6'''<br />
*: ''Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?'' ([[2009 AMC 12A Problems/Problem 23|Solution]])<br />
<br />
=== [[AIME]] ===<br />
<br />
* Problem 1 - 5: '''3-3.5'''<br />
*: ''Consider the integer <cmath>N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>.'' ([[2019 AIME I Problems/Problem 1|Solution]])<br />
* Problem 6 - 9: '''4-4.5''' <br />
*: ''How many positive integers <math>N</math> less than <math>1000</math> are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>?'' ([[2009 AIME I Problems/Problem 6|Solution]])<br />
* Problem 10 - 12: '''5-5.5'''<br />
*: Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by <math>1000</math>.Let <math>S</math> be the sum of all elements in <math>R</math>. Find the remainder when <math>S</math> is divided by <math>1000</math> ([[2011 AIME I Problems/Problem 11|Solution]])<br />
* Problem 13 - 15: '''6-6.5'''<br />
*: ''Let <cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath> Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let <cmath>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</cmath> where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>.'' ([[2003 AIME II Problems/Problem 15|Solution]])<br />
<br />
=== [[ARML]] ===<br />
<br />
* Individuals, Problem 1: '''2'''<br />
<br />
* Individuals, Problems 2, 3, 4, 5, 7, and 9: '''3'''<br />
<br />
* Individuals, Problems 6 and 8: '''4''' <br />
<br />
* Individuals, Problem 10: '''5.5'''<br />
<br />
* Team/power, Problem 1-5: '''3.5''' <br />
<br />
* Team/power, Problem 6-10: '''5'''<br />
<br />
===[[HMMT|HMMT (November)]]===<br />
* Individual Round, Problem 6-8: '''4'''<br />
* Individual Round, Problem 10: '''4.5'''<br />
* Team Round: '''4-5'''<br />
* Guts: '''3.5-5.25'''<br />
<br />
===[[CEMC|CEMC Euclid]]===<br />
<br />
* Problem 1-6: '''1-3'''<br />
* Problem 7-10: '''3-5'''<br />
<br />
===[[Purple Comet! Math Meet|Purple Comet]]===<br />
<br />
* Problems 1-10 (MS): '''1.5-3'''<br />
* Problems 11-20 (MS): '''3-4.5'''<br />
* Problems 1-10 (HS): '''2-3.5'''<br />
* Problems 11-20 (HS): '''3.5'''<br />
* Problems 21-30 (HS): '''4.5-6'''<br />
<br />
=== [[Philippine Mathematical Olympiad Qualifying Round]] ===<br />
<br />
* Problem 1-15: '''2'''<br />
* Problem 16-25: '''3'''<br />
* Problem 26-30: '''4'''<br />
<br />
===[[Lexington Math Tournament|LMT]]===<br />
<br />
* Easy Problems: '''1-2'''<br />
*: ''Let trapezoid <math>ABCD</math> be such that <math>AB||CD</math>. Additionally, <math>AC = AD = 5</math>, <math>CD = 6</math>, and <math>AB = 3</math>. Find <math>BC</math>. ''<br />
* Medium Problems: '''2-4'''<br />
*: ''Let <math>\triangle LMN</math> have side lengths <math>LM = 15</math>, <math>MN = 14</math>, and <math>NL = 13</math>. Let the angle bisector of <math>\angle MLN</math> meet the circumcircle of <math>\triangle LMN</math> at a point <math>T \ne L</math>. Determine the area of <math>\triangle LMT</math>. ''<br />
* Hard Problems: '''5-7'''<br />
*: ''A magic <math>3 \times 5</math> board can toggle its cells between black and white. Define a pattern to be an assignment of black or white to each of the board’s <math>15</math> cells (so there are <math>2^{15}</math> patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than <math>3</math> cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day 1, compute the maximum number of days it can stay alive.''<br />
<br />
==Problem Solving Books for Intermediate Students==<br />
<br />
Remark: As stated above, there are many books for Intermediate students that have not been published by AoPS. Below is a list of intermediate books that AoPS has published and their difficulty. The left-hand number corresponds to the difficulty of the easiest review problems, while the right-hand number corresponds to the difficulty of the hardest challenge problems.<br />
<br />
===[[Intermediate Algebra by AoPS]]===<br />
'''2.5-6.5/7''', may vary across chapters<br />
<br />
===[[Intermediate Counting & Probability by AoPS]]===<br />
'''3.5-7.5/8''', may vary across chapters<br />
<br />
===[[Precalculus by AoPS]]===<br />
'''2-8''', may vary across chapters<br />
<br />
==Beginner Olympiad Competitions==<br />
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3ABeginner+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMTS]] ===<br />
USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:<br />
* Problem 1-2: '''3-4'''<br />
*: ''Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter.'' ([http://usamts.org/Solutions/Solution2_3_16.pdf Solution])<br />
* Problem 3-5: '''4-6'''<br />
*: ''Call a positive real number groovy if it can be written in the form <math>\sqrt{n} + \sqrt{n + 1}</math> for some positive integer <math>n</math>. Show that if <math>x</math> is groovy, then for any positive integer <math>r</math>, the number <math>x^r</math> is groovy as well.'' ([http://usamts.org/Solutions/Solutions_20_1.pdf Solution])<br />
<br />
=== [[Indonesia Mathematical Olympiad|Indonesia MO]] ===<br />
* Problem 1/5: '''3.5'''<br />
*: '' In a drawer, there are at most <math>2009</math> balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is <math>\frac12</math>. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?'' ([https://artofproblemsolving.com/community/c6h294065 Solution])<br />
* Problem 2/6: '''4.5'''<br />
*: ''Find the lowest possible values from the function <cmath>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</cmath> for any real numbers <math>x</math>.'' ([https://artofproblemsolving.com/community/c6h294067 Solution])<br />
* Problem 3/7: '''5'''<br />
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if <cmath>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</cmath> Given 2 positive integers <math>a,b > 1</math> which are relatively prime, prove that there exists a ''good'' pair <math>(m,n)</math> with <math>a\mid m</math> and <math>b\mid n</math>, but <math>a\nmid n</math> and <math>b\nmid m</math>.'' ([https://artofproblemsolving.com/community/c6h294068 Solution])<br />
* Problem 4/8: '''6'''<br />
*: ''Given an acute triangle <math>ABC</math>. The incircle of triangle <math>ABC</math> touches <math>BC,CA,AB</math> respectively at <math>D,E,F</math>. The angle bisector of <math>\angle A</math> cuts <math>DE</math> and <math>DF</math> respectively at <math>K</math> and <math>L</math>. Suppose <math>AA_1</math> is one of the altitudes of triangle <math>ABC</math>, and <math>M</math> be the midpoint of <math>BC</math>.''<br />
<br />
::''(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.''<br />
<br />
::''(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' ([https://artofproblemsolving.com/community/c6h294069 Solution])<br />
<br />
=== [[Central American Olympiad]] ===<br />
* Problem 1: '''4'''<br />
*: ''Find all three-digit numbers <math>abc</math> (with <math>a \neq 0</math>) such that <math>a^{2} + b^{2} + c^{2}</math> is a divisor of 26.'' ([https://artofproblemsolving.com/community/c6h161957p903856 Solution])<br />
* Problem 2,4,5: '''5-6'''<br />
*: ''Show that the equation <math>a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005</math> has no integer solutions.'' ([https://artofproblemsolving.com/community/c6h46028p291301 Solution])<br />
* Problem 3/6: '''6.5''' <br />
*: ''Let <math>ABCD</math> be a convex quadrilateral. <math>I = AC\cap BD</math>, and <math>E</math>, <math>H</math>, <math>F</math> and <math>G</math> are points on <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively, such that <math>EF \cap GH = I</math>. If <math>M = EG \cap AC</math>, <math>N = HF \cap AC</math>, show that <math>\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}</math>.'' ([https://artofproblemsolving.com/community/c6h146421p828841 Solution])<br />
<br />
=== [[JBMO]] ===<br />
<br />
* Problem 1: '''4'''<br />
*: ''Find all real numbers <math>a,b,c,d</math> such that <br />
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''<br />
* Problem 2: '''4.5-5'''<br />
*: ''Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAC=\angle BDC=36^\circ</math>, <math>\angle CBD=18^\circ</math> and <math>\angle BAC=72^\circ</math>. The diagonals intersect at point <math>P</math>. Determine the measure of <math>\angle APD</math>.''<br />
* Problem 3: '''5'''<br />
*: ''Find all prime numbers <math>p,q,r</math>, such that <math>\frac pq-\frac4{r+1}=1</math>.''<br />
* Problem 4: '''6'''<br />
*: ''A <math>4\times4</math> table is divided into <math>16</math> white unit square cells. Two cells are called neighbors if they share a common side. A '''move''' consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly <math>n</math> moves all the <math>16</math> cells were black. Find all possible values of <math>n</math>.''<br />
<br />
==Olympiad Competitions==<br />
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AOlympiad+mathematics+competitions here].<br />
<br />
=== [[USAJMO]] ===<br />
* Problem 1/4: '''5'''<br />
*: ''There are <math>a+b</math> bowls arranged in a row, numbered <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive integers. Initially, each of the first <math>a</math> bowls contains an apple, and each of the last <math>b</math> bowls contains a pear.''<br />
<br />
::''A legal move consists of moving an apple from bowl <math>i</math> to bowl <math>i+1</math> and a pear from bowl <math>j</math> to bowl <math>j-1</math>, provided that the difference <math>i-j</math> is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first <math>b</math> bowls each containing a pear and the last <math>a</math> bowls each containing an apple. Show that this is possible if and only if the product <math>ab</math> is even.'' ([[2019 USAJMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''6-6.5'''<br />
*: ''Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>'' ([[2018 USAJMO Problems/Problem 2|Solution]])<br />
<br />
* Problem 3/6: '''7'''<br />
*: ''Two rational numbers <math>\tfrac{m}{n}</math> and <math>\tfrac{n}{m}</math> are written on a blackboard, where <math>m</math> and <math>n</math> are relatively prime positive integers. At any point, Evan may pick two of the numbers <math>x</math> and <math>y</math> written on the board and write either their arithmetic mean <math>\tfrac{x+y}{2}</math> or their harmonic mean <math>\tfrac{2xy}{x+y}</math> on the board as well. Find all pairs <math>(m,n)</math> such that Evan can write <math>1</math> on the board in finitely many steps.'' ([[2019 USAJMO Problems/Problem 6|Solution]])<br />
<br />
===[[HMMT|HMMT (February)]]===<br />
* Individual Round, Problem 1-5: '''5'''<br />
* Individual Round, Problem 6-10: '''5.5-6'''<br />
* Team Round: '''7.5'''<br />
* HMIC: '''8'''<br />
<br />
=== [[Canadian MO]] ===<br />
<br />
* Problem 1: '''5.5'''<br />
* Problem 2: '''6'''<br />
* Problem 3: '''6.5''' <br />
* Problem 4: '''7-7.5'''<br />
* Problem 5: '''7.5-8'''<br />
<br />
=== Austrian MO ===<br />
<br />
* Regional Competition for Advanced Students, Problems 1-4: '''5''' <br />
* Federal Competition for Advanced Students, Part 1. Problems 1-4: '''6''' <br />
* Federal Competition for Advanced Students, Part 2, Problems 1-6: '''7'''<br />
<br />
=== [[Iberoamerican Math Olympiad]] ===<br />
<br />
* Problem 1/4: '''5.5'''<br />
* Problem 2/5: '''6.5'''<br />
* Problem 3/6: '''7.5'''<br />
<br />
=== [[APMO]] ===<br />
*Problem 1: '''6'''<br />
*Problem 2: '''7'''<br />
*Problem 3: '''7'''<br />
*Problem 4: '''7.5'''<br />
*Problem 5: '''8.5'''<br />
<br />
=== Balkan MO ===<br />
<br />
* Problem 1: '''6'''<br />
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. '' <br />
* Problem 2: '''6.5'''<br />
*: '' Let <math>MN</math> be a line parallel to the side <math>BC</math> of a triangle <math>ABC</math>, with <math>M</math> on the side <math>AB</math> and <math>N</math> on the side <math>AC</math>. The lines <math>BN</math> and <math>CM</math> meet at point <math>P</math>. The circumcircles of triangles <math>BMP</math> and <math>CNP</math> meet at two distinct points <math>P</math> and <math>Q</math>. Prove that <math>\angle BAQ = \angle CAP</math>. ''<br />
* Problem 3: '''7.5'''<br />
*: '' A <math>9 \times 12</math> rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres <math>C_1,C_2...,C_{96}</math> in such way that the following to conditions are both fulfilled''<br />
<br />
::<math>(\rm i)</math> ''the distances <math>C_1C_2,...C_{95}C_{96}, C_{96}C_{1}</math> are all equal to <math>\sqrt {13}</math>''<br />
<br />
::<math>(\rm ii)</math> ''the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry?''<br />
* Problem 4: '''8'''<br />
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that'' <cmath>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2\text{ for all }m,n \in S.</cmath><br />
<br />
==Hard Olympiad Competitions==<br />
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AHard+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMO]] ===<br />
* Problem 1/4: '''6-7'''<br />
*: ''Let <math>\mathcal{P}</math> be a convex polygon with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n - 3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.'' ([[2008 USAMO Problems/Problem 4|Solution]]) <br />
* Problem 2/5: '''7-8'''<br />
*: ''Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x \le y</math>, then erase <math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.'' ([[2008 USAMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''8-9'''<br />
*: ''Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n </math> real roots.'' ([[2002 USAMO Problems/Problem 3|Solution]])<br />
<br />
=== [[USA TST]] ===<br />
<br />
<br />
<br />
* Problem 1/4/7: '''6.5-7'''<br />
* Problem 2/5/8: '''7.5-8'''<br />
* Problem 3/6/9: '''8.5-9'''<br />
<br />
=== [[Putnam]] ===<br />
<br />
* Problem A/B,1-2: '''7'''<br />
*: ''Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola <math>xyz = 1</math> and both branches of the hyperbola <math>xwy = - 1.</math> (A set <math>S</math> in the plane is called ''convex'' if for any two points in <math>S</math> the line segment connecting them is contained in <math>S.</math>)'' ([https://artofproblemsolving.com/community/c7h177227p978383 Solution])<br />
* Problem A/B,3-4: '''8'''<br />
*: ''Let <math>H</math> be an <math>n\times n</math> matrix all of whose entries are <math>\pm1</math> and whose rows are mutually orthogonal. Suppose <math>H</math> has an <math>a\times b</math> submatrix whose entries are all <math>1.</math> Show that <math>ab\le n</math>.'' ([https://artofproblemsolving.com/community/c7h64435p383280 Solution])<br />
* Problem A/B,5-6: '''9'''<br />
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution])<br />
<br />
=== [[China TST]] ===<br />
<br />
* Problem 1/4: '''8-8.5''' <br />
*: ''Given an integer <math>m,</math> prove that there exist odd integers <math>a,b</math> and a positive integer <math>k</math> such that <cmath>2m=a^{19}+b^{99}+k*2^{1000}.</cmath>''<br />
* Problem 2/5: '''9''' <br />
*: ''Given a positive integer <math>n=1</math> and real numbers <math>a_1 < a_2 < \ldots < a_n,</math> such that <math>\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,</math> prove that for any positive real number <math>x,</math> <cmath>\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.</cmath>''<br />
* Problem 3/6: '''9.5-10'''<br />
*: ''Let <math>n>1</math> be an integer and let <math>a_0,a_1,\ldots,a_n</math> be non-negative real numbers. Define <math>S_k=\sum_{i=0}^k \binom{k}{i}a_i</math> for <math>k=0,1,\ldots,n</math>. Prove that<cmath>\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.</cmath>''<br />
<br />
=== [[IMO]] ===<br />
<br />
* Problem 1/4: '''5.5-7'''<br />
*: ''Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.'' ([[2018 IMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''7-8'''<br />
*: ''Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.'' ([[2006 IMO Problems/Problem 5|Solution]])<br />
<br />
* Problem 3/6: '''9-10'''<br />
*: ''Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.'' ([https://artofproblemsolving.com/community/c6h101488p572824 Solution])<br />
<br />
=== [[IMO Shortlist]] ===<br />
<br />
* Problem 1-2: '''5.5-7'''<br />
* Problem 3-4: '''7-8'''<br />
* Problem 5+: '''8-10'''<br />
<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Competition_ratings&diff=147985AoPS Wiki:Competition ratings2021-02-26T14:56:24Z<p>Myh2910: /* USAJMO */</p>
<hr />
<div>This page contains an approximate estimation of the difficulty level of various [[List of mathematics competitions|competitions]]. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.<br />
<br />
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution. <br />
<br />
As you may have guessed with time many competitions got more challenging because many countries got more access to books targeted at olympiad preparation. But especially web site where one can discuss Olympiads such as our very own AoPS!<br />
<br />
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC problems] and 10 is hardest level, e.g. [http://www.mathlinks.ro/resources.php?c=37&cid=47 China IMO Team Selection Test.] When considering problem difficulty '''put more emphasis on problem-solving aspects and less so on technical skill requirements'''.<br />
<br />
= Scale =<br />
All levels are estimated and refer to ''averages''. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this. <br />
# Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, #1-10 AMC 10s, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problems<br />
# For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #11-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc), traditional middle/high school word problems with extremely complex problem solving<br />
# Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, easier #1-5 on AIMEs, etc.)<br />
# Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.<br />
# More difficult AIME problems (#10-12), simple proof-based problems (JBMO), etc<br />
# High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1,4s).<br />
# Tougher Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.<br />
# High-level Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.<br />
# Expert Olympiad-level questions, eg #3,6s on difficult Olympiad contests.<br />
# Super Expert problems, problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).<br />
<br />
= Competitions =<br />
<br />
==Introductory Competitions==<br />
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntroductory+mathematics+competitions here].<br />
<br />
=== [[MOEMS]] ===<br />
*Division E: '''1'''<br />
*: ''The whole number <math>N</math> is divisible by <math>7</math>. <math>N</math> leaves a remainder of <math>1</math> when divided by <math>2,3,4,</math> or <math>5</math>. What is the smallest value that <math>N</math> can be?'' ([http://www.moems.org/sample_files/SampleE.pdf Solution])<br />
*Division M: '''1'''<br />
*: ''The value of a two-digit number is <math>10</math> times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.'' ([http://www.moems.org/sample_files/SampleM.pdf Solution])<br />
<br />
=== [[AMC 8]] ===<br />
<br />
* Problem 1 - Problem 12: '''1''' <br />
*: ''The <math>\emph{harmonic mean}</math> of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?'' ([[2018 AMC 8 Problems/Problem 10|Solution]])<br />
* Problem 13 - Problem 25: '''1.5-2'''<br />
*: ''How many positive factors does <math>23,232</math> have?'' ([[2018 AMC 8 Problems/Problem 18|Solution]])<br />
