Difference between revisions of "AoPS Wiki:Competition ratings"

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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.  
 
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.  
  
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 olympiad as our very own ML/AoPS! Thus when judging the difficulty level consider the last 10-15 years with more priority.
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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'''.
  
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, i.e. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC] problems and 10 is hardest level, i.e. [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.{{ref|1}}
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= Scale =
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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.  
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# 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
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# 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
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# Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20 on AMC 12, easier #1-5 on AIMEs, etc.)
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# Intermediate-level problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.
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# More difficult AIME problems (#10-12), simple proof-based problems (JBMO)
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# High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1-4).
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# Tougher Olympiad-level questions, #1-4 that require more technical knowledge than new students to Olympiad-type questions    have, easier #2-5, etc.
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# High-level Olympiad-level questions, #2-5s on difficult Olympiad contest and easier #3,6s, etc.
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# Expert Olympiad-level questions, #3-6 on difficult Olympiad contests.
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# Super Expert problems, problems occasionally even unsuitable for very hard competitions (such as the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).
  
== Competitions ==
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For reference, here are problems from each of the difficulty levels 1-10:
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'''<1:''' Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? (2015 AMC 8, Problem 1)
  
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'''1:''' How many integer values of <math>x</math> satisfy <math>|x| < 3\pi</math>? (2021 AMC 10B, Problem 1)
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'''1.5:''' A number is called flippy if its digits alternate between two distinct digits. For example, <math>2020</math> and <math>37373</math> are flippy, but <math>3883</math> and <math>123123</math> are not. How many five-digit flippy numbers are divisible by <math>15?</math> (2020 AMC 8, Problem 19)
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'''2:''' For all positive integers <math>n</math>, let <math>f(n)=\log_{2002} n^2</math>. Let <math>N=f(11)+f(13)+f(14)</math>. Which of the following relations is true? (2002 AMC 12A, Problem 14)
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'''2.5:''' Triangle <math>ABC</math> with <math>AB=50</math> and <math>AC=10</math> has area <math>120</math>. Let <math>D</math> be the midpoint of <math>\overline{AB}</math>, and let <math>E</math> be the midpoint of <math>\overline{AC}</math>. The angle bisector of <math>\angle BAC</math> intersects <math>\overline{DE}</math> and <math>\overline{BC}</math> at <math>F</math> and <math>G</math>, respectively. What is the area of quadrilateral <math>FDBG</math>? (2018 AMC 10A, Problem 24)
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'''3:''' <math>A</math>, <math>B</math>, <math>C</math> are three piles of rocks. The mean weight of the rocks in <math>A</math> is <math>40</math> pounds, the mean weight of the rocks in <math>B</math> is <math>50</math> pounds, the mean weight of the rocks in the combined piles <math>A</math> and <math>B</math> is <math>43</math> pounds, and the mean weight of the rocks in the combined piles <math>A</math> and <math>C</math> is <math>44</math> pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles <math>B</math> and <math>C</math>? (2013 AMC 12A, Problem 16)
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'''3.5:''' Find the number of integer values of <math>k</math> in the closed interval <math>[-500,500]</math> for which the equation <math>\log(kx)=2\log(x+2)</math> has exactly one real solution. (2017 AIME II, Problem 7)