<br />
=== [[Mathcounts]] ===<br />
<br />
* Countdown: '''1-2.'''<br />
* Sprint: '''1-1.5''' (school/chapter), '''1.5-2''' (State), '''2-2.5''' (National)<br />
* Target: '''1-2''' (school/chapter), '''1.5-2.5''' (State), '''2.5-3.5''' (National)<br />
<br />
=== [[AMC 10]] ===<br />
<br />
* Problem 1 - 10: '''1-2'''<br />
*: ''A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box?'' ([[2016 AMC 10A Problems/Problem 5|Solution]])<br />
* Problem 11 - 20: '''2-3'''<br />
*: ''For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?'' ([[2019 AMC 10A Problems/Problem 18|Solution]])<br />
* Problem 21 - 25: '''3.5-4.5'''<br />
*: ''The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?'' ([[2017 AMC 10B Problems/Problem 24|Solution]])<br />
<br />
===[[CEMC|CEMC Multiple Choice Tests]]===<br />
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.<br />
<br />
* Part A: '''1-1.5'''<br />
*: ''How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number?'' (2015 Gauss 7 Problem 10)<br />
* Part B: '''1-2'''<br />
*: ''Two lines with slopes <math>\tfrac14</math> and <math>\tfrac54</math> intersect at <math>(1,1)</math>. What is the area of the triangle formed by these two lines and the vertical line <math>x = 5</math>?'' (2017 Cayley Problem 19)<br />
* Part C (Gauss/Pascal): '''2-2.5'''<br />
*: ''Suppose that <math>\tfrac{2009}{2014} + \tfrac{2019}{n} = \tfrac{a}{b}</math>, where <math>a</math>, <math>b</math>, and <math>n</math> are positive integers with <math>\tfrac{a}{b}</math> in lowest terms. What is the sum of the digits of the smallest positive integer <math>n</math> for which <math>a</math> is a multiple of 1004?'' (2014 Pascal Problem 25)<br />
* Part C (Cayley/Fermat): '''2.5-3'''<br />
*: ''Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets?'' (2018 Fermat Problem 24)<br />
<br />
===[[CEMC|CEMC Fryer/Galois/Hypatia]]===<br />
<br />
* Problem 1-2: '''1-2'''<br />
* Problem 3-4 (early parts): '''2-3'''<br />
* Problem 3-4 (later parts): '''3-5'''<br />
<br />
===Problem Solving Books for Introductory Students===<br />
<br />
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.<br />
<br />
===[[Prealgebra by AoPS]]===<br />
1-2<br />
===[[Introduction to Algebra by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Counting and Probability by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Number Theory by AoPS]]===<br />
1-3<br />
===[[Introduction to Geometry by AoPS]]===<br />
1-4<br />
<br />
==Intermediate Competitions==<br />
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntermediate+mathematics+competitions here].<br />
<br />
=== [[AMC 12]] ===<br />
<br />
* Problem 1-10: '''1.5-2'''<br />
*: ''What is the value of <cmath>\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?</cmath>'' ([[2018 AMC 12B Problems/Problem 7|Solution]])<br />
* Problem 11-20: '''2.5-3.5'''<br />
*: ''An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?'' ([[2006 AMC 12B Problems/Problem 18|Solution]])<br />
* Problem 21-25: '''4.5-6'''<br />
*: ''Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?'' ([[2009 AMC 12A Problems/Problem 23|Solution]])<br />
<br />
=== [[AIME]] ===<br />
<br />
* Problem 1 - 5: '''3-3.5'''<br />
*: ''Consider the integer <cmath>N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>.'' ([[2019 AIME I Problems/Problem 1|Solution]])<br />
* Problem 6 - 9: '''4-4.5''' <br />
*: ''How many positive integers <math>N</math> less than <math>1000</math> are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>?'' ([[2009 AIME I Problems/Problem 6|Solution]])<br />
* Problem 10 - 12: '''5-5.5'''<br />
*: Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by <math>1000</math>.Let <math>S</math> be the sum of all elements in <math>R</math>. Find the remainder when <math>S</math> is divided by <math>1000</math> ([[2011 AIME I Problems/Problem 11|Solution]])<br />
* Problem 13 - 15: '''6-6.5'''<br />
*: ''Let <cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath> Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let <cmath>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</cmath> where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>.'' ([[2003 AIME II Problems/Problem 15|Solution]])<br />
<br />
=== [[ARML]] ===<br />
<br />
* Individuals, Problem 1: '''2'''<br />
<br />
* Individuals, Problems 2, 3, 4, 5, 7, and 9: '''3'''<br />
<br />
* Individuals, Problems 6 and 8: '''4''' <br />
<br />
* Individuals, Problem 10: '''5.5'''<br />
<br />
* Team/power, Problem 1-5: '''3.5''' <br />
<br />
* Team/power, Problem 6-10: '''5'''<br />
<br />
===[[HMMT|HMMT (November)]]===<br />
* Individual Round, Problem 6-8: '''4'''<br />
* Individual Round, Problem 10: '''4.5'''<br />
* Team Round: '''4-5'''<br />
* Guts: '''3.5-5.25'''<br />
<br />
===[[CEMC|CEMC Euclid]]===<br />
<br />
* Problem 1-6: '''1-3'''<br />
* Problem 7-10: '''3-5'''<br />
<br />
===[[Purple Comet! Math Meet|Purple Comet]]===<br />
<br />
* Problems 1-10 (MS): '''1.5-3'''<br />
* Problems 11-20 (MS): '''3-4.5'''<br />
* Problems 1-10 (HS): '''2-3.5'''<br />
* Problems 11-20 (HS): '''3.5'''<br />
* Problems 21-30 (HS): '''4.5-6'''<br />
<br />
=== [[Philippine Mathematical Olympiad Qualifying Round]] ===<br />
<br />
* Problem 1-15: '''2'''<br />
* Problem 16-25: '''3'''<br />
* Problem 26-30: '''4'''<br />
<br />
===[[Lexington Math Tournament|LMT]]===<br />
<br />
* Easy Problems: '''1-2'''<br />
*: ''Let trapezoid <math>ABCD</math> be such that <math>AB||CD</math>. Additionally, <math>AC = AD = 5</math>, <math>CD = 6</math>, and <math>AB = 3</math>. Find <math>BC</math>. ''<br />
* Medium Problems: '''2-4'''<br />
*: ''Let <math>\triangle LMN</math> have side lengths <math>LM = 15</math>, <math>MN = 14</math>, and <math>NL = 13</math>. Let the angle bisector of <math>\angle MLN</math> meet the circumcircle of <math>\triangle LMN</math> at a point <math>T \ne L</math>. Determine the area of <math>\triangle LMT</math>. ''<br />
* Hard Problems: '''5-7'''<br />
*: ''A magic <math>3 \times 5</math> board can toggle its cells between black and white. Define a pattern to be an assignment of black or white to each of the board’s <math>15</math> cells (so there are <math>2^{15}</math> patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than <math>3</math> cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day 1, compute the maximum number of days it can stay alive.''<br />
<br />
==Problem Solving Books for Intermediate Students==<br />
<br />
Remark: As stated above, there are many books for Intermediate students that have not been published by AoPS. Below is a list of intermediate books that AoPS has published and their difficulty. The left-hand number corresponds to the difficulty of the easiest review problems, while the right-hand number corresponds to the difficulty of the hardest challenge problems.<br />
<br />
===[[Intermediate Algebra by AoPS]]===<br />
'''2.5-6.5/7''', may vary across chapters<br />
<br />
===[[Intermediate Counting & Probability by AoPS]]===<br />
'''3.5-7.5/8''', may vary across chapters<br />
<br />
===[[Precalculus by AoPS]]===<br />
'''2-8''', may vary across chapters<br />
<br />
==Beginner Olympiad Competitions==<br />
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3ABeginner+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMTS]] ===<br />
USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:<br />
* Problem 1-2: '''3-4'''<br />
*: ''Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter.'' ([http://usamts.org/Solutions/Solution2_3_16.pdf Solution])<br />
* Problem 3-5: '''4-6'''<br />
*: ''Call a positive real number groovy if it can be written in the form <math>\sqrt{n} + \sqrt{n + 1}</math> for some positive integer <math>n</math>. Show that if <math>x</math> is groovy, then for any positive integer <math>r</math>, the number <math>x^r</math> is groovy as well.'' ([http://usamts.org/Solutions/Solutions_20_1.pdf Solution])<br />
<br />
=== [[Indonesia Mathematical Olympiad|Indonesia MO]] ===<br />
* Problem 1/5: '''3.5'''<br />
*: '' In a drawer, there are at most <math>2009</math> balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is <math>\frac12</math>. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?'' ([https://artofproblemsolving.com/community/c6h294065 Solution])<br />
* Problem 2/6: '''4.5'''<br />
*: ''Find the lowest possible values from the function <cmath>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</cmath> for any real numbers <math>x</math>.'' ([https://artofproblemsolving.com/community/c6h294067 Solution])<br />
* Problem 3/7: '''5'''<br />
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if <cmath>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</cmath> Given 2 positive integers <math>a,b > 1</math> which are relatively prime, prove that there exists a ''good'' pair <math>(m,n)</math> with <math>a\mid m</math> and <math>b\mid n</math>, but <math>a\nmid n</math> and <math>b\nmid m</math>.'' ([https://artofproblemsolving.com/community/c6h294068 Solution])<br />
* Problem 4/8: '''6'''<br />
*: ''Given an acute triangle <math>ABC</math>. The incircle of triangle <math>ABC</math> touches <math>BC,CA,AB</math> respectively at <math>D,E,F</math>. The angle bisector of <math>\angle A</math> cuts <math>DE</math> and <math>DF</math> respectively at <math>K</math> and <math>L</math>. Suppose <math>AA_1</math> is one of the altitudes of triangle <math>ABC</math>, and <math>M</math> be the midpoint of <math>BC</math>.''<br />
<br />
::''(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.''<br />
<br />
::''(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' ([https://artofproblemsolving.com/community/c6h294069 Solution])<br />
<br />
=== [[Central American Olympiad]] ===<br />
* Problem 1: '''4'''<br />
*: ''Find all three-digit numbers <math>abc</math> (with <math>a \neq 0</math>) such that <math>a^{2} + b^{2} + c^{2}</math> is a divisor of 26.'' ([https://artofproblemsolving.com/community/c6h161957p903856 Solution])<br />
* Problem 2,4,5: '''5-6'''<br />
*: ''Show that the equation <math>a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005</math> has no integer solutions.'' ([https://artofproblemsolving.com/community/c6h46028p291301 Solution])<br />
* Problem 3/6: '''6.5''' <br />
*: ''Let <math>ABCD</math> be a convex quadrilateral. <math>I = AC\cap BD</math>, and <math>E</math>, <math>H</math>, <math>F</math> and <math>G</math> are points on <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively, such that <math>EF \cap GH = I</math>. If <math>M = EG \cap AC</math>, <math>N = HF \cap AC</math>, show that <math>\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}</math>.'' ([https://artofproblemsolving.com/community/c6h146421p828841 Solution])<br />
<br />
=== [[JBMO]] ===<br />
<br />
* Problem 1: '''4'''<br />
*: ''Find all real numbers <math>a,b,c,d</math> such that <br />
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''<br />
* Problem 2: '''4.5-5'''<br />
*: ''Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAC=\angle BDC=36^\circ</math>, <math>\angle CBD=18^\circ</math> and <math>\angle BAC=72^\circ</math>. The diagonals intersect at point <math>P</math>. Determine the measure of <math>\angle APD</math>.''<br />
* Problem 3: '''5'''<br />
*: ''Find all prime numbers <math>p,q,r</math>, such that <math>\frac pq-\frac4{r+1}=1</math>.''<br />
* Problem 4: '''6'''<br />
*: ''A <math>4\times4</math> table is divided into <math>16</math> white unit square cells. Two cells are called neighbors if they share a common side. A '''move''' consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly <math>n</math> moves all the <math>16</math> cells were black. Find all possible values of <math>n</math>.''<br />
<br />
==Olympiad Competitions==<br />
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AOlympiad+mathematics+competitions here].<br />
<br />
=== [[USAJMO]] ===<br />
* Problem 1/4: '''5'''<br />
*: ''There are <math>a+b</math> bowls arranged in a row, numbered <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive integers. Initially, each of the first <math>a</math> bowls contains an apple, and each of the last <math>b</math> bowls contains a pear.''<br />
<br />
::''A legal move consists of moving an apple from bowl <math>i</math> to bowl <math>i+1</math> and a pear from bowl <math>j</math> to bowl <math>j-1</math>, provided that the difference <math>i-j</math> is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first <math>b</math> bowls each containing a pear and the last <math>a</math> bowls each containing an apple. Show that this is possible if and only if the product <math>ab</math> is even.'' ([[2019 USAJMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''6-6.5'''<br />
*: ''Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>'' ([[2018 USAJMO Problems/Problem 2|Solution]])<br />
<br />
* Problem 3/6: '''7'''<br />
*: ''Two rational numbers <math>\tfrac{m}{n}</math> and <math>\tfrac{n}{m}</math> are written on a blackboard, where <math>m</math> and <math>n</math> are relatively prime positive integers. At any point, Evan may pick two of the numbers <math>x</math> and <math>y</math> written on the board and write either their arithmetic mean <math>\tfrac{x+y}{2}</math> or their harmonic mean <math>\tfrac{2xy}{x+y}</math> on the board as well. Find all pairs <math>(m,n)</math> such that Evan can write <math>1</math> on the board in finitely many steps.'' ([[2019 USAJMO Problems/Problem 6|Solution]])<br />
<br />
===[[HMMT|HMMT (February)]]===<br />
* Individual Round, Problem 1-5: '''5'''<br />
* Individual Round, Problem 6-10: '''5.5-6'''<br />
* Team Round: '''7.5'''<br />
* HMIC: '''8'''<br />
<br />
=== [[Canadian MO]] ===<br />
<br />
* Problem 1: '''5.5'''<br />
* Problem 2: '''6'''<br />
* Problem 3: '''6.5''' <br />
* Problem 4: '''7-7.5'''<br />
* Problem 5: '''7.5-8'''<br />
<br />
=== Austrian MO ===<br />
<br />
* Regional Competition for Advanced Students, Problems 1-4: '''5''' <br />
* Federal Competition for Advanced Students, Part 1. Problems 1-4: '''6''' <br />
* Federal Competition for Advanced Students, Part 2, Problems 1-6: '''7'''<br />
<br />
=== [[Iberoamerican Math Olympiad]] ===<br />
<br />
* Problem 1/4: '''5.5'''<br />
* Problem 2/5: '''6.5'''<br />
* Problem 3/6: '''7.5'''<br />
<br />
=== [[APMO]] ===<br />
*Problem 1: '''6'''<br />
*Problem 2: '''7'''<br />
*Problem 3: '''7'''<br />
*Problem 4: '''7.5'''<br />
*Problem 5: '''8.5'''<br />
<br />
=== Balkan MO ===<br />
<br />
* Problem 1: '''6'''<br />
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. '' <br />
* Problem 2: '''6.5'''<br />
*: '' Let <math>MN</math> be a line parallel to the side <math>BC</math> of a triangle <math>ABC</math>, with <math>M</math> on the side <math>AB</math> and <math>N</math> on the side <math>AC</math>. The lines <math>BN</math> and <math>CM</math> meet at point <math>P</math>. The circumcircles of triangles <math>BMP</math> and <math>CNP</math> meet at two distinct points <math>P</math> and <math>Q</math>. Prove that <math>\angle BAQ = \angle CAP</math>. ''<br />
* Problem 3: '''7.5'''<br />
*: '' A <math>9 \times 12</math> rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres <math>C_1,C_2...,C_{96}</math> in such way that the following to conditions are both fulfilled<br />
<br />
<math>(i)</math> the distances <math>C_1C_2,...C_{95}C_{96}, C_{96}C_{1}</math> are all equal to <math>\sqrt {13}</math><br />
<br />
<math>(ii)</math> the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry? ''<br />
* Problem 4: '''8'''<br />
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that<br />
<br />
<math>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2</math> for all <math>m,n \in S</math>. '<br />
<br />
==Hard Olympiad Competitions==<br />
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AHard+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMO]] ===<br />
* Problem 1/4: '''6-7'''<br />
*: ''Let <math>\mathcal{P}</math> be a convex polygon with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n - 3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.'' ([[2008 USAMO Problems/Problem 4|Solution]]) <br />
* Problem 2/5: '''7-8'''<br />
*: ''Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x \le y</math>, then erase <math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.'' ([[2008 USAMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''8-9'''<br />
*: ''Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n </math> real roots.'' ([[2002 USAMO Problems/Problem 3|Solution]])<br />
<br />
=== [[USA TST]] ===<br />
<br />
<br />
<br />
* Problem 1/4/7: '''6.5-7'''<br />
* Problem 2/5/8: '''7.5-8'''<br />
* Problem 3/6/9: '''8.5-9'''<br />
<br />
=== [[Putnam]] ===<br />
<br />
* Problem A/B,1-2: '''7'''<br />
*: ''Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola <math>xyz = 1</math> and both branches of the hyperbola <math>xwy = - 1.</math> (A set <math>S</math> in the plane is called ''convex'' if for any two points in <math>S</math> the line segment connecting them is contained in <math>S.</math>)'' ([https://artofproblemsolving.com/community/c7h177227p978383 Solution])<br />
* Problem A/B,3-4: '''8'''<br />
*: ''Let <math>H</math> be an <math>n\times n</math> matrix all of whose entries are <math>\pm1</math> and whose rows are mutually orthogonal. Suppose <math>H</math> has an <math>a\times b</math> submatrix whose entries are all <math>1.</math> Show that <math>ab\le n</math>.'' ([https://artofproblemsolving.com/community/c7h64435p383280 Solution])<br />
* Problem A/B,5-6: '''9'''<br />
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution])<br />
<br />
=== [[China TST]] ===<br />
<br />
* Problem 1/4: '''8-8.5''' <br />
*: ''Given an integer <math>m,</math> prove that there exist odd integers <math>a,b</math> and a positive integer <math>k</math> such that <cmath>2m=a^{19}+b^{99}+k*2^{1000}.</cmath>''<br />
* Problem 2/5: '''9''' <br />
*: ''Given a positive integer <math>n=1</math> and real numbers <math>a_1 < a_2 < \ldots < a_n,</math> such that <math>\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,</math> prove that for any positive real number <math>x,</math> <cmath>\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.</cmath>''<br />
* Problem 3/6: '''9.5-10'''<br />
*: ''Let <math>n>1</math> be an integer and let <math>a_0,a_1,\ldots,a_n</math> be non-negative real numbers. Define <math>S_k=\sum_{i=0}^k \binom{k}{i}a_i</math> for <math>k=0,1,\ldots,n</math>. Prove that<cmath>\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.</cmath>''<br />
<br />
=== [[IMO]] ===<br />
<br />
* Problem 1/4: '''5.5-7'''<br />
*: ''Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.'' ([[2018 IMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''7-8'''<br />
*: ''Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.'' ([[2006 IMO Problems/Problem 5|Solution]])<br />
<br />
* Problem 3/6: '''9-10'''<br />
*: ''Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.'' ([https://artofproblemsolving.com/community/c6h101488p572824 Solution])<br />
<br />
=== [[IMO Shortlist]] ===<br />
<br />
* Problem 1-2: '''5.5-7'''<br />
* Problem 3-4: '''7-8'''<br />
* Problem 5+: '''8-10'''<br />
<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Competition_ratings&diff=147984AoPS Wiki:Competition ratings2021-02-26T14:55:49Z<p>Myh2910: /* Central American Olympiad */ Fixed links</p>
<hr />
<div>This page contains an approximate estimation of the difficulty level of various [[List of mathematics competitions|competitions]]. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.<br />
<br />
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution. <br />
<br />
As you may have guessed with time many competitions got more challenging because many countries got more access to books targeted at olympiad preparation. But especially web site where one can discuss Olympiads such as our very own AoPS!<br />
<br />
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC problems] and 10 is hardest level, e.g. [http://www.mathlinks.ro/resources.php?c=37&cid=47 China IMO Team Selection Test.] When considering problem difficulty '''put more emphasis on problem-solving aspects and less so on technical skill requirements'''.<br />
<br />
= Scale =<br />
All levels are estimated and refer to ''averages''. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this. <br />
# Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, #1-10 AMC 10s, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problems<br />
# For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #11-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc), traditional middle/high school word problems with extremely complex problem solving<br />
# Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, easier #1-5 on AIMEs, etc.)<br />
# Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.<br />
# More difficult AIME problems (#10-12), simple proof-based problems (JBMO), etc<br />
# High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1,4s).<br />
# Tougher Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.<br />
# High-level Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.<br />
# Expert Olympiad-level questions, eg #3,6s on difficult Olympiad contests.<br />
# Super Expert problems, problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).<br />
<br />
= Competitions =<br />
<br />
==Introductory Competitions==<br />
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntroductory+mathematics+competitions here].<br />
<br />
=== [[MOEMS]] ===<br />
*Division E: '''1'''<br />
*: ''The whole number <math>N</math> is divisible by <math>7</math>. <math>N</math> leaves a remainder of <math>1</math> when divided by <math>2,3,4,</math> or <math>5</math>. What is the smallest value that <math>N</math> can be?'' ([http://www.moems.org/sample_files/SampleE.pdf Solution])<br />
*Division M: '''1'''<br />
*: ''The value of a two-digit number is <math>10</math> times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.'' ([http://www.moems.org/sample_files/SampleM.pdf Solution])<br />
<br />
=== [[AMC 8]] ===<br />
<br />
* Problem 1 - Problem 12: '''1''' <br />
*: ''The <math>\emph{harmonic mean}</math> of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?'' ([[2018 AMC 8 Problems/Problem 10|Solution]])<br />
* Problem 13 - Problem 25: '''1.5-2'''<br />
*: ''How many positive factors does <math>23,232</math> have?'' ([[2018 AMC 8 Problems/Problem 18|Solution]])<br />
<br />
=== [[Mathcounts]] ===<br />
<br />
* Countdown: '''1-2.'''<br />
* Sprint: '''1-1.5''' (school/chapter), '''1.5-2''' (State), '''2-2.5''' (National)<br />
* Target: '''1-2''' (school/chapter), '''1.5-2.5''' (State), '''2.5-3.5''' (National)<br />
<br />
=== [[AMC 10]] ===<br />
<br />
* Problem 1 - 10: '''1-2'''<br />
*: ''A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box?'' ([[2016 AMC 10A Problems/Problem 5|Solution]])<br />
* Problem 11 - 20: '''2-3'''<br />
*: ''For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?'' ([[2019 AMC 10A Problems/Problem 18|Solution]])<br />
* Problem 21 - 25: '''3.5-4.5'''<br />
*: ''The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?'' ([[2017 AMC 10B Problems/Problem 24|Solution]])<br />
<br />
===[[CEMC|CEMC Multiple Choice Tests]]===<br />
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.<br />
<br />
* Part A: '''1-1.5'''<br />
*: ''How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number?'' (2015 Gauss 7 Problem 10)<br />
* Part B: '''1-2'''<br />
*: ''Two lines with slopes <math>\tfrac14</math> and <math>\tfrac54</math> intersect at <math>(1,1)</math>. What is the area of the triangle formed by these two lines and the vertical line <math>x = 5</math>?'' (2017 Cayley Problem 19)<br />
* Part C (Gauss/Pascal): '''2-2.5'''<br />
*: ''Suppose that <math>\tfrac{2009}{2014} + \tfrac{2019}{n} = \tfrac{a}{b}</math>, where <math>a</math>, <math>b</math>, and <math>n</math> are positive integers with <math>\tfrac{a}{b}</math> in lowest terms. What is the sum of the digits of the smallest positive integer <math>n</math> for which <math>a</math> is a multiple of 1004?'' (2014 Pascal Problem 25)<br />
* Part C (Cayley/Fermat): '''2.5-3'''<br />
*: ''Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets?'' (2018 Fermat Problem 24)<br />
<br />
===[[CEMC|CEMC Fryer/Galois/Hypatia]]===<br />
<br />
* Problem 1-2: '''1-2'''<br />
* Problem 3-4 (early parts): '''2-3'''<br />
* Problem 3-4 (later parts): '''3-5'''<br />
<br />
===Problem Solving Books for Introductory Students===<br />
<br />
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.<br />
<br />
===[[Prealgebra by AoPS]]===<br />
1-2<br />
===[[Introduction to Algebra by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Counting and Probability by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Number Theory by AoPS]]===<br />
1-3<br />
===[[Introduction to Geometry by AoPS]]===<br />
1-4<br />
<br />
==Intermediate Competitions==<br />
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntermediate+mathematics+competitions here].<br />
<br />
=== [[AMC 12]] ===<br />
<br />
* Problem 1-10: '''1.5-2'''<br />
*: ''What is the value of <cmath>\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?</cmath>'' ([[2018 AMC 12B Problems/Problem 7|Solution]])<br />
* Problem 11-20: '''2.5-3.5'''<br />
*: ''An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?'' ([[2006 AMC 12B Problems/Problem 18|Solution]])<br />
* Problem 21-25: '''4.5-6'''<br />
*: ''Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?'' ([[2009 AMC 12A Problems/Problem 23|Solution]])<br />
<br />
=== [[AIME]] ===<br />
<br />
* Problem 1 - 5: '''3-3.5'''<br />
*: ''Consider the integer <cmath>N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>.'' ([[2019 AIME I Problems/Problem 1|Solution]])<br />
* Problem 6 - 9: '''4-4.5''' <br />
*: ''How many positive integers <math>N</math> less than <math>1000</math> are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>?'' ([[2009 AIME I Problems/Problem 6|Solution]])<br />
* Problem 10 - 12: '''5-5.5'''<br />
*: Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by <math>1000</math>.Let <math>S</math> be the sum of all elements in <math>R</math>. Find the remainder when <math>S</math> is divided by <math>1000</math> ([[2011 AIME I Problems/Problem 11|Solution]])<br />
* Problem 13 - 15: '''6-6.5'''<br />
*: ''Let <cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath> Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let <cmath>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</cmath> where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>.'' ([[2003 AIME II Problems/Problem 15|Solution]])<br />
<br />
=== [[ARML]] ===<br />
<br />
* Individuals, Problem 1: '''2'''<br />
<br />
* Individuals, Problems 2, 3, 4, 5, 7, and 9: '''3'''<br />
<br />
* Individuals, Problems 6 and 8: '''4''' <br />
<br />
* Individuals, Problem 10: '''5.5'''<br />
<br />
* Team/power, Problem 1-5: '''3.5''' <br />
<br />
* Team/power, Problem 6-10: '''5'''<br />
<br />
===[[HMMT|HMMT (November)]]===<br />
* Individual Round, Problem 6-8: '''4'''<br />
* Individual Round, Problem 10: '''4.5'''<br />
* Team Round: '''4-5'''<br />
* Guts: '''3.5-5.25'''<br />
<br />
===[[CEMC|CEMC Euclid]]===<br />
<br />
* Problem 1-6: '''1-3'''<br />
* Problem 7-10: '''3-5'''<br />
<br />
===[[Purple Comet! Math Meet|Purple Comet]]===<br />
<br />
* Problems 1-10 (MS): '''1.5-3'''<br />
* Problems 11-20 (MS): '''3-4.5'''<br />
* Problems 1-10 (HS): '''2-3.5'''<br />
* Problems 11-20 (HS): '''3.5'''<br />
* Problems 21-30 (HS): '''4.5-6'''<br />
<br />
=== [[Philippine Mathematical Olympiad Qualifying Round]] ===<br />
<br />
* Problem 1-15: '''2'''<br />
* Problem 16-25: '''3'''<br />
* Problem 26-30: '''4'''<br />
<br />
===[[Lexington Math Tournament|LMT]]===<br />
<br />
* Easy Problems: '''1-2'''<br />
*: ''Let trapezoid <math>ABCD</math> be such that <math>AB||CD</math>. Additionally, <math>AC = AD = 5</math>, <math>CD = 6</math>, and <math>AB = 3</math>. Find <math>BC</math>. ''<br />
* Medium Problems: '''2-4'''<br />
*: ''Let <math>\triangle LMN</math> have side lengths <math>LM = 15</math>, <math>MN = 14</math>, and <math>NL = 13</math>. Let the angle bisector of <math>\angle MLN</math> meet the circumcircle of <math>\triangle LMN</math> at a point <math>T \ne L</math>. Determine the area of <math>\triangle LMT</math>. ''<br />
* Hard Problems: '''5-7'''<br />
*: ''A magic <math>3 \times 5</math> board can toggle its cells between black and white. Define a pattern to be an assignment of black or white to each of the board’s <math>15</math> cells (so there are <math>2^{15}</math> patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than <math>3</math> cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day 1, compute the maximum number of days it can stay alive.''<br />
<br />
==Problem Solving Books for Intermediate Students==<br />
<br />
Remark: As stated above, there are many books for Intermediate students that have not been published by AoPS. Below is a list of intermediate books that AoPS has published and their difficulty. The left-hand number corresponds to the difficulty of the easiest review problems, while the right-hand number corresponds to the difficulty of the hardest challenge problems.<br />
<br />
===[[Intermediate Algebra by AoPS]]===<br />
'''2.5-6.5/7''', may vary across chapters<br />
<br />
===[[Intermediate Counting & Probability by AoPS]]===<br />
'''3.5-7.5/8''', may vary across chapters<br />
<br />
===[[Precalculus by AoPS]]===<br />
'''2-8''', may vary across chapters<br />
<br />
==Beginner Olympiad Competitions==<br />
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3ABeginner+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMTS]] ===<br />
USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:<br />
* Problem 1-2: '''3-4'''<br />
*: ''Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter.'' ([http://usamts.org/Solutions/Solution2_3_16.pdf Solution])<br />
* Problem 3-5: '''4-6'''<br />
*: ''Call a positive real number groovy if it can be written in the form <math>\sqrt{n} + \sqrt{n + 1}</math> for some positive integer <math>n</math>. Show that if <math>x</math> is groovy, then for any positive integer <math>r</math>, the number <math>x^r</math> is groovy as well.'' ([http://usamts.org/Solutions/Solutions_20_1.pdf Solution])<br />
<br />
=== [[Indonesia Mathematical Olympiad|Indonesia MO]] ===<br />
* Problem 1/5: '''3.5'''<br />
*: '' In a drawer, there are at most <math>2009</math> balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is <math>\frac12</math>. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?'' ([https://artofproblemsolving.com/community/c6h294065 Solution])<br />
* Problem 2/6: '''4.5'''<br />
*: ''Find the lowest possible values from the function <cmath>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</cmath> for any real numbers <math>x</math>.'' ([https://artofproblemsolving.com/community/c6h294067 Solution])<br />
* Problem 3/7: '''5'''<br />
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if <cmath>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</cmath> Given 2 positive integers <math>a,b > 1</math> which are relatively prime, prove that there exists a ''good'' pair <math>(m,n)</math> with <math>a\mid m</math> and <math>b\mid n</math>, but <math>a\nmid n</math> and <math>b\nmid m</math>.'' ([https://artofproblemsolving.com/community/c6h294068 Solution])<br />
* Problem 4/8: '''6'''<br />
*: ''Given an acute triangle <math>ABC</math>. The incircle of triangle <math>ABC</math> touches <math>BC,CA,AB</math> respectively at <math>D,E,F</math>. The angle bisector of <math>\angle A</math> cuts <math>DE</math> and <math>DF</math> respectively at <math>K</math> and <math>L</math>. Suppose <math>AA_1</math> is one of the altitudes of triangle <math>ABC</math>, and <math>M</math> be the midpoint of <math>BC</math>.''<br />
<br />
::''(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.''<br />
<br />
::''(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' ([https://artofproblemsolving.com/community/c6h294069 Solution])<br />
<br />
=== [[Central American Olympiad]] ===<br />
* Problem 1: '''4'''<br />
*: ''Find all three-digit numbers <math>abc</math> (with <math>a \neq 0</math>) such that <math>a^{2} + b^{2} + c^{2}</math> is a divisor of 26.'' ([https://artofproblemsolving.com/community/c6h161957p903856 Solution])<br />
* Problem 2,4,5: '''5-6'''<br />
*: ''Show that the equation <math>a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005</math> has no integer solutions.'' ([https://artofproblemsolving.com/community/c6h46028p291301 Solution])<br />
* Problem 3/6: '''6.5''' <br />
*: ''Let <math>ABCD</math> be a convex quadrilateral. <math>I = AC\cap BD</math>, and <math>E</math>, <math>H</math>, <math>F</math> and <math>G</math> are points on <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively, such that <math>EF \cap GH = I</math>. If <math>M = EG \cap AC</math>, <math>N = HF \cap AC</math>, show that <math>\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}</math>.'' ([https://artofproblemsolving.com/community/c6h146421p828841 Solution])<br />
<br />
=== [[JBMO]] ===<br />
<br />
* Problem 1: '''4'''<br />
*: ''Find all real numbers <math>a,b,c,d</math> such that <br />
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''<br />
* Problem 2: '''4.5-5'''<br />
*: ''Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAC=\angle BDC=36^\circ</math>, <math>\angle CBD=18^\circ</math> and <math>\angle BAC=72^\circ</math>. The diagonals intersect at point <math>P</math>. Determine the measure of <math>\angle APD</math>.''<br />
* Problem 3: '''5'''<br />
*: ''Find all prime numbers <math>p,q,r</math>, such that <math>\frac pq-\frac4{r+1}=1</math>.''<br />
* Problem 4: '''6'''<br />
*: ''A <math>4\times4</math> table is divided into <math>16</math> white unit square cells. Two cells are called neighbors if they share a common side. A '''move''' consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly <math>n</math> moves all the <math>16</math> cells were black. Find all possible values of <math>n</math>.''<br />
<br />
==Olympiad Competitions==<br />
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AOlympiad+mathematics+competitions here].<br />
<br />
=== [[USAJMO]] ===<br />
* Problem 1/4: '''5'''<br />
*: ''There are <math>a+b</math> bowls arranged in a row, numbered <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive integers. Initially, each of the first <math>a</math> bowls contains an apple, and each of the last <math>b</math> bowls contains a pear.''<br />
<br />
''A legal move consists of moving an apple from bowl <math>i</math> to bowl <math>i+1</math> and a pear from bowl <math>j</math> to bowl <math>j-1</math>, provided that the difference <math>i-j</math> is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first <math>b</math> bowls each containing a pear and the last <math>a</math> bowls each containing an apple. Show that this is possible if and only if the product <math>ab</math> is even.'' ([[2019 USAJMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''6-6.5'''<br />
*: ''Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>'' ([[2018 USAJMO Problems/Problem 2|Solution]])<br />
<br />
* Problem 3/6: '''7'''<br />
*: ''Two rational numbers <math>\tfrac{m}{n}</math> and <math>\tfrac{n}{m}</math> are written on a blackboard, where <math>m</math> and <math>n</math> are relatively prime positive integers. At any point, Evan may pick two of the numbers <math>x</math> and <math>y</math> written on the board and write either their arithmetic mean <math>\tfrac{x+y}{2}</math> or their harmonic mean <math>\tfrac{2xy}{x+y}</math> on the board as well. Find all pairs <math>(m,n)</math> such that Evan can write <math>1</math> on the board in finitely many steps.'' ([[2019 USAJMO Problems/Problem 6|Solution]])<br />
<br />
===[[HMMT|HMMT (February)]]===<br />
* Individual Round, Problem 1-5: '''5'''<br />
* Individual Round, Problem 6-10: '''5.5-6'''<br />
* Team Round: '''7.5'''<br />
* HMIC: '''8'''<br />
<br />
=== [[Canadian MO]] ===<br />
<br />
* Problem 1: '''5.5'''<br />
* Problem 2: '''6'''<br />
* Problem 3: '''6.5''' <br />
* Problem 4: '''7-7.5'''<br />
* Problem 5: '''7.5-8'''<br />
<br />
=== Austrian MO ===<br />
<br />
* Regional Competition for Advanced Students, Problems 1-4: '''5''' <br />
* Federal Competition for Advanced Students, Part 1. Problems 1-4: '''6''' <br />
* Federal Competition for Advanced Students, Part 2, Problems 1-6: '''7'''<br />
<br />
=== [[Iberoamerican Math Olympiad]] ===<br />
<br />
* Problem 1/4: '''5.5'''<br />
* Problem 2/5: '''6.5'''<br />
* Problem 3/6: '''7.5'''<br />
<br />
=== [[APMO]] ===<br />
*Problem 1: '''6'''<br />
*Problem 2: '''7'''<br />
*Problem 3: '''7'''<br />
*Problem 4: '''7.5'''<br />
*Problem 5: '''8.5'''<br />
<br />
=== Balkan MO ===<br />
<br />
* Problem 1: '''6'''<br />
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. '' <br />
* Problem 2: '''6.5'''<br />
*: '' Let <math>MN</math> be a line parallel to the side <math>BC</math> of a triangle <math>ABC</math>, with <math>M</math> on the side <math>AB</math> and <math>N</math> on the side <math>AC</math>. The lines <math>BN</math> and <math>CM</math> meet at point <math>P</math>. The circumcircles of triangles <math>BMP</math> and <math>CNP</math> meet at two distinct points <math>P</math> and <math>Q</math>. Prove that <math>\angle BAQ = \angle CAP</math>. ''<br />
* Problem 3: '''7.5'''<br />
*: '' A <math>9 \times 12</math> rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres <math>C_1,C_2...,C_{96}</math> in such way that the following to conditions are both fulfilled<br />
<br />
<math>(i)</math> the distances <math>C_1C_2,...C_{95}C_{96}, C_{96}C_{1}</math> are all equal to <math>\sqrt {13}</math><br />
<br />
<math>(ii)</math> the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry? ''<br />
* Problem 4: '''8'''<br />
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that<br />
<br />
<math>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2</math> for all <math>m,n \in S</math>. '<br />
<br />
==Hard Olympiad Competitions==<br />
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AHard+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMO]] ===<br />
* Problem 1/4: '''6-7'''<br />
*: ''Let <math>\mathcal{P}</math> be a convex polygon with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n - 3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.'' ([[2008 USAMO Problems/Problem 4|Solution]]) <br />
* Problem 2/5: '''7-8'''<br />
*: ''Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x \le y</math>, then erase <math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.'' ([[2008 USAMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''8-9'''<br />
*: ''Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n </math> real roots.'' ([[2002 USAMO Problems/Problem 3|Solution]])<br />
<br />
=== [[USA TST]] ===<br />
<br />
<br />
<br />
* Problem 1/4/7: '''6.5-7'''<br />
* Problem 2/5/8: '''7.5-8'''<br />
* Problem 3/6/9: '''8.5-9'''<br />
<br />
=== [[Putnam]] ===<br />
<br />
* Problem A/B,1-2: '''7'''<br />
*: ''Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola <math>xyz = 1</math> and both branches of the hyperbola <math>xwy = - 1.</math> (A set <math>S</math> in the plane is called ''convex'' if for any two points in <math>S</math> the line segment connecting them is contained in <math>S.</math>)'' ([https://artofproblemsolving.com/community/c7h177227p978383 Solution])<br />
* Problem A/B,3-4: '''8'''<br />
*: ''Let <math>H</math> be an <math>n\times n</math> matrix all of whose entries are <math>\pm1</math> and whose rows are mutually orthogonal. Suppose <math>H</math> has an <math>a\times b</math> submatrix whose entries are all <math>1.</math> Show that <math>ab\le n</math>.'' ([https://artofproblemsolving.com/community/c7h64435p383280 Solution])<br />
* Problem A/B,5-6: '''9'''<br />
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution])<br />
<br />
=== [[China TST]] ===<br />
<br />
* Problem 1/4: '''8-8.5''' <br />
*: ''Given an integer <math>m,</math> prove that there exist odd integers <math>a,b</math> and a positive integer <math>k</math> such that <cmath>2m=a^{19}+b^{99}+k*2^{1000}.</cmath>''<br />
* Problem 2/5: '''9''' <br />
*: ''Given a positive integer <math>n=1</math> and real numbers <math>a_1 < a_2 < \ldots < a_n,</math> such that <math>\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,</math> prove that for any positive real number <math>x,</math> <cmath>\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.</cmath>''<br />
* Problem 3/6: '''9.5-10'''<br />
*: ''Let <math>n>1</math> be an integer and let <math>a_0,a_1,\ldots,a_n</math> be non-negative real numbers. Define <math>S_k=\sum_{i=0}^k \binom{k}{i}a_i</math> for <math>k=0,1,\ldots,n</math>. Prove that<cmath>\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.</cmath>''<br />
<br />
=== [[IMO]] ===<br />
<br />
* Problem 1/4: '''5.5-7'''<br />
*: ''Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.'' ([[2018 IMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''7-8'''<br />
*: ''Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.'' ([[2006 IMO Problems/Problem 5|Solution]])<br />
<br />
* Problem 3/6: '''9-10'''<br />
*: ''Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.'' ([https://artofproblemsolving.com/community/c6h101488p572824 Solution])<br />
<br />
=== [[IMO Shortlist]] ===<br />
<br />
* Problem 1-2: '''5.5-7'''<br />
* Problem 3-4: '''7-8'''<br />
* Problem 5+: '''8-10'''<br />
<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Competition_ratings&diff=147983AoPS Wiki:Competition ratings2021-02-26T14:53:47Z<p>Myh2910: /* AIME */</p>
<hr />
<div>This page contains an approximate estimation of the difficulty level of various [[List of mathematics competitions|competitions]]. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.<br />
<br />
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution. <br />
<br />
As you may have guessed with time many competitions got more challenging because many countries got more access to books targeted at olympiad preparation. But especially web site where one can discuss Olympiads such as our very own AoPS!<br />
<br />
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC problems] and 10 is hardest level, e.g. [http://www.mathlinks.ro/resources.php?c=37&cid=47 China IMO Team Selection Test.] When considering problem difficulty '''put more emphasis on problem-solving aspects and less so on technical skill requirements'''.<br />
<br />
= Scale =<br />
All levels are estimated and refer to ''averages''. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this. <br />
# Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, #1-10 AMC 10s, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problems<br />
# For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #11-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc), traditional middle/high school word problems with extremely complex problem solving<br />
# Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, easier #1-5 on AIMEs, etc.)<br />
# Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.<br />
# More difficult AIME problems (#10-12), simple proof-based problems (JBMO), etc<br />
# High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1,4s).<br />
# Tougher Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.<br />
# High-level Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.<br />
# Expert Olympiad-level questions, eg #3,6s on difficult Olympiad contests.<br />
# Super Expert problems, problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).<br />
<br />
= Competitions =<br />
<br />
==Introductory Competitions==<br />
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntroductory+mathematics+competitions here].<br />
<br />
=== [[MOEMS]] ===<br />
*Division E: '''1'''<br />
*: ''The whole number <math>N</math> is divisible by <math>7</math>. <math>N</math> leaves a remainder of <math>1</math> when divided by <math>2,3,4,</math> or <math>5</math>. What is the smallest value that <math>N</math> can be?'' ([http://www.moems.org/sample_files/SampleE.pdf Solution])<br />
*Division M: '''1'''<br />
*: ''The value of a two-digit number is <math>10</math> times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.'' ([http://www.moems.org/sample_files/SampleM.pdf Solution])<br />
<br />
=== [[AMC 8]] ===<br />
<br />
* Problem 1 - Problem 12: '''1''' <br />
*: ''The <math>\emph{harmonic mean}</math> of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?'' ([[2018 AMC 8 Problems/Problem 10|Solution]])<br />
* Problem 13 - Problem 25: '''1.5-2'''<br />
*: ''How many positive factors does <math>23,232</math> have?'' ([[2018 AMC 8 Problems/Problem 18|Solution]])<br />
<br />
=== [[Mathcounts]] ===<br />
<br />
* Countdown: '''1-2.'''<br />
* Sprint: '''1-1.5''' (school/chapter), '''1.5-2''' (State), '''2-2.5''' (National)<br />
* Target: '''1-2''' (school/chapter), '''1.5-2.5''' (State), '''2.5-3.5''' (National)<br />
<br />
=== [[AMC 10]] ===<br />
<br />
* Problem 1 - 10: '''1-2'''<br />
*: ''A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box?'' ([[2016 AMC 10A Problems/Problem 5|Solution]])<br />
* Problem 11 - 20: '''2-3'''<br />
*: ''For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?'' ([[2019 AMC 10A Problems/Problem 18|Solution]])<br />
* Problem 21 - 25: '''3.5-4.5'''<br />
*: ''The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?'' ([[2017 AMC 10B Problems/Problem 24|Solution]])<br />
<br />
===[[CEMC|CEMC Multiple Choice Tests]]===<br />
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.<br />
<br />
* Part A: '''1-1.5'''<br />
*: ''How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number?'' (2015 Gauss 7 Problem 10)<br />
* Part B: '''1-2'''<br />
*: ''Two lines with slopes <math>\tfrac14</math> and <math>\tfrac54</math> intersect at <math>(1,1)</math>. What is the area of the triangle formed by these two lines and the vertical line <math>x = 5</math>?'' (2017 Cayley Problem 19)<br />
* Part C (Gauss/Pascal): '''2-2.5'''<br />
*: ''Suppose that <math>\tfrac{2009}{2014} + \tfrac{2019}{n} = \tfrac{a}{b}</math>, where <math>a</math>, <math>b</math>, and <math>n</math> are positive integers with <math>\tfrac{a}{b}</math> in lowest terms. What is the sum of the digits of the smallest positive integer <math>n</math> for which <math>a</math> is a multiple of 1004?'' (2014 Pascal Problem 25)<br />
* Part C (Cayley/Fermat): '''2.5-3'''<br />
*: ''Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets?'' (2018 Fermat Problem 24)<br />
<br />
===[[CEMC|CEMC Fryer/Galois/Hypatia]]===<br />
<br />
* Problem 1-2: '''1-2'''<br />
* Problem 3-4 (early parts): '''2-3'''<br />
* Problem 3-4 (later parts): '''3-5'''<br />
<br />
===Problem Solving Books for Introductory Students===<br />
<br />
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.<br />
<br />
===[[Prealgebra by AoPS]]===<br />
1-2<br />
===[[Introduction to Algebra by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Counting and Probability by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Number Theory by AoPS]]===<br />
1-3<br />
===[[Introduction to Geometry by AoPS]]===<br />
1-4<br />
<br />
==Intermediate Competitions==<br />
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntermediate+mathematics+competitions here].<br />
<br />
=== [[AMC 12]] ===<br />
<br />
* Problem 1-10: '''1.5-2'''<br />
*: ''What is the value of <cmath>\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?</cmath>'' ([[2018 AMC 12B Problems/Problem 7|Solution]])<br />
* Problem 11-20: '''2.5-3.5'''<br />
*: ''An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?'' ([[2006 AMC 12B Problems/Problem 18|Solution]])<br />
* Problem 21-25: '''4.5-6'''<br />
*: ''Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?'' ([[2009 AMC 12A Problems/Problem 23|Solution]])<br />
<br />
=== [[AIME]] ===<br />
<br />
* Problem 1 - 5: '''3-3.5'''<br />
*: ''Consider the integer <cmath>N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>.'' ([[2019 AIME I Problems/Problem 1|Solution]])<br />
* Problem 6 - 9: '''4-4.5''' <br />
*: ''How many positive integers <math>N</math> less than <math>1000</math> are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>?'' ([[2009 AIME I Problems/Problem 6|Solution]])<br />
* Problem 10 - 12: '''5-5.5'''<br />
*: Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by <math>1000</math>.Let <math>S</math> be the sum of all elements in <math>R</math>. Find the remainder when <math>S</math> is divided by <math>1000</math> ([[2011 AIME I Problems/Problem 11|Solution]])<br />
* Problem 13 - 15: '''6-6.5'''<br />
*: ''Let <cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath> Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let <cmath>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</cmath> where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>.'' ([[2003 AIME II Problems/Problem 15|Solution]])<br />
<br />
=== [[ARML]] ===<br />
<br />
* Individuals, Problem 1: '''2'''<br />
<br />
* Individuals, Problems 2, 3, 4, 5, 7, and 9: '''3'''<br />
<br />
* Individuals, Problems 6 and 8: '''4''' <br />
<br />
* Individuals, Problem 10: '''5.5'''<br />
<br />
* Team/power, Problem 1-5: '''3.5''' <br />
<br />
* Team/power, Problem 6-10: '''5'''<br />
<br />
===[[HMMT|HMMT (November)]]===<br />
* Individual Round, Problem 6-8: '''4'''<br />
* Individual Round, Problem 10: '''4.5'''<br />
* Team Round: '''4-5'''<br />
* Guts: '''3.5-5.25'''<br />
<br />
===[[CEMC|CEMC Euclid]]===<br />
<br />
* Problem 1-6: '''1-3'''<br />
* Problem 7-10: '''3-5'''<br />
<br />
===[[Purple Comet! Math Meet|Purple Comet]]===<br />
<br />
* Problems 1-10 (MS): '''1.5-3'''<br />
* Problems 11-20 (MS): '''3-4.5'''<br />
* Problems 1-10 (HS): '''2-3.5'''<br />
* Problems 11-20 (HS): '''3.5'''<br />
* Problems 21-30 (HS): '''4.5-6'''<br />
<br />
=== [[Philippine Mathematical Olympiad Qualifying Round]] ===<br />
<br />
* Problem 1-15: '''2'''<br />
* Problem 16-25: '''3'''<br />
* Problem 26-30: '''4'''<br />
<br />
===[[Lexington Math Tournament|LMT]]===<br />
<br />
* Easy Problems: '''1-2'''<br />
*: ''Let trapezoid <math>ABCD</math> be such that <math>AB||CD</math>. Additionally, <math>AC = AD = 5</math>, <math>CD = 6</math>, and <math>AB = 3</math>. Find <math>BC</math>. ''<br />
* Medium Problems: '''2-4'''<br />
*: ''Let <math>\triangle LMN</math> have side lengths <math>LM = 15</math>, <math>MN = 14</math>, and <math>NL = 13</math>. Let the angle bisector of <math>\angle MLN</math> meet the circumcircle of <math>\triangle LMN</math> at a point <math>T \ne L</math>. Determine the area of <math>\triangle LMT</math>. ''<br />
* Hard Problems: '''5-7'''<br />
*: ''A magic <math>3 \times 5</math> board can toggle its cells between black and white. Define a pattern to be an assignment of black or white to each of the board’s <math>15</math> cells (so there are <math>2^{15}</math> patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than <math>3</math> cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day 1, compute the maximum number of days it can stay alive.''<br />
<br />
==Problem Solving Books for Intermediate Students==<br />
<br />
Remark: As stated above, there are many books for Intermediate students that have not been published by AoPS. Below is a list of intermediate books that AoPS has published and their difficulty. The left-hand number corresponds to the difficulty of the easiest review problems, while the right-hand number corresponds to the difficulty of the hardest challenge problems.<br />
<br />
===[[Intermediate Algebra by AoPS]]===<br />
'''2.5-6.5/7''', may vary across chapters<br />
<br />
===[[Intermediate Counting & Probability by AoPS]]===<br />
'''3.5-7.5/8''', may vary across chapters<br />
<br />
===[[Precalculus by AoPS]]===<br />
'''2-8''', may vary across chapters<br />
<br />
==Beginner Olympiad Competitions==<br />
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3ABeginner+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMTS]] ===<br />
USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:<br />
* Problem 1-2: '''3-4'''<br />
*: ''Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter.'' ([http://usamts.org/Solutions/Solution2_3_16.pdf Solution])<br />
* Problem 3-5: '''4-6'''<br />
*: ''Call a positive real number groovy if it can be written in the form <math>\sqrt{n} + \sqrt{n + 1}</math> for some positive integer <math>n</math>. Show that if <math>x</math> is groovy, then for any positive integer <math>r</math>, the number <math>x^r</math> is groovy as well.'' ([http://usamts.org/Solutions/Solutions_20_1.pdf Solution])<br />
<br />
=== [[Indonesia Mathematical Olympiad|Indonesia MO]] ===<br />
* Problem 1/5: '''3.5'''<br />
*: '' In a drawer, there are at most <math>2009</math> balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is <math>\frac12</math>. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?'' ([https://artofproblemsolving.com/community/c6h294065 Solution])<br />
* Problem 2/6: '''4.5'''<br />
*: ''Find the lowest possible values from the function <cmath>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</cmath> for any real numbers <math>x</math>.'' ([https://artofproblemsolving.com/community/c6h294067 Solution])<br />
* Problem 3/7: '''5'''<br />
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if <cmath>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</cmath> Given 2 positive integers <math>a,b > 1</math> which are relatively prime, prove that there exists a ''good'' pair <math>(m,n)</math> with <math>a\mid m</math> and <math>b\mid n</math>, but <math>a\nmid n</math> and <math>b\nmid m</math>.'' ([https://artofproblemsolving.com/community/c6h294068 Solution])<br />
* Problem 4/8: '''6'''<br />
*: ''Given an acute triangle <math>ABC</math>. The incircle of triangle <math>ABC</math> touches <math>BC,CA,AB</math> respectively at <math>D,E,F</math>. The angle bisector of <math>\angle A</math> cuts <math>DE</math> and <math>DF</math> respectively at <math>K</math> and <math>L</math>. Suppose <math>AA_1</math> is one of the altitudes of triangle <math>ABC</math>, and <math>M</math> be the midpoint of <math>BC</math>.''<br />
<br />
::''(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.''<br />
<br />
::''(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' ([https://artofproblemsolving.com/community/c6h294069 Solution])<br />
<br />
=== [[Central American Olympiad]] ===<br />
* Problem 1: '''4'''<br />
*: ''Find all three-digit numbers <math>abc</math> (with <math>a \neq 0</math>) such that <math>a^{2} + b^{2} + c^{2}</math> is a divisor of 26.'' (<url>viewtopic.php?p=903856#903856 Solution</url>)<br />
* Problem 2,4,5: '''5-6'''<br />
*: ''Show that the equation <math>a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005</math> has no integer solutions.'' (<url>viewtopic.php?p=291301#291301 Solution</url>)<br />
* Problem 3/6: '''6.5''' <br />
*: ''Let <math>ABCD</math> be a convex quadrilateral. <math>I = AC\cap BD</math>, and <math>E</math>, <math>H</math>, <math>F</math> and <math>G</math> are points on <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively, such that <math>EF \cap GH = I</math>. If <math>M = EG \cap AC</math>, <math>N = HF \cap AC</math>, show that <math>\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}</math>.'' (<url>viewtopic.php?p=828841#p828841 Solution</url><br />
<br />
=== [[JBMO]] ===<br />
<br />
* Problem 1: '''4'''<br />
*: ''Find all real numbers <math>a,b,c,d</math> such that <br />
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''<br />
* Problem 2: '''4.5-5'''<br />
*: ''Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAC=\angle BDC=36^\circ</math>, <math>\angle CBD=18^\circ</math> and <math>\angle BAC=72^\circ</math>. The diagonals intersect at point <math>P</math>. Determine the measure of <math>\angle APD</math>.''<br />
* Problem 3: '''5'''<br />
*: ''Find all prime numbers <math>p,q,r</math>, such that <math>\frac pq-\frac4{r+1}=1</math>.''<br />
* Problem 4: '''6'''<br />
*: ''A <math>4\times4</math> table is divided into <math>16</math> white unit square cells. Two cells are called neighbors if they share a common side. A '''move''' consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly <math>n</math> moves all the <math>16</math> cells were black. Find all possible values of <math>n</math>.''<br />
<br />
==Olympiad Competitions==<br />
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AOlympiad+mathematics+competitions here].<br />
<br />
=== [[USAJMO]] ===<br />
* Problem 1/4: '''5'''<br />
*: ''There are <math>a+b</math> bowls arranged in a row, numbered <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive integers. Initially, each of the first <math>a</math> bowls contains an apple, and each of the last <math>b</math> bowls contains a pear.''<br />
<br />
''A legal move consists of moving an apple from bowl <math>i</math> to bowl <math>i+1</math> and a pear from bowl <math>j</math> to bowl <math>j-1</math>, provided that the difference <math>i-j</math> is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first <math>b</math> bowls each containing a pear and the last <math>a</math> bowls each containing an apple. Show that this is possible if and only if the product <math>ab</math> is even.'' ([[2019 USAJMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''6-6.5'''<br />
*: ''Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>'' ([[2018 USAJMO Problems/Problem 2|Solution]])<br />
<br />
* Problem 3/6: '''7'''<br />
*: ''Two rational numbers <math>\tfrac{m}{n}</math> and <math>\tfrac{n}{m}</math> are written on a blackboard, where <math>m</math> and <math>n</math> are relatively prime positive integers. At any point, Evan may pick two of the numbers <math>x</math> and <math>y</math> written on the board and write either their arithmetic mean <math>\tfrac{x+y}{2}</math> or their harmonic mean <math>\tfrac{2xy}{x+y}</math> on the board as well. Find all pairs <math>(m,n)</math> such that Evan can write <math>1</math> on the board in finitely many steps.'' ([[2019 USAJMO Problems/Problem 6|Solution]])<br />
<br />
===[[HMMT|HMMT (February)]]===<br />
* Individual Round, Problem 1-5: '''5'''<br />
* Individual Round, Problem 6-10: '''5.5-6'''<br />
* Team Round: '''7.5'''<br />
* HMIC: '''8'''<br />
<br />
=== [[Canadian MO]] ===<br />
<br />
* Problem 1: '''5.5'''<br />
* Problem 2: '''6'''<br />
* Problem 3: '''6.5''' <br />
* Problem 4: '''7-7.5'''<br />
* Problem 5: '''7.5-8'''<br />
<br />
=== Austrian MO ===<br />
<br />
* Regional Competition for Advanced Students, Problems 1-4: '''5''' <br />
* Federal Competition for Advanced Students, Part 1. Problems 1-4: '''6''' <br />
* Federal Competition for Advanced Students, Part 2, Problems 1-6: '''7'''<br />
<br />
=== [[Iberoamerican Math Olympiad]] ===<br />
<br />
* Problem 1/4: '''5.5'''<br />
* Problem 2/5: '''6.5'''<br />
* Problem 3/6: '''7.5'''<br />
<br />
=== [[APMO]] ===<br />
*Problem 1: '''6'''<br />
*Problem 2: '''7'''<br />
*Problem 3: '''7'''<br />
*Problem 4: '''7.5'''<br />
*Problem 5: '''8.5'''<br />
<br />
=== Balkan MO ===<br />
<br />
* Problem 1: '''6'''<br />
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. '' <br />
* Problem 2: '''6.5'''<br />
*: '' Let <math>MN</math> be a line parallel to the side <math>BC</math> of a triangle <math>ABC</math>, with <math>M</math> on the side <math>AB</math> and <math>N</math> on the side <math>AC</math>. The lines <math>BN</math> and <math>CM</math> meet at point <math>P</math>. The circumcircles of triangles <math>BMP</math> and <math>CNP</math> meet at two distinct points <math>P</math> and <math>Q</math>. Prove that <math>\angle BAQ = \angle CAP</math>. ''<br />
* Problem 3: '''7.5'''<br />
*: '' A <math>9 \times 12</math> rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres <math>C_1,C_2...,C_{96}</math> in such way that the following to conditions are both fulfilled<br />
<br />
<math>(i)</math> the distances <math>C_1C_2,...C_{95}C_{96}, C_{96}C_{1}</math> are all equal to <math>\sqrt {13}</math><br />
<br />
<math>(ii)</math> the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry? ''<br />
* Problem 4: '''8'''<br />
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that<br />
<br />
<math>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2</math> for all <math>m,n \in S</math>. '<br />
<br />
==Hard Olympiad Competitions==<br />
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AHard+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMO]] ===<br />
* Problem 1/4: '''6-7'''<br />
*: ''Let <math>\mathcal{P}</math> be a convex polygon with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n - 3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.'' ([[2008 USAMO Problems/Problem 4|Solution]]) <br />
* Problem 2/5: '''7-8'''<br />
*: ''Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x \le y</math>, then erase <math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.'' ([[2008 USAMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''8-9'''<br />
*: ''Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n </math> real roots.'' ([[2002 USAMO Problems/Problem 3|Solution]])<br />
<br />
=== [[USA TST]] ===<br />
<br />
<br />
<br />
* Problem 1/4/7: '''6.5-7'''<br />
* Problem 2/5/8: '''7.5-8'''<br />
* Problem 3/6/9: '''8.5-9'''<br />
<br />
=== [[Putnam]] ===<br />
<br />
* Problem A/B,1-2: '''7'''<br />
*: ''Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola <math>xyz = 1</math> and both branches of the hyperbola <math>xwy = - 1.</math> (A set <math>S</math> in the plane is called ''convex'' if for any two points in <math>S</math> the line segment connecting them is contained in <math>S.</math>)'' ([https://artofproblemsolving.com/community/c7h177227p978383 Solution])<br />
* Problem A/B,3-4: '''8'''<br />
*: ''Let <math>H</math> be an <math>n\times n</math> matrix all of whose entries are <math>\pm1</math> and whose rows are mutually orthogonal. Suppose <math>H</math> has an <math>a\times b</math> submatrix whose entries are all <math>1.</math> Show that <math>ab\le n</math>.'' ([https://artofproblemsolving.com/community/c7h64435p383280 Solution])<br />
* Problem A/B,5-6: '''9'''<br />
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution])<br />
<br />
=== [[China TST]] ===<br />
<br />
* Problem 1/4: '''8-8.5''' <br />
*: ''Given an integer <math>m,</math> prove that there exist odd integers <math>a,b</math> and a positive integer <math>k</math> such that <cmath>2m=a^{19}+b^{99}+k*2^{1000}.</cmath>''<br />
* Problem 2/5: '''9''' <br />
*: ''Given a positive integer <math>n=1</math> and real numbers <math>a_1 < a_2 < \ldots < a_n,</math> such that <math>\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,</math> prove that for any positive real number <math>x,</math> <cmath>\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.</cmath>''<br />
* Problem 3/6: '''9.5-10'''<br />
*: ''Let <math>n>1</math> be an integer and let <math>a_0,a_1,\ldots,a_n</math> be non-negative real numbers. Define <math>S_k=\sum_{i=0}^k \binom{k}{i}a_i</math> for <math>k=0,1,\ldots,n</math>. Prove that<cmath>\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.</cmath>''<br />
<br />
=== [[IMO]] ===<br />
<br />
* Problem 1/4: '''5.5-7'''<br />
*: ''Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.'' ([[2018 IMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''7-8'''<br />
*: ''Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.'' ([[2006 IMO Problems/Problem 5|Solution]])<br />
<br />
* Problem 3/6: '''9-10'''<br />
*: ''Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.'' ([https://artofproblemsolving.com/community/c6h101488p572824 Solution])<br />
<br />
=== [[IMO Shortlist]] ===<br />
<br />
* Problem 1-2: '''5.5-7'''<br />
* Problem 3-4: '''7-8'''<br />
* Problem 5+: '''8-10'''<br />
<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Competition_ratings&diff=147982AoPS Wiki:Competition ratings2021-02-26T14:51:50Z<p>Myh2910: /* Indonesia MO */</p>
<hr />
<div>This page contains an approximate estimation of the difficulty level of various [[List of mathematics competitions|competitions]]. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.<br />
<br />
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution. <br />
<br />
As you may have guessed with time many competitions got more challenging because many countries got more access to books targeted at olympiad preparation. But especially web site where one can discuss Olympiads such as our very own AoPS!<br />
<br />
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC problems] and 10 is hardest level, e.g. [http://www.mathlinks.ro/resources.php?c=37&cid=47 China IMO Team Selection Test.] When considering problem difficulty '''put more emphasis on problem-solving aspects and less so on technical skill requirements'''.<br />
<br />
= Scale =<br />
All levels are estimated and refer to ''averages''. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this. <br />
# Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, #1-10 AMC 10s, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problems<br />
# For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #11-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc), traditional middle/high school word problems with extremely complex problem solving<br />
# Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, easier #1-5 on AIMEs, etc.)<br />
# Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.<br />
# More difficult AIME problems (#10-12), simple proof-based problems (JBMO), etc<br />
# High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1,4s).<br />
# Tougher Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.<br />
# High-level Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.<br />
# Expert Olympiad-level questions, eg #3,6s on difficult Olympiad contests.<br />
# Super Expert problems, problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).<br />
<br />
= Competitions =<br />
<br />
==Introductory Competitions==<br />
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntroductory+mathematics+competitions here].<br />
<br />
=== [[MOEMS]] ===<br />
*Division E: '''1'''<br />
*: ''The whole number <math>N</math> is divisible by <math>7</math>. <math>N</math> leaves a remainder of <math>1</math> when divided by <math>2,3,4,</math> or <math>5</math>. What is the smallest value that <math>N</math> can be?'' ([http://www.moems.org/sample_files/SampleE.pdf Solution])<br />
*Division M: '''1'''<br />
*: ''The value of a two-digit number is <math>10</math> times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.'' ([http://www.moems.org/sample_files/SampleM.pdf Solution])<br />
<br />
=== [[AMC 8]] ===<br />
<br />
* Problem 1 - Problem 12: '''1''' <br />
*: ''The <math>\emph{harmonic mean}</math> of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?'' ([[2018 AMC 8 Problems/Problem 10|Solution]])<br />
* Problem 13 - Problem 25: '''1.5-2'''<br />
*: ''How many positive factors does <math>23,232</math> have?'' ([[2018 AMC 8 Problems/Problem 18|Solution]])<br />
<br />
=== [[Mathcounts]] ===<br />
<br />
* Countdown: '''1-2.'''<br />
* Sprint: '''1-1.5''' (school/chapter), '''1.5-2''' (State), '''2-2.5''' (National)<br />
* Target: '''1-2''' (school/chapter), '''1.5-2.5''' (State), '''2.5-3.5''' (National)<br />
<br />
=== [[AMC 10]] ===<br />
<br />
* Problem 1 - 10: '''1-2'''<br />
*: ''A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box?'' ([[2016 AMC 10A Problems/Problem 5|Solution]])<br />
* Problem 11 - 20: '''2-3'''<br />
*: ''For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?'' ([[2019 AMC 10A Problems/Problem 18|Solution]])<br />
* Problem 21 - 25: '''3.5-4.5'''<br />
*: ''The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?'' ([[2017 AMC 10B Problems/Problem 24|Solution]])<br />
<br />
===[[CEMC|CEMC Multiple Choice Tests]]===<br />
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.<br />
<br />
* Part A: '''1-1.5'''<br />
*: ''How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number?'' (2015 Gauss 7 Problem 10)<br />
* Part B: '''1-2'''<br />
*: ''Two lines with slopes <math>\tfrac14</math> and <math>\tfrac54</math> intersect at <math>(1,1)</math>. What is the area of the triangle formed by these two lines and the vertical line <math>x = 5</math>?'' (2017 Cayley Problem 19)<br />
* Part C (Gauss/Pascal): '''2-2.5'''<br />
*: ''Suppose that <math>\tfrac{2009}{2014} + \tfrac{2019}{n} = \tfrac{a}{b}</math>, where <math>a</math>, <math>b</math>, and <math>n</math> are positive integers with <math>\tfrac{a}{b}</math> in lowest terms. What is the sum of the digits of the smallest positive integer <math>n</math> for which <math>a</math> is a multiple of 1004?'' (2014 Pascal Problem 25)<br />
* Part C (Cayley/Fermat): '''2.5-3'''<br />
*: ''Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets?'' (2018 Fermat Problem 24)<br />
<br />
===[[CEMC|CEMC Fryer/Galois/Hypatia]]===<br />
<br />
* Problem 1-2: '''1-2'''<br />
* Problem 3-4 (early parts): '''2-3'''<br />
* Problem 3-4 (later parts): '''3-5'''<br />
<br />
===Problem Solving Books for Introductory Students===<br />
<br />
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.<br />
<br />
===[[Prealgebra by AoPS]]===<br />
1-2<br />
===[[Introduction to Algebra by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Counting and Probability by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Number Theory by AoPS]]===<br />
1-3<br />
===[[Introduction to Geometry by AoPS]]===<br />
1-4<br />
<br />
==Intermediate Competitions==<br />
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntermediate+mathematics+competitions here].<br />
<br />
=== [[AMC 12]] ===<br />
<br />
* Problem 1-10: '''1.5-2'''<br />
*: ''What is the value of <cmath>\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?</cmath>'' ([[2018 AMC 12B Problems/Problem 7|Solution]])<br />
* Problem 11-20: '''2.5-3.5'''<br />
*: ''An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?'' ([[2006 AMC 12B Problems/Problem 18|Solution]])<br />
* Problem 21-25: '''4.5-6'''<br />
*: ''Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?'' ([[2009 AMC 12A Problems/Problem 23|Solution]])<br />
<br />
=== [[AIME]] ===<br />
<br />
* Problem 1 - 5: '''3-3.5'''<br />
*: ''Consider the integer <cmath>N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>.'' ([[2019 AIME I Problems/Problem 1|Solution]])<br />
* Problem 6 - 9: '''4-4.5''' <br />
*: ''How many positive integers <math>N</math> less than <math>1000</math> are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>?'' ([[2009 AIME I Problems/Problem 6|Solution]])<br />
* Problem 10 - 12: '''5-5.5'''<br />
*: Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by <math>1000</math>.Let <math>S</math> be the sum of all elements in <math>R</math>. Find the remainder when <math>S</math> is divided by <math>1000</math> ([[2011 AIME I Problems/Problem 11|Solution]])<br />
* Problem 13 - 15: '''6-6.5'''<br />
*: ''Let<br />
<br />
<cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath><br />
Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let<br />
<br />
<cmath>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</cmath><br />
where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>.'' ([[2003 AIME II Problems/Problem 15|Solution]])<br />
<br />
=== [[ARML]] ===<br />
<br />
* Individuals, Problem 1: '''2'''<br />
<br />
* Individuals, Problems 2, 3, 4, 5, 7, and 9: '''3'''<br />
<br />
* Individuals, Problems 6 and 8: '''4''' <br />
<br />
* Individuals, Problem 10: '''5.5'''<br />
<br />
* Team/power, Problem 1-5: '''3.5''' <br />
<br />
* Team/power, Problem 6-10: '''5'''<br />
<br />
===[[HMMT|HMMT (November)]]===<br />
* Individual Round, Problem 6-8: '''4'''<br />
* Individual Round, Problem 10: '''4.5'''<br />
* Team Round: '''4-5'''<br />
* Guts: '''3.5-5.25'''<br />
<br />
===[[CEMC|CEMC Euclid]]===<br />
<br />
* Problem 1-6: '''1-3'''<br />
* Problem 7-10: '''3-5'''<br />
<br />
===[[Purple Comet! Math Meet|Purple Comet]]===<br />
<br />
* Problems 1-10 (MS): '''1.5-3'''<br />
* Problems 11-20 (MS): '''3-4.5'''<br />
* Problems 1-10 (HS): '''2-3.5'''<br />
* Problems 11-20 (HS): '''3.5'''<br />
* Problems 21-30 (HS): '''4.5-6'''<br />
<br />
=== [[Philippine Mathematical Olympiad Qualifying Round]] ===<br />
<br />
* Problem 1-15: '''2'''<br />
* Problem 16-25: '''3'''<br />
* Problem 26-30: '''4'''<br />
<br />
===[[Lexington Math Tournament|LMT]]===<br />
<br />
* Easy Problems: '''1-2'''<br />
*: ''Let trapezoid <math>ABCD</math> be such that <math>AB||CD</math>. Additionally, <math>AC = AD = 5</math>, <math>CD = 6</math>, and <math>AB = 3</math>. Find <math>BC</math>. ''<br />
* Medium Problems: '''2-4'''<br />
*: ''Let <math>\triangle LMN</math> have side lengths <math>LM = 15</math>, <math>MN = 14</math>, and <math>NL = 13</math>. Let the angle bisector of <math>\angle MLN</math> meet the circumcircle of <math>\triangle LMN</math> at a point <math>T \ne L</math>. Determine the area of <math>\triangle LMT</math>. ''<br />
* Hard Problems: '''5-7'''<br />
*: ''A magic <math>3 \times 5</math> board can toggle its cells between black and white. Define a pattern to be an assignment of black or white to each of the board’s <math>15</math> cells (so there are <math>2^{15}</math> patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than <math>3</math> cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day 1, compute the maximum number of days it can stay alive.''<br />
<br />
==Problem Solving Books for Intermediate Students==<br />
<br />
Remark: As stated above, there are many books for Intermediate students that have not been published by AoPS. Below is a list of intermediate books that AoPS has published and their difficulty. The left-hand number corresponds to the difficulty of the easiest review problems, while the right-hand number corresponds to the difficulty of the hardest challenge problems.<br />
<br />
===[[Intermediate Algebra by AoPS]]===<br />
'''2.5-6.5/7''', may vary across chapters<br />
<br />
===[[Intermediate Counting & Probability by AoPS]]===<br />
'''3.5-7.5/8''', may vary across chapters<br />
<br />
===[[Precalculus by AoPS]]===<br />
'''2-8''', may vary across chapters<br />
<br />
==Beginner Olympiad Competitions==<br />
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3ABeginner+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMTS]] ===<br />
USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:<br />
* Problem 1-2: '''3-4'''<br />
*: ''Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter.'' ([http://usamts.org/Solutions/Solution2_3_16.pdf Solution])<br />
* Problem 3-5: '''4-6'''<br />
*: ''Call a positive real number groovy if it can be written in the form <math>\sqrt{n} + \sqrt{n + 1}</math> for some positive integer <math>n</math>. Show that if <math>x</math> is groovy, then for any positive integer <math>r</math>, the number <math>x^r</math> is groovy as well.'' ([http://usamts.org/Solutions/Solutions_20_1.pdf Solution])<br />
<br />
=== [[Indonesia Mathematical Olympiad|Indonesia MO]] ===<br />
* Problem 1/5: '''3.5'''<br />
*: '' In a drawer, there are at most <math>2009</math> balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is <math>\frac12</math>. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?'' ([https://artofproblemsolving.com/community/c6h294065 Solution])<br />
* Problem 2/6: '''4.5'''<br />
*: ''Find the lowest possible values from the function <cmath>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</cmath> for any real numbers <math>x</math>.'' ([https://artofproblemsolving.com/community/c6h294067 Solution])<br />
* Problem 3/7: '''5'''<br />
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if <cmath>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</cmath> Given 2 positive integers <math>a,b > 1</math> which are relatively prime, prove that there exists a ''good'' pair <math>(m,n)</math> with <math>a\mid m</math> and <math>b\mid n</math>, but <math>a\nmid n</math> and <math>b\nmid m</math>.'' ([https://artofproblemsolving.com/community/c6h294068 Solution])<br />
* Problem 4/8: '''6'''<br />
*: ''Given an acute triangle <math>ABC</math>. The incircle of triangle <math>ABC</math> touches <math>BC,CA,AB</math> respectively at <math>D,E,F</math>. The angle bisector of <math>\angle A</math> cuts <math>DE</math> and <math>DF</math> respectively at <math>K</math> and <math>L</math>. Suppose <math>AA_1</math> is one of the altitudes of triangle <math>ABC</math>, and <math>M</math> be the midpoint of <math>BC</math>.''<br />
<br />
::''(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.''<br />
<br />
::''(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' ([https://artofproblemsolving.com/community/c6h294069 Solution])<br />
<br />
=== [[Central American Olympiad]] ===<br />
* Problem 1: '''4'''<br />
*: ''Find all three-digit numbers <math>abc</math> (with <math>a \neq 0</math>) such that <math>a^{2} + b^{2} + c^{2}</math> is a divisor of 26.'' (<url>viewtopic.php?p=903856#903856 Solution</url>)<br />
* Problem 2,4,5: '''5-6'''<br />
*: ''Show that the equation <math>a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005</math> has no integer solutions.'' (<url>viewtopic.php?p=291301#291301 Solution</url>)<br />
* Problem 3/6: '''6.5''' <br />
*: ''Let <math>ABCD</math> be a convex quadrilateral. <math>I = AC\cap BD</math>, and <math>E</math>, <math>H</math>, <math>F</math> and <math>G</math> are points on <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively, such that <math>EF \cap GH = I</math>. If <math>M = EG \cap AC</math>, <math>N = HF \cap AC</math>, show that <math>\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}</math>.'' (<url>viewtopic.php?p=828841#p828841 Solution</url><br />
<br />
=== [[JBMO]] ===<br />
<br />
* Problem 1: '''4'''<br />
*: ''Find all real numbers <math>a,b,c,d</math> such that <br />
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''<br />
* Problem 2: '''4.5-5'''<br />
*: ''Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAC=\angle BDC=36^\circ</math>, <math>\angle CBD=18^\circ</math> and <math>\angle BAC=72^\circ</math>. The diagonals intersect at point <math>P</math>. Determine the measure of <math>\angle APD</math>.''<br />
* Problem 3: '''5'''<br />
*: ''Find all prime numbers <math>p,q,r</math>, such that <math>\frac pq-\frac4{r+1}=1</math>.''<br />
* Problem 4: '''6'''<br />
*: ''A <math>4\times4</math> table is divided into <math>16</math> white unit square cells. Two cells are called neighbors if they share a common side. A '''move''' consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly <math>n</math> moves all the <math>16</math> cells were black. Find all possible values of <math>n</math>.''<br />
<br />
==Olympiad Competitions==<br />
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AOlympiad+mathematics+competitions here].<br />
<br />
=== [[USAJMO]] ===<br />
* Problem 1/4: '''5'''<br />
*: ''There are <math>a+b</math> bowls arranged in a row, numbered <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive integers. Initially, each of the first <math>a</math> bowls contains an apple, and each of the last <math>b</math> bowls contains a pear.''<br />
<br />
''A legal move consists of moving an apple from bowl <math>i</math> to bowl <math>i+1</math> and a pear from bowl <math>j</math> to bowl <math>j-1</math>, provided that the difference <math>i-j</math> is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first <math>b</math> bowls each containing a pear and the last <math>a</math> bowls each containing an apple. Show that this is possible if and only if the product <math>ab</math> is even.'' ([[2019 USAJMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''6-6.5'''<br />
*: ''Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>'' ([[2018 USAJMO Problems/Problem 2|Solution]])<br />
<br />
* Problem 3/6: '''7'''<br />
*: ''Two rational numbers <math>\tfrac{m}{n}</math> and <math>\tfrac{n}{m}</math> are written on a blackboard, where <math>m</math> and <math>n</math> are relatively prime positive integers. At any point, Evan may pick two of the numbers <math>x</math> and <math>y</math> written on the board and write either their arithmetic mean <math>\tfrac{x+y}{2}</math> or their harmonic mean <math>\tfrac{2xy}{x+y}</math> on the board as well. Find all pairs <math>(m,n)</math> such that Evan can write <math>1</math> on the board in finitely many steps.'' ([[2019 USAJMO Problems/Problem 6|Solution]])<br />
<br />
===[[HMMT|HMMT (February)]]===<br />
* Individual Round, Problem 1-5: '''5'''<br />
* Individual Round, Problem 6-10: '''5.5-6'''<br />
* Team Round: '''7.5'''<br />
* HMIC: '''8'''<br />
<br />
=== [[Canadian MO]] ===<br />
<br />
* Problem 1: '''5.5'''<br />
* Problem 2: '''6'''<br />
* Problem 3: '''6.5''' <br />
* Problem 4: '''7-7.5'''<br />
* Problem 5: '''7.5-8'''<br />
<br />
=== Austrian MO ===<br />
<br />
* Regional Competition for Advanced Students, Problems 1-4: '''5''' <br />
* Federal Competition for Advanced Students, Part 1. Problems 1-4: '''6''' <br />
* Federal Competition for Advanced Students, Part 2, Problems 1-6: '''7'''<br />
<br />
=== [[Iberoamerican Math Olympiad]] ===<br />
<br />
* Problem 1/4: '''5.5'''<br />
* Problem 2/5: '''6.5'''<br />
* Problem 3/6: '''7.5'''<br />
<br />
=== [[APMO]] ===<br />
*Problem 1: '''6'''<br />
*Problem 2: '''7'''<br />
*Problem 3: '''7'''<br />
*Problem 4: '''7.5'''<br />
*Problem 5: '''8.5'''<br />
<br />
=== Balkan MO ===<br />
<br />
* Problem 1: '''6'''<br />
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. '' <br />
* Problem 2: '''6.5'''<br />
*: '' Let <math>MN</math> be a line parallel to the side <math>BC</math> of a triangle <math>ABC</math>, with <math>M</math> on the side <math>AB</math> and <math>N</math> on the side <math>AC</math>. The lines <math>BN</math> and <math>CM</math> meet at point <math>P</math>. The circumcircles of triangles <math>BMP</math> and <math>CNP</math> meet at two distinct points <math>P</math> and <math>Q</math>. Prove that <math>\angle BAQ = \angle CAP</math>. ''<br />
* Problem 3: '''7.5'''<br />
*: '' A <math>9 \times 12</math> rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres <math>C_1,C_2...,C_{96}</math> in such way that the following to conditions are both fulfilled<br />
<br />
<math>(i)</math> the distances <math>C_1C_2,...C_{95}C_{96}, C_{96}C_{1}</math> are all equal to <math>\sqrt {13}</math><br />
<br />
<math>(ii)</math> the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry? ''<br />
* Problem 4: '''8'''<br />
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that<br />
<br />
<math>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2</math> for all <math>m,n \in S</math>. '<br />
<br />
==Hard Olympiad Competitions==<br />
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AHard+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMO]] ===<br />
* Problem 1/4: '''6-7'''<br />
*: ''Let <math>\mathcal{P}</math> be a convex polygon with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n - 3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.'' ([[2008 USAMO Problems/Problem 4|Solution]]) <br />
* Problem 2/5: '''7-8'''<br />
*: ''Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x \le y</math>, then erase <math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.'' ([[2008 USAMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''8-9'''<br />
*: ''Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n </math> real roots.'' ([[2002 USAMO Problems/Problem 3|Solution]])<br />
<br />
=== [[USA TST]] ===<br />
<br />
<br />
<br />
* Problem 1/4/7: '''6.5-7'''<br />
* Problem 2/5/8: '''7.5-8'''<br />
* Problem 3/6/9: '''8.5-9'''<br />
<br />
=== [[Putnam]] ===<br />
<br />
* Problem A/B,1-2: '''7'''<br />
*: ''Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola <math>xyz = 1</math> and both branches of the hyperbola <math>xwy = - 1.</math> (A set <math>S</math> in the plane is called ''convex'' if for any two points in <math>S</math> the line segment connecting them is contained in <math>S.</math>)'' ([https://artofproblemsolving.com/community/c7h177227p978383 Solution])<br />
* Problem A/B,3-4: '''8'''<br />
*: ''Let <math>H</math> be an <math>n\times n</math> matrix all of whose entries are <math>\pm1</math> and whose rows are mutually orthogonal. Suppose <math>H</math> has an <math>a\times b</math> submatrix whose entries are all <math>1.</math> Show that <math>ab\le n</math>.'' ([https://artofproblemsolving.com/community/c7h64435p383280 Solution])<br />
* Problem A/B,5-6: '''9'''<br />
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution])<br />
<br />
=== [[China TST]] ===<br />
<br />
* Problem 1/4: '''8-8.5''' <br />
*: ''Given an integer <math>m,</math> prove that there exist odd integers <math>a,b</math> and a positive integer <math>k</math> such that <cmath>2m=a^{19}+b^{99}+k*2^{1000}.</cmath>''<br />
* Problem 2/5: '''9''' <br />
*: ''Given a positive integer <math>n=1</math> and real numbers <math>a_1 < a_2 < \ldots < a_n,</math> such that <math>\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,</math> prove that for any positive real number <math>x,</math> <cmath>\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.</cmath>''<br />
* Problem 3/6: '''9.5-10'''<br />
*: ''Let <math>n>1</math> be an integer and let <math>a_0,a_1,\ldots,a_n</math> be non-negative real numbers. Define <math>S_k=\sum_{i=0}^k \binom{k}{i}a_i</math> for <math>k=0,1,\ldots,n</math>. Prove that<cmath>\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.</cmath>''<br />
<br />
=== [[IMO]] ===<br />
<br />
* Problem 1/4: '''5.5-7'''<br />
*: ''Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.'' ([[2018 IMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''7-8'''<br />
*: ''Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.'' ([[2006 IMO Problems/Problem 5|Solution]])<br />
<br />
* Problem 3/6: '''9-10'''<br />
*: ''Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.'' ([https://artofproblemsolving.com/community/c6h101488p572824 Solution])<br />
<br />
=== [[IMO Shortlist]] ===<br />
<br />
* Problem 1-2: '''5.5-7'''<br />
* Problem 3-4: '''7-8'''<br />
* Problem 5+: '''8-10'''<br />
<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Competition_ratings&diff=147981AoPS Wiki:Competition ratings2021-02-26T14:41:03Z<p>Myh2910: /* IMO */ Fixed link</p>
<hr />
<div>This page contains an approximate estimation of the difficulty level of various [[List of mathematics competitions|competitions]]. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.<br />
<br />
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution. <br />
<br />
As you may have guessed with time many competitions got more challenging because many countries got more access to books targeted at olympiad preparation. But especially web site where one can discuss Olympiads such as our very own AoPS!<br />
<br />
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC problems] and 10 is hardest level, e.g. [http://www.mathlinks.ro/resources.php?c=37&cid=47 China IMO Team Selection Test.] When considering problem difficulty '''put more emphasis on problem-solving aspects and less so on technical skill requirements'''.<br />
<br />
= Scale =<br />
All levels are estimated and refer to ''averages''. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this. <br />
# Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, #1-10 AMC 10s, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problems<br />
# For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #11-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc), traditional middle/high school word problems with extremely complex problem solving<br />
# Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, easier #1-5 on AIMEs, etc.)<br />
# Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.<br />
# More difficult AIME problems (#10-12), simple proof-based problems (JBMO), etc<br />
# High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1,4s).<br />
# Tougher Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.<br />
# High-level Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.<br />
# Expert Olympiad-level questions, eg #3,6s on difficult Olympiad contests.<br />
# Super Expert problems, problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).<br />
<br />
= Competitions =<br />
<br />
==Introductory Competitions==<br />
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntroductory+mathematics+competitions here].<br />
<br />
=== [[MOEMS]] ===<br />
*Division E: '''1'''<br />
*: ''The whole number <math>N</math> is divisible by <math>7</math>. <math>N</math> leaves a remainder of <math>1</math> when divided by <math>2,3,4,</math> or <math>5</math>. What is the smallest value that <math>N</math> can be?'' ([http://www.moems.org/sample_files/SampleE.pdf Solution])<br />
*Division M: '''1'''<br />
*: ''The value of a two-digit number is <math>10</math> times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.'' ([http://www.moems.org/sample_files/SampleM.pdf Solution])<br />
<br />
=== [[AMC 8]] ===<br />
<br />
* Problem 1 - Problem 12: '''1''' <br />
*: ''The <math>\emph{harmonic mean}</math> of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?'' ([[2018 AMC 8 Problems/Problem 10|Solution]])<br />
* Problem 13 - Problem 25: '''1.5-2'''<br />
*: ''How many positive factors does <math>23,232</math> have?'' ([[2018 AMC 8 Problems/Problem 18|Solution]])<br />
<br />
=== [[Mathcounts]] ===<br />
<br />
* Countdown: '''1-2.'''<br />
* Sprint: '''1-1.5''' (school/chapter), '''1.5-2''' (State), '''2-2.5''' (National)<br />
* Target: '''1-2''' (school/chapter), '''1.5-2.5''' (State), '''2.5-3.5''' (National)<br />
<br />
=== [[AMC 10]] ===<br />
<br />
* Problem 1 - 10: '''1-2'''<br />
*: ''A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box?'' ([[2016 AMC 10A Problems/Problem 5|Solution]])<br />
* Problem 11 - 20: '''2-3'''<br />
*: ''For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?'' ([[2019 AMC 10A Problems/Problem 18|Solution]])<br />
* Problem 21 - 25: '''3.5-4.5'''<br />
*: ''The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?'' ([[2017 AMC 10B Problems/Problem 24|Solution]])<br />
<br />
===[[CEMC|CEMC Multiple Choice Tests]]===<br />
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.<br />
<br />
* Part A: '''1-1.5'''<br />
*: ''How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number?'' (2015 Gauss 7 Problem 10)<br />
* Part B: '''1-2'''<br />
*: ''Two lines with slopes <math>\tfrac14</math> and <math>\tfrac54</math> intersect at <math>(1,1)</math>. What is the area of the triangle formed by these two lines and the vertical line <math>x = 5</math>?'' (2017 Cayley Problem 19)<br />
* Part C (Gauss/Pascal): '''2-2.5'''<br />
*: ''Suppose that <math>\tfrac{2009}{2014} + \tfrac{2019}{n} = \tfrac{a}{b}</math>, where <math>a</math>, <math>b</math>, and <math>n</math> are positive integers with <math>\tfrac{a}{b}</math> in lowest terms. What is the sum of the digits of the smallest positive integer <math>n</math> for which <math>a</math> is a multiple of 1004?'' (2014 Pascal Problem 25)<br />
* Part C (Cayley/Fermat): '''2.5-3'''<br />
*: ''Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets?'' (2018 Fermat Problem 24)<br />
<br />
===[[CEMC|CEMC Fryer/Galois/Hypatia]]===<br />
<br />
* Problem 1-2: '''1-2'''<br />
* Problem 3-4 (early parts): '''2-3'''<br />
* Problem 3-4 (later parts): '''3-5'''<br />
<br />
===Problem Solving Books for Introductory Students===<br />
<br />
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.<br />
<br />
===[[Prealgebra by AoPS]]===<br />
1-2<br />
===[[Introduction to Algebra by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Counting and Probability by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Number Theory by AoPS]]===<br />
1-3<br />
===[[Introduction to Geometry by AoPS]]===<br />
1-4<br />
<br />
==Intermediate Competitions==<br />
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntermediate+mathematics+competitions here].<br />
<br />
=== [[AMC 12]] ===<br />
<br />
* Problem 1-10: '''1.5-2'''<br />
*: ''What is the value of <cmath>\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?</cmath>'' ([[2018 AMC 12B Problems/Problem 7|Solution]])<br />
* Problem 11-20: '''2.5-3.5'''<br />
*: ''An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?'' ([[2006 AMC 12B Problems/Problem 18|Solution]])<br />
* Problem 21-25: '''4.5-6'''<br />
*: ''Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?'' ([[2009 AMC 12A Problems/Problem 23|Solution]])<br />
<br />
=== [[AIME]] ===<br />
<br />
* Problem 1 - 5: '''3-3.5'''<br />
*: ''Consider the integer <cmath>N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>.'' ([[2019 AIME I Problems/Problem 1|Solution]])<br />
* Problem 6 - 9: '''4-4.5''' <br />
*: ''How many positive integers <math>N</math> less than <math>1000</math> are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>?'' ([[2009 AIME I Problems/Problem 6|Solution]])<br />
* Problem 10 - 12: '''5-5.5'''<br />
*: Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by <math>1000</math>.Let <math>S</math> be the sum of all elements in <math>R</math>. Find the remainder when <math>S</math> is divided by <math>1000</math> ([[2011 AIME I Problems/Problem 11|Solution]])<br />
* Problem 13 - 15: '''6-6.5'''<br />
*: ''Let<br />
<br />
<cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath><br />
Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let<br />
<br />
<cmath>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</cmath><br />
where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>.'' ([[2003 AIME II Problems/Problem 15|Solution]])<br />
<br />
=== [[ARML]] ===<br />
<br />
* Individuals, Problem 1: '''2'''<br />
<br />
* Individuals, Problems 2, 3, 4, 5, 7, and 9: '''3'''<br />
<br />
* Individuals, Problems 6 and 8: '''4''' <br />
<br />
* Individuals, Problem 10: '''5.5'''<br />
<br />
* Team/power, Problem 1-5: '''3.5''' <br />
<br />
* Team/power, Problem 6-10: '''5'''<br />
<br />
===[[HMMT|HMMT (November)]]===<br />
* Individual Round, Problem 6-8: '''4'''<br />
* Individual Round, Problem 10: '''4.5'''<br />
* Team Round: '''4-5'''<br />
* Guts: '''3.5-5.25'''<br />
<br />
===[[CEMC|CEMC Euclid]]===<br />
<br />
* Problem 1-6: '''1-3'''<br />
* Problem 7-10: '''3-5'''<br />
<br />
===[[Purple Comet! Math Meet|Purple Comet]]===<br />
<br />
* Problems 1-10 (MS): '''1.5-3'''<br />
* Problems 11-20 (MS): '''3-4.5'''<br />
* Problems 1-10 (HS): '''2-3.5'''<br />
* Problems 11-20 (HS): '''3.5'''<br />
* Problems 21-30 (HS): '''4.5-6'''<br />
<br />
=== [[Philippine Mathematical Olympiad Qualifying Round]] ===<br />
<br />
* Problem 1-15: '''2'''<br />
* Problem 16-25: '''3'''<br />
* Problem 26-30: '''4'''<br />
<br />
===[[Lexington Math Tournament|LMT]]===<br />
<br />
* Easy Problems: '''1-2'''<br />
*: ''Let trapezoid <math>ABCD</math> be such that <math>AB||CD</math>. Additionally, <math>AC = AD = 5</math>, <math>CD = 6</math>, and <math>AB = 3</math>. Find <math>BC</math>. ''<br />
* Medium Problems: '''2-4'''<br />
*: ''Let <math>\triangle LMN</math> have side lengths <math>LM = 15</math>, <math>MN = 14</math>, and <math>NL = 13</math>. Let the angle bisector of <math>\angle MLN</math> meet the circumcircle of <math>\triangle LMN</math> at a point <math>T \ne L</math>. Determine the area of <math>\triangle LMT</math>. ''<br />
* Hard Problems: '''5-7'''<br />
*: ''A magic <math>3 \times 5</math> board can toggle its cells between black and white. Define a pattern to be an assignment of black or white to each of the board’s <math>15</math> cells (so there are <math>2^{15}</math> patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than <math>3</math> cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day 1, compute the maximum number of days it can stay alive.''<br />
<br />
==Problem Solving Books for Intermediate Students==<br />
<br />
Remark: As stated above, there are many books for Intermediate students that have not been published by AoPS. Below is a list of intermediate books that AoPS has published and their difficulty. The left-hand number corresponds to the difficulty of the easiest review problems, while the right-hand number corresponds to the difficulty of the hardest challenge problems.<br />
<br />
===[[Intermediate Algebra by AoPS]]===<br />
'''2.5-6.5/7''', may vary across chapters<br />
<br />
===[[Intermediate Counting & Probability by AoPS]]===<br />
'''3.5-7.5/8''', may vary across chapters<br />
<br />
===[[Precalculus by AoPS]]===<br />
'''2-8''', may vary across chapters<br />
<br />
==Beginner Olympiad Competitions==<br />
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3ABeginner+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMTS]] ===<br />
USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:<br />
* Problem 1-2: '''3-4'''<br />
*: ''Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter.'' ([http://usamts.org/Solutions/Solution2_3_16.pdf Solution])<br />
* Problem 3-5: '''4-6'''<br />
*: ''Call a positive real number groovy if it can be written in the form <math>\sqrt{n} + \sqrt{n + 1}</math> for some positive integer <math>n</math>. Show that if <math>x</math> is groovy, then for any positive integer <math>r</math>, the number <math>x^r</math> is groovy as well.'' ([http://usamts.org/Solutions/Solutions_20_1.pdf Solution])<br />
<br />
=== [[Indonesia Mathematical Olympiad|Indonesia MO]] ===<br />
* Problem 1/5: '''3.5'''<br />
*: '' In a drawer, there are at most <math>2009</math> balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is <math>\frac12</math>. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?'' <url>viewtopic.php?t=294065 (Solution)</url><br />
* Problem 2/6: '''4.5'''<br />
*: ''Find the lowest possible values from the function<br />
<math>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</math><br />
<br />
for any real numbers <math>x</math>.''<url>viewtopic.php?t=294067 (Solution)</url><br />
* Problem 3/7: '''5'''<br />
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if<br />
<math>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</math><br />
<br />
Given 2 positive integers <math>a,b > 1</math> which are relatively prime, prove that there exists a ''good'' pair <math>(m,n)</math> with <math>a\mid m</math> and <math>b\mid n</math>, but <math>a\nmid n</math> and <math>b\nmid m</math>.'' <url>viewtopic.php?t=294068 (Solution)</url><br />
* Problem 4/8: '''6'''<br />
*: ''Given an acute triangle <math>ABC</math>. The incircle of triangle <math>ABC</math> touches <math>BC,CA,AB</math> respectively at <math>D,E,F</math>. The angle bisector of <math>\angle A</math> cuts <math>DE</math> and <math>DF</math> respectively at <math>K</math> and <math>L</math>. Suppose <math>AA_1</math> is one of the altitudes of triangle <math>ABC</math>, and <math>M</math> be the midpoint of <math>BC</math>.<br />
<br />
(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.<br />
<br />
(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' <url>viewtopic.php?t=294069 (Solution)</url><br />
<br />
=== [[Central American Olympiad]] ===<br />
* Problem 1: '''4'''<br />
*: ''Find all three-digit numbers <math>abc</math> (with <math>a \neq 0</math>) such that <math>a^{2} + b^{2} + c^{2}</math> is a divisor of 26.'' (<url>viewtopic.php?p=903856#903856 Solution</url>)<br />
* Problem 2,4,5: '''5-6'''<br />
*: ''Show that the equation <math>a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005</math> has no integer solutions.'' (<url>viewtopic.php?p=291301#291301 Solution</url>)<br />
* Problem 3/6: '''6.5''' <br />
*: ''Let <math>ABCD</math> be a convex quadrilateral. <math>I = AC\cap BD</math>, and <math>E</math>, <math>H</math>, <math>F</math> and <math>G</math> are points on <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively, such that <math>EF \cap GH = I</math>. If <math>M = EG \cap AC</math>, <math>N = HF \cap AC</math>, show that <math>\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}</math>.'' (<url>viewtopic.php?p=828841#p828841 Solution</url><br />
<br />
=== [[JBMO]] ===<br />
<br />
* Problem 1: '''4'''<br />
*: ''Find all real numbers <math>a,b,c,d</math> such that <br />
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''<br />
* Problem 2: '''4.5-5'''<br />
*: ''Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAC=\angle BDC=36^\circ</math>, <math>\angle CBD=18^\circ</math> and <math>\angle BAC=72^\circ</math>. The diagonals intersect at point <math>P</math>. Determine the measure of <math>\angle APD</math>.''<br />
* Problem 3: '''5'''<br />
*: ''Find all prime numbers <math>p,q,r</math>, such that <math>\frac pq-\frac4{r+1}=1</math>.''<br />
* Problem 4: '''6'''<br />
*: ''A <math>4\times4</math> table is divided into <math>16</math> white unit square cells. Two cells are called neighbors if they share a common side. A '''move''' consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly <math>n</math> moves all the <math>16</math> cells were black. Find all possible values of <math>n</math>.''<br />
<br />
==Olympiad Competitions==<br />
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AOlympiad+mathematics+competitions here].<br />
<br />
=== [[USAJMO]] ===<br />
* Problem 1/4: '''5'''<br />
*: ''There are <math>a+b</math> bowls arranged in a row, numbered <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive integers. Initially, each of the first <math>a</math> bowls contains an apple, and each of the last <math>b</math> bowls contains a pear.''<br />
<br />
''A legal move consists of moving an apple from bowl <math>i</math> to bowl <math>i+1</math> and a pear from bowl <math>j</math> to bowl <math>j-1</math>, provided that the difference <math>i-j</math> is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first <math>b</math> bowls each containing a pear and the last <math>a</math> bowls each containing an apple. Show that this is possible if and only if the product <math>ab</math> is even.'' ([[2019 USAJMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''6-6.5'''<br />
*: ''Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>'' ([[2018 USAJMO Problems/Problem 2|Solution]])<br />
<br />
* Problem 3/6: '''7'''<br />
*: ''Two rational numbers <math>\tfrac{m}{n}</math> and <math>\tfrac{n}{m}</math> are written on a blackboard, where <math>m</math> and <math>n</math> are relatively prime positive integers. At any point, Evan may pick two of the numbers <math>x</math> and <math>y</math> written on the board and write either their arithmetic mean <math>\tfrac{x+y}{2}</math> or their harmonic mean <math>\tfrac{2xy}{x+y}</math> on the board as well. Find all pairs <math>(m,n)</math> such that Evan can write <math>1</math> on the board in finitely many steps.'' ([[2019 USAJMO Problems/Problem 6|Solution]])<br />
<br />
===[[HMMT|HMMT (February)]]===<br />
* Individual Round, Problem 1-5: '''5'''<br />
* Individual Round, Problem 6-10: '''5.5-6'''<br />
* Team Round: '''7.5'''<br />
* HMIC: '''8'''<br />
<br />
=== [[Canadian MO]] ===<br />
<br />
* Problem 1: '''5.5'''<br />
* Problem 2: '''6'''<br />
* Problem 3: '''6.5''' <br />
* Problem 4: '''7-7.5'''<br />
* Problem 5: '''7.5-8'''<br />
<br />
=== Austrian MO ===<br />
<br />
* Regional Competition for Advanced Students, Problems 1-4: '''5''' <br />
* Federal Competition for Advanced Students, Part 1. Problems 1-4: '''6''' <br />
* Federal Competition for Advanced Students, Part 2, Problems 1-6: '''7'''<br />
<br />
=== [[Iberoamerican Math Olympiad]] ===<br />
<br />
* Problem 1/4: '''5.5'''<br />
* Problem 2/5: '''6.5'''<br />
* Problem 3/6: '''7.5'''<br />
<br />
=== [[APMO]] ===<br />
*Problem 1: '''6'''<br />
*Problem 2: '''7'''<br />
*Problem 3: '''7'''<br />
*Problem 4: '''7.5'''<br />
*Problem 5: '''8.5'''<br />
<br />
=== Balkan MO ===<br />
<br />
* Problem 1: '''6'''<br />
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. '' <br />
* Problem 2: '''6.5'''<br />
*: '' Let <math>MN</math> be a line parallel to the side <math>BC</math> of a triangle <math>ABC</math>, with <math>M</math> on the side <math>AB</math> and <math>N</math> on the side <math>AC</math>. The lines <math>BN</math> and <math>CM</math> meet at point <math>P</math>. The circumcircles of triangles <math>BMP</math> and <math>CNP</math> meet at two distinct points <math>P</math> and <math>Q</math>. Prove that <math>\angle BAQ = \angle CAP</math>. ''<br />
* Problem 3: '''7.5'''<br />
*: '' A <math>9 \times 12</math> rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres <math>C_1,C_2...,C_{96}</math> in such way that the following to conditions are both fulfilled<br />
<br />
<math>(i)</math> the distances <math>C_1C_2,...C_{95}C_{96}, C_{96}C_{1}</math> are all equal to <math>\sqrt {13}</math><br />
<br />
<math>(ii)</math> the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry? ''<br />
* Problem 4: '''8'''<br />
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that<br />
<br />
<math>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2</math> for all <math>m,n \in S</math>. '<br />
<br />
==Hard Olympiad Competitions==<br />
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AHard+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMO]] ===<br />
* Problem 1/4: '''6-7'''<br />
*: ''Let <math>\mathcal{P}</math> be a convex polygon with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n - 3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.'' ([[2008 USAMO Problems/Problem 4|Solution]]) <br />
* Problem 2/5: '''7-8'''<br />
*: ''Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x \le y</math>, then erase <math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.'' ([[2008 USAMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''8-9'''<br />
*: ''Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n </math> real roots.'' ([[2002 USAMO Problems/Problem 3|Solution]])<br />
<br />
=== [[USA TST]] ===<br />
<br />
<br />
<br />
* Problem 1/4/7: '''6.5-7'''<br />
* Problem 2/5/8: '''7.5-8'''<br />
* Problem 3/6/9: '''8.5-9'''<br />
<br />
=== [[Putnam]] ===<br />
<br />
* Problem A/B,1-2: '''7'''<br />
*: ''Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola <math>xyz = 1</math> and both branches of the hyperbola <math>xwy = - 1.</math> (A set <math>S</math> in the plane is called ''convex'' if for any two points in <math>S</math> the line segment connecting them is contained in <math>S.</math>)'' ([https://artofproblemsolving.com/community/c7h177227p978383 Solution])<br />
* Problem A/B,3-4: '''8'''<br />
*: ''Let <math>H</math> be an <math>n\times n</math> matrix all of whose entries are <math>\pm1</math> and whose rows are mutually orthogonal. Suppose <math>H</math> has an <math>a\times b</math> submatrix whose entries are all <math>1.</math> Show that <math>ab\le n</math>.'' ([https://artofproblemsolving.com/community/c7h64435p383280 Solution])<br />
* Problem A/B,5-6: '''9'''<br />
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution])<br />
<br />
=== [[China TST]] ===<br />
<br />
* Problem 1/4: '''8-8.5''' <br />
*: ''Given an integer <math>m,</math> prove that there exist odd integers <math>a,b</math> and a positive integer <math>k</math> such that <cmath>2m=a^{19}+b^{99}+k*2^{1000}.</cmath>''<br />
* Problem 2/5: '''9''' <br />
*: ''Given a positive integer <math>n=1</math> and real numbers <math>a_1 < a_2 < \ldots < a_n,</math> such that <math>\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,</math> prove that for any positive real number <math>x,</math> <cmath>\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.</cmath>''<br />
* Problem 3/6: '''9.5-10'''<br />
*: ''Let <math>n>1</math> be an integer and let <math>a_0,a_1,\ldots,a_n</math> be non-negative real numbers. Define <math>S_k=\sum_{i=0}^k \binom{k}{i}a_i</math> for <math>k=0,1,\ldots,n</math>. Prove that<cmath>\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.</cmath>''<br />
<br />
=== [[IMO]] ===<br />
<br />
* Problem 1/4: '''5.5-7'''<br />
*: ''Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.'' ([[2018 IMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''7-8'''<br />
*: ''Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.'' ([[2006 IMO Problems/Problem 5|Solution]])<br />
<br />
* Problem 3/6: '''9-10'''<br />
*: ''Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.'' ([https://artofproblemsolving.com/community/c6h101488p572824 Solution])<br />
<br />
=== [[IMO Shortlist]] ===<br />
<br />
* Problem 1-2: '''5.5-7'''<br />
* Problem 3-4: '''7-8'''<br />
* Problem 5+: '''8-10'''<br />
<br />
[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Competition_ratings&diff=147980AoPS Wiki:Competition ratings2021-02-26T14:38:37Z<p>Myh2910: /* Putnam */ Fixed link</p>
<hr />
<div>This page contains an approximate estimation of the difficulty level of various [[List of mathematics competitions|competitions]]. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.<br />
<br />
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution. <br />
<br />
As you may have guessed with time many competitions got more challenging because many countries got more access to books targeted at olympiad preparation. But especially web site where one can discuss Olympiads such as our very own AoPS!<br />
<br />
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC problems] and 10 is hardest level, e.g. [http://www.mathlinks.ro/resources.php?c=37&cid=47 China IMO Team Selection Test.] When considering problem difficulty '''put more emphasis on problem-solving aspects and less so on technical skill requirements'''.<br />
<br />
= Scale =<br />
All levels are estimated and refer to ''averages''. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this. <br />
# Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, #1-10 AMC 10s, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problems<br />
# For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #11-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc), traditional middle/high school word problems with extremely complex problem solving<br />
# Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, easier #1-5 on AIMEs, etc.)<br />
# Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.<br />
# More difficult AIME problems (#10-12), simple proof-based problems (JBMO), etc<br />
# High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1,4s).<br />
# Tougher Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.<br />
# High-level Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.<br />
# Expert Olympiad-level questions, eg #3,6s on difficult Olympiad contests.<br />
# Super Expert problems, problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).<br />
<br />
= Competitions =<br />
<br />
==Introductory Competitions==<br />
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntroductory+mathematics+competitions here].<br />
<br />
=== [[MOEMS]] ===<br />
*Division E: '''1'''<br />
*: ''The whole number <math>N</math> is divisible by <math>7</math>. <math>N</math> leaves a remainder of <math>1</math> when divided by <math>2,3,4,</math> or <math>5</math>. What is the smallest value that <math>N</math> can be?'' ([http://www.moems.org/sample_files/SampleE.pdf Solution])<br />
*Division M: '''1'''<br />
*: ''The value of a two-digit number is <math>10</math> times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.'' ([http://www.moems.org/sample_files/SampleM.pdf Solution])<br />
<br />
=== [[AMC 8]] ===<br />
<br />
* Problem 1 - Problem 12: '''1''' <br />
*: ''The <math>\emph{harmonic mean}</math> of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?'' ([[2018 AMC 8 Problems/Problem 10|Solution]])<br />
* Problem 13 - Problem 25: '''1.5-2'''<br />
*: ''How many positive factors does <math>23,232</math> have?'' ([[2018 AMC 8 Problems/Problem 18|Solution]])<br />
<br />
=== [[Mathcounts]] ===<br />
<br />
* Countdown: '''1-2.'''<br />
* Sprint: '''1-1.5''' (school/chapter), '''1.5-2''' (State), '''2-2.5''' (National)<br />
* Target: '''1-2''' (school/chapter), '''1.5-2.5''' (State), '''2.5-3.5''' (National)<br />
<br />
=== [[AMC 10]] ===<br />
<br />
* Problem 1 - 10: '''1-2'''<br />
*: ''A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box?'' ([[2016 AMC 10A Problems/Problem 5|Solution]])<br />
* Problem 11 - 20: '''2-3'''<br />
*: ''For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?'' ([[2019 AMC 10A Problems/Problem 18|Solution]])<br />
* Problem 21 - 25: '''3.5-4.5'''<br />
*: ''The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?'' ([[2017 AMC 10B Problems/Problem 24|Solution]])<br />
<br />
===[[CEMC|CEMC Multiple Choice Tests]]===<br />
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.<br />
<br />
* Part A: '''1-1.5'''<br />
*: ''How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number?'' (2015 Gauss 7 Problem 10)<br />
* Part B: '''1-2'''<br />
*: ''Two lines with slopes <math>\tfrac14</math> and <math>\tfrac54</math> intersect at <math>(1,1)</math>. What is the area of the triangle formed by these two lines and the vertical line <math>x = 5</math>?'' (2017 Cayley Problem 19)<br />
* Part C (Gauss/Pascal): '''2-2.5'''<br />
*: ''Suppose that <math>\tfrac{2009}{2014} + \tfrac{2019}{n} = \tfrac{a}{b}</math>, where <math>a</math>, <math>b</math>, and <math>n</math> are positive integers with <math>\tfrac{a}{b}</math> in lowest terms. What is the sum of the digits of the smallest positive integer <math>n</math> for which <math>a</math> is a multiple of 1004?'' (2014 Pascal Problem 25)<br />
* Part C (Cayley/Fermat): '''2.5-3'''<br />
*: ''Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets?'' (2018 Fermat Problem 24)<br />
<br />
===[[CEMC|CEMC Fryer/Galois/Hypatia]]===<br />
<br />
* Problem 1-2: '''1-2'''<br />
* Problem 3-4 (early parts): '''2-3'''<br />
* Problem 3-4 (later parts): '''3-5'''<br />
<br />
===Problem Solving Books for Introductory Students===<br />
<br />
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.<br />
<br />
===[[Prealgebra by AoPS]]===<br />
1-2<br />
===[[Introduction to Algebra by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Counting and Probability by AoPS]]===<br />
1-3.5<br />
===[[Introduction to Number Theory by AoPS]]===<br />
1-3<br />
===[[Introduction to Geometry by AoPS]]===<br />
1-4<br />
<br />
==Intermediate Competitions==<br />
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AIntermediate+mathematics+competitions here].<br />
<br />
=== [[AMC 12]] ===<br />
<br />
* Problem 1-10: '''1.5-2'''<br />
*: ''What is the value of <cmath>\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?</cmath>'' ([[2018 AMC 12B Problems/Problem 7|Solution]])<br />
* Problem 11-20: '''2.5-3.5'''<br />
*: ''An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?'' ([[2006 AMC 12B Problems/Problem 18|Solution]])<br />
* Problem 21-25: '''4.5-6'''<br />
*: ''Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?'' ([[2009 AMC 12A Problems/Problem 23|Solution]])<br />
<br />
=== [[AIME]] ===<br />
<br />
* Problem 1 - 5: '''3-3.5'''<br />
*: ''Consider the integer <cmath>N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>.'' ([[2019 AIME I Problems/Problem 1|Solution]])<br />
* Problem 6 - 9: '''4-4.5''' <br />
*: ''How many positive integers <math>N</math> less than <math>1000</math> are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>?'' ([[2009 AIME I Problems/Problem 6|Solution]])<br />
* Problem 10 - 12: '''5-5.5'''<br />
*: Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by <math>1000</math>.Let <math>S</math> be the sum of all elements in <math>R</math>. Find the remainder when <math>S</math> is divided by <math>1000</math> ([[2011 AIME I Problems/Problem 11|Solution]])<br />
* Problem 13 - 15: '''6-6.5'''<br />
*: ''Let<br />
<br />
<cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath><br />
Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let<br />
<br />
<cmath>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</cmath><br />
where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>.'' ([[2003 AIME II Problems/Problem 15|Solution]])<br />
<br />
=== [[ARML]] ===<br />
<br />
* Individuals, Problem 1: '''2'''<br />
<br />
* Individuals, Problems 2, 3, 4, 5, 7, and 9: '''3'''<br />
<br />
* Individuals, Problems 6 and 8: '''4''' <br />
<br />
* Individuals, Problem 10: '''5.5'''<br />
<br />
* Team/power, Problem 1-5: '''3.5''' <br />
<br />
* Team/power, Problem 6-10: '''5'''<br />
<br />
===[[HMMT|HMMT (November)]]===<br />
* Individual Round, Problem 6-8: '''4'''<br />
* Individual Round, Problem 10: '''4.5'''<br />
* Team Round: '''4-5'''<br />
* Guts: '''3.5-5.25'''<br />
<br />
===[[CEMC|CEMC Euclid]]===<br />
<br />
* Problem 1-6: '''1-3'''<br />
* Problem 7-10: '''3-5'''<br />
<br />
===[[Purple Comet! Math Meet|Purple Comet]]===<br />
<br />
* Problems 1-10 (MS): '''1.5-3'''<br />
* Problems 11-20 (MS): '''3-4.5'''<br />
* Problems 1-10 (HS): '''2-3.5'''<br />
* Problems 11-20 (HS): '''3.5'''<br />
* Problems 21-30 (HS): '''4.5-6'''<br />
<br />
=== [[Philippine Mathematical Olympiad Qualifying Round]] ===<br />
<br />
* Problem 1-15: '''2'''<br />
* Problem 16-25: '''3'''<br />
* Problem 26-30: '''4'''<br />
<br />
===[[Lexington Math Tournament|LMT]]===<br />
<br />
* Easy Problems: '''1-2'''<br />
*: ''Let trapezoid <math>ABCD</math> be such that <math>AB||CD</math>. Additionally, <math>AC = AD = 5</math>, <math>CD = 6</math>, and <math>AB = 3</math>. Find <math>BC</math>. ''<br />
* Medium Problems: '''2-4'''<br />
*: ''Let <math>\triangle LMN</math> have side lengths <math>LM = 15</math>, <math>MN = 14</math>, and <math>NL = 13</math>. Let the angle bisector of <math>\angle MLN</math> meet the circumcircle of <math>\triangle LMN</math> at a point <math>T \ne L</math>. Determine the area of <math>\triangle LMT</math>. ''<br />
* Hard Problems: '''5-7'''<br />
*: ''A magic <math>3 \times 5</math> board can toggle its cells between black and white. Define a pattern to be an assignment of black or white to each of the board’s <math>15</math> cells (so there are <math>2^{15}</math> patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than <math>3</math> cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day 1, compute the maximum number of days it can stay alive.''<br />
<br />
==Problem Solving Books for Intermediate Students==<br />
<br />
Remark: As stated above, there are many books for Intermediate students that have not been published by AoPS. Below is a list of intermediate books that AoPS has published and their difficulty. The left-hand number corresponds to the difficulty of the easiest review problems, while the right-hand number corresponds to the difficulty of the hardest challenge problems.<br />
<br />
===[[Intermediate Algebra by AoPS]]===<br />
'''2.5-6.5/7''', may vary across chapters<br />
<br />
===[[Intermediate Counting & Probability by AoPS]]===<br />
'''3.5-7.5/8''', may vary across chapters<br />
<br />
===[[Precalculus by AoPS]]===<br />
'''2-8''', may vary across chapters<br />
<br />
==Beginner Olympiad Competitions==<br />
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3ABeginner+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMTS]] ===<br />
USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:<br />
* Problem 1-2: '''3-4'''<br />
*: ''Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter.'' ([http://usamts.org/Solutions/Solution2_3_16.pdf Solution])<br />
* Problem 3-5: '''4-6'''<br />
*: ''Call a positive real number groovy if it can be written in the form <math>\sqrt{n} + \sqrt{n + 1}</math> for some positive integer <math>n</math>. Show that if <math>x</math> is groovy, then for any positive integer <math>r</math>, the number <math>x^r</math> is groovy as well.'' ([http://usamts.org/Solutions/Solutions_20_1.pdf Solution])<br />
<br />
=== [[Indonesia Mathematical Olympiad|Indonesia MO]] ===<br />
* Problem 1/5: '''3.5'''<br />
*: '' In a drawer, there are at most <math>2009</math> balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is <math>\frac12</math>. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?'' <url>viewtopic.php?t=294065 (Solution)</url><br />
* Problem 2/6: '''4.5'''<br />
*: ''Find the lowest possible values from the function<br />
<math>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</math><br />
<br />
for any real numbers <math>x</math>.''<url>viewtopic.php?t=294067 (Solution)</url><br />
* Problem 3/7: '''5'''<br />
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if<br />
<math>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</math><br />
<br />
Given 2 positive integers <math>a,b > 1</math> which are relatively prime, prove that there exists a ''good'' pair <math>(m,n)</math> with <math>a\mid m</math> and <math>b\mid n</math>, but <math>a\nmid n</math> and <math>b\nmid m</math>.'' <url>viewtopic.php?t=294068 (Solution)</url><br />
* Problem 4/8: '''6'''<br />
*: ''Given an acute triangle <math>ABC</math>. The incircle of triangle <math>ABC</math> touches <math>BC,CA,AB</math> respectively at <math>D,E,F</math>. The angle bisector of <math>\angle A</math> cuts <math>DE</math> and <math>DF</math> respectively at <math>K</math> and <math>L</math>. Suppose <math>AA_1</math> is one of the altitudes of triangle <math>ABC</math>, and <math>M</math> be the midpoint of <math>BC</math>.<br />
<br />
(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.<br />
<br />
(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' <url>viewtopic.php?t=294069 (Solution)</url><br />
<br />
=== [[Central American Olympiad]] ===<br />
* Problem 1: '''4'''<br />
*: ''Find all three-digit numbers <math>abc</math> (with <math>a \neq 0</math>) such that <math>a^{2} + b^{2} + c^{2}</math> is a divisor of 26.'' (<url>viewtopic.php?p=903856#903856 Solution</url>)<br />
* Problem 2,4,5: '''5-6'''<br />
*: ''Show that the equation <math>a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005</math> has no integer solutions.'' (<url>viewtopic.php?p=291301#291301 Solution</url>)<br />
* Problem 3/6: '''6.5''' <br />
*: ''Let <math>ABCD</math> be a convex quadrilateral. <math>I = AC\cap BD</math>, and <math>E</math>, <math>H</math>, <math>F</math> and <math>G</math> are points on <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively, such that <math>EF \cap GH = I</math>. If <math>M = EG \cap AC</math>, <math>N = HF \cap AC</math>, show that <math>\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}</math>.'' (<url>viewtopic.php?p=828841#p828841 Solution</url><br />
<br />
=== [[JBMO]] ===<br />
<br />
* Problem 1: '''4'''<br />
*: ''Find all real numbers <math>a,b,c,d</math> such that <br />
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''<br />
* Problem 2: '''4.5-5'''<br />
*: ''Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAC=\angle BDC=36^\circ</math>, <math>\angle CBD=18^\circ</math> and <math>\angle BAC=72^\circ</math>. The diagonals intersect at point <math>P</math>. Determine the measure of <math>\angle APD</math>.''<br />
* Problem 3: '''5'''<br />
*: ''Find all prime numbers <math>p,q,r</math>, such that <math>\frac pq-\frac4{r+1}=1</math>.''<br />
* Problem 4: '''6'''<br />
*: ''A <math>4\times4</math> table is divided into <math>16</math> white unit square cells. Two cells are called neighbors if they share a common side. A '''move''' consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly <math>n</math> moves all the <math>16</math> cells were black. Find all possible values of <math>n</math>.''<br />
<br />
==Olympiad Competitions==<br />
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AOlympiad+mathematics+competitions here].<br />
<br />
=== [[USAJMO]] ===<br />
* Problem 1/4: '''5'''<br />
*: ''There are <math>a+b</math> bowls arranged in a row, numbered <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive integers. Initially, each of the first <math>a</math> bowls contains an apple, and each of the last <math>b</math> bowls contains a pear.''<br />
<br />
''A legal move consists of moving an apple from bowl <math>i</math> to bowl <math>i+1</math> and a pear from bowl <math>j</math> to bowl <math>j-1</math>, provided that the difference <math>i-j</math> is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first <math>b</math> bowls each containing a pear and the last <math>a</math> bowls each containing an apple. Show that this is possible if and only if the product <math>ab</math> is even.'' ([[2019 USAJMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''6-6.5'''<br />
*: ''Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>'' ([[2018 USAJMO Problems/Problem 2|Solution]])<br />
<br />
* Problem 3/6: '''7'''<br />
*: ''Two rational numbers <math>\tfrac{m}{n}</math> and <math>\tfrac{n}{m}</math> are written on a blackboard, where <math>m</math> and <math>n</math> are relatively prime positive integers. At any point, Evan may pick two of the numbers <math>x</math> and <math>y</math> written on the board and write either their arithmetic mean <math>\tfrac{x+y}{2}</math> or their harmonic mean <math>\tfrac{2xy}{x+y}</math> on the board as well. Find all pairs <math>(m,n)</math> such that Evan can write <math>1</math> on the board in finitely many steps.'' ([[2019 USAJMO Problems/Problem 6|Solution]])<br />
<br />
===[[HMMT|HMMT (February)]]===<br />
* Individual Round, Problem 1-5: '''5'''<br />
* Individual Round, Problem 6-10: '''5.5-6'''<br />
* Team Round: '''7.5'''<br />
* HMIC: '''8'''<br />
<br />
=== [[Canadian MO]] ===<br />
<br />
* Problem 1: '''5.5'''<br />
* Problem 2: '''6'''<br />
* Problem 3: '''6.5''' <br />
* Problem 4: '''7-7.5'''<br />
* Problem 5: '''7.5-8'''<br />
<br />
=== Austrian MO ===<br />
<br />
* Regional Competition for Advanced Students, Problems 1-4: '''5''' <br />
* Federal Competition for Advanced Students, Part 1. Problems 1-4: '''6''' <br />
* Federal Competition for Advanced Students, Part 2, Problems 1-6: '''7'''<br />
<br />
=== [[Iberoamerican Math Olympiad]] ===<br />
<br />
* Problem 1/4: '''5.5'''<br />
* Problem 2/5: '''6.5'''<br />
* Problem 3/6: '''7.5'''<br />
<br />
=== [[APMO]] ===<br />
*Problem 1: '''6'''<br />
*Problem 2: '''7'''<br />
*Problem 3: '''7'''<br />
*Problem 4: '''7.5'''<br />
*Problem 5: '''8.5'''<br />
<br />
=== Balkan MO ===<br />
<br />
* Problem 1: '''6'''<br />
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. '' <br />
* Problem 2: '''6.5'''<br />
*: '' Let <math>MN</math> be a line parallel to the side <math>BC</math> of a triangle <math>ABC</math>, with <math>M</math> on the side <math>AB</math> and <math>N</math> on the side <math>AC</math>. The lines <math>BN</math> and <math>CM</math> meet at point <math>P</math>. The circumcircles of triangles <math>BMP</math> and <math>CNP</math> meet at two distinct points <math>P</math> and <math>Q</math>. Prove that <math>\angle BAQ = \angle CAP</math>. ''<br />
* Problem 3: '''7.5'''<br />
*: '' A <math>9 \times 12</math> rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres <math>C_1,C_2...,C_{96}</math> in such way that the following to conditions are both fulfilled<br />
<br />
<math>(i)</math> the distances <math>C_1C_2,...C_{95}C_{96}, C_{96}C_{1}</math> are all equal to <math>\sqrt {13}</math><br />
<br />
<math>(ii)</math> the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry? ''<br />
* Problem 4: '''8'''<br />
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that<br />
<br />
<math>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2</math> for all <math>m,n \in S</math>. '<br />
<br />
==Hard Olympiad Competitions==<br />
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available [https://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AHard+Olympiad+mathematics+competitions here].<br />
<br />
=== [[USAMO]] ===<br />
* Problem 1/4: '''6-7'''<br />
*: ''Let <math>\mathcal{P}</math> be a convex polygon with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n - 3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.'' ([[2008 USAMO Problems/Problem 4|Solution]]) <br />
* Problem 2/5: '''7-8'''<br />
*: ''Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x \le y</math>, then erase <math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.'' ([[2008 USAMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''8-9'''<br />
*: ''Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n </math> real roots.'' ([[2002 USAMO Problems/Problem 3|Solution]])<br />
<br />
=== [[USA TST]] ===<br />
<br />
<br />
<br />
* Problem 1/4/7: '''6.5-7'''<br />
* Problem 2/5/8: '''7.5-8'''<br />
* Problem 3/6/9: '''8.5-9'''<br />
<br />
=== [[Putnam]] ===<br />
<br />
* Problem A/B,1-2: '''7'''<br />
*: ''Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola <math>xyz = 1</math> and both branches of the hyperbola <math>xwy = - 1.</math> (A set <math>S</math> in the plane is called ''convex'' if for any two points in <math>S</math> the line segment connecting them is contained in <math>S.</math>)'' ([https://artofproblemsolving.com/community/c7h177227p978383 Solution])<br />
* Problem A/B,3-4: '''8'''<br />
*: ''Let <math>H</math> be an <math>n\times n</math> matrix all of whose entries are <math>\pm1</math> and whose rows are mutually orthogonal. Suppose <math>H</math> has an <math>a\times b</math> submatrix whose entries are all <math>1.</math> Show that <math>ab\le n</math>.'' ([https://artofproblemsolving.com/community/c7h64435p383280 Solution])<br />
* Problem A/B,5-6: '''9'''<br />
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution])<br />
<br />
=== [[China TST]] ===<br />
<br />
* Problem 1/4: '''8-8.5''' <br />
*: ''Given an integer <math>m,</math> prove that there exist odd integers <math>a,b</math> and a positive integer <math>k</math> such that <cmath>2m=a^{19}+b^{99}+k*2^{1000}.</cmath>''<br />
* Problem 2/5: '''9''' <br />
*: ''Given a positive integer <math>n=1</math> and real numbers <math>a_1 < a_2 < \ldots < a_n,</math> such that <math>\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,</math> prove that for any positive real number <math>x,</math> <cmath>\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.</cmath>''<br />
* Problem 3/6: '''9.5-10'''<br />
*: ''Let <math>n>1</math> be an integer and let <math>a_0,a_1,\ldots,a_n</math> be non-negative real numbers. Define <math>S_k=\sum_{i=0}^k \binom{k}{i}a_i</math> for <math>k=0,1,\ldots,n</math>. Prove that<cmath>\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.</cmath>''<br />
<br />
=== [[IMO]] ===<br />
<br />
* Problem 1/4: '''5.5-7'''<br />
*: ''Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.'' ([[2018 IMO Problems/Problem 1|Solution]])<br />
<br />
* Problem 2/5: '''7-8'''<br />
*: ''Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.'' ([[2006 IMO Problems/Problem 5|Solution]])<br />
<br />
* Problem 3/6: '''9-10'''<br />
*: ''Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.'' (<url>viewtopic.php?p=572824#572824 Solution</url>)<br />
<br />
=== [[IMO Shortlist]] ===<br />
<br />
* Problem 1-2: '''5.5-7'''<br />
* Problem 3-4: '''7-8'''<br />
* Problem 5+: '''8-10'''<br />
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[[Category:Mathematics competitions]]</div>Myh2910https://artofproblemsolving.com/wiki/index.php?title=User:Piphi&diff=134841User:Piphi2020-10-09T21:21:44Z<p>Myh2910: /* User Count */</p>
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==<font color="black" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">User Count</div></font>==<br />
<div style="margin-left: 10px; margin-bottom:10px"><font color="black">If this is your first time visiting this page, edit it by incrementing the user count below by one.</font></div><br />
<center><font size="100px">347</font></center><br />
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==<font color="black" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">About Me</div></font>==<br />
<div style="margin-left: 10px; margin-bottom:10px"><font color="black">Piphi is legendary and made the USA IMO team in 2019.<br><br />
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Piphi is the creator of the [[User:Piphi/Games|AoPS Wiki Games by Piphi]], the future of games on AoPS.<br><br />
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Piphi started the signature trend at around May 2020.<br><br />
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Piphi is an extremely OP person - LJCoder619. <br><br />
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Piphi is OP --[[User:Aray10|Aray10]] ([[User talk:Aray10|talk]]) 23:22, 17 June 2020 (EDT) <br><br />
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Piphi is 100 percent OP compared to the rest of us --[[User:Aops-g5-gethsemanea2|Aops-g5-gethsemanea2]] ([[User talk:Aops-g5-gethsemanea2|talk]]) 09:24, 4 September 2020 (EDT) <br><br />
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According to my studies Piphi is 50 percent asymptote, 50 percent AoPS Wiki, and 100 percent fun. --[[User:CreativeHedgehog|Creativehedgehog]] ([[User talk:CreativeHedgehog|talk]]) 16:14, 19 September 2020 (EDT) <br><br />
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Piphi has been very close to winning multiple [[Greed Control]] games, piphi placed 5th in game #18 and 2nd in game #19. Thanks to piphi, Greed Control games have started to be kept track of. Piphi made a spreadsheet that has all of Greed Control history [https://artofproblemsolving.com/community/c19451h2126208p15569802 here].<br><br />
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Piphi also found out who won [[Reaper]] games #1 and #2 as seen [https://artofproblemsolving.com/community/c19451h1826745p15526330 here].<br><br />
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Piphi has been called op by many AoPSers, including the legendary [[User:Radio2|Radio2]] himself [https://artofproblemsolving.com/community/c19451h1826745p15526800 here]. (note: Radio2 calls many users op.)<br><br />
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Piphi created the [[AoPS Administrators]] page, added most of the AoPS Admins to it, and created the scrollable table.<br><br />
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Piphi has also added a lot of the info that is in the [[Reaper Archives]].<br><br />
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Piphi has a side-project that is making the Wiki's [[Main Page]] look better, you can check that out [[User:Piphi/AoPS Wiki|here]].<br><br />
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Piphi published Greed Control Game 19 statistics [https://artofproblemsolving.com/community/c19451h2126212 here].<br />
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Piphi has a post that was made an announcement on a official AoPS Forum [https://artofproblemsolving.com/community/c68h2175116 here].<br />
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Piphi is a proud member of [https://artofproblemsolving.com/community/c562043 The Interuniversal GMAAS Society].<br />
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==<font color="black" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">Goals</div></font>==<br />
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You can check out more goals/statistics [[User:Piphi/Statistics|here]].<br />
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A User Count of 500<br />
{{User:Piphi/Template:Progress_Bar|68.6|width=100%}}<br />
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200 subpages of [[User:Piphi]]<br />
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200 signups for [[User:Piphi/Games|AoPS Wiki Games by Piphi]]<br />
{{User:Piphi/Template:Progress_Bar|46.5|width=100%}}<br />
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Make 10,000 edits<br />
{{User:Piphi/Template:Progress_Bar|19.72|width=100%}}</div><br />
</div></div>Myh2910