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'''4:''' Define a sequence recursively by <math>x_0=5</math> and<cmath>x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}</cmath>for all nonnegative integers <math>n.</math> Let <math>m</math> be the least positive integer such that<cmath>x_m\leq 4+\frac{1}{2^{20}}.</cmath>In which of the following intervals does <math>m</math> lie?
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<math>\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty)</math> (2019 AMC 10B, Problem 24 and 2019 AMC 12B, Problem 22)
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'''4.5:''' Find, with proof, all positive integers <math>n</math> for which <math>2^n + 12^n + 2011^n</math> is a perfect square. (USAJMO 2011, Problem 1)
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'''5:''' Find all triples <math>(a,b,c)</math> of real numbers such that the following system holds:
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<cmath>\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}</cmath> (JBMO 2020, Problem 1)
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'''5.5:''' Triangle <math>ABC</math> has <math>\angle BAC = 60^{\circ}</math>, <math>\angle CBA \leq 90^{\circ}</math>, <math>BC=1</math>, and <math>AC \geq AB</math>. Let <math>H</math>, <math>I</math>, and <math>O</math> be the orthocenter, incenter, and circumcenter of <math>\triangle ABC</math>, respectively. Assume that the area of pentagon <math>BCOIH</math> is the maximum possible. What is <math>\angle CBA</math>? (2011 AMC 12A, Problem 25)
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'''6:''' Let <math>\triangle ABC</math> be an acute triangle with circumcircle <math>\omega,</math> and let <math>H</math> be the intersection of the altitudes of <math>\triangle ABC.</math> Suppose the tangent to the circumcircle of <math>\triangle HBC</math> at <math>H</math> intersects <math>\omega</math> at points <math>X</math> and <math>Y</math> with <math>HA=3,HX=2,</math> and <math>HY=6.</math> The area of <math>\triangle ABC</math> can be written in the form <math>m\sqrt{n},</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math> (2020 AIME I, Problem 15)
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'''6.5:''' Rectangles <math>BCC_1B_2,</math> <math>CAA_1C_2,</math> and <math>ABB_1A_2</math> are erected outside an acute triangle <math>ABC.</math> Suppose that<cmath>\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.</cmath>Prove that lines <math>B_1C_2,</math> <math>C_1A_2,</math> and <math>A_1B_2</math> are concurrent. (2021 USAMO Problem 1, 2021 USAJMO Problem 2)
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'''7:''' Let <math>ABC</math> be an acute triangle with orthocenter <math>H</math>, and let <math>W</math> be a point on the side <math>BC</math>, lying strictly between <math>B</math> and <math>C</math>. The points <math>M</math> and <math>N</math> are the feet of the altitudes from <math>B</math> and <math>C</math>, respectively. Denote by <math>\omega_1</math> is the circumcircle of <math>BWN</math>, and let <math>X</math> be the point on <math>\omega_1</math> such that <math>WX</math> is a diameter of <math>\omega_1</math>. Analogously, denote by <math>\omega_2</math> the circumcircle of triangle <math>CWM</math>, and let <math>Y</math> be the point such that <math>WY</math> is a diameter of <math>\omega_2</math>. Prove that <math>X,Y</math> and <math>H</math> are collinear. (2013 IMO, Problem 1)
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'''7.5:''' A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other. Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges <math>\overrightarrow{uv}</math> and <math>\overrightarrow{vw}</math>, those two edges are in different colors. Note that it is permissible for <math>\overrightarrow{uv}</math> and <math>\overrightarrow{uw}</math> to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each <math>n</math>, determine the minimum directed-edge-chromatic-number over all tournaments on <math>n</math> vertices. (2015 USA TST, Problem 5)
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'''8:''' Let <math>\mathbb{Z}</math> be the set of integers. Find all functions <math>f : \mathbb{Z} \rightarrow \mathbb{Z}</math> such that<cmath>xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))</cmath>for all <math>x, y \in \mathbb{Z}</math> with <math>x \neq 0</math>. (2014 USAMO, Problem 2)
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'''8.5:''' Determine all functions <math>f:(0,\infty)\to\mathbb{R}</math> satisfying<cmath>\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)</cmath>for all <math>x,y>0</math>. (2018 IMO Shortlist, Problem A5)
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'''9:''' Let <math>ABC</math> be a scalene triangle with circumcircle <math>\Omega</math> and incenter <math>I</math>. Ray <math>AI</math> meets <math>\overline{BC}</math> at <math>D</math> and meets <math>\Omega</math> again at <math>M</math>; the circle with diameter <math>\overline{DM}</math> cuts <math>\Omega</math> again at <math>K</math>. Lines <math>MK</math> and <math>BC</math> meet at <math>S</math>, and <math>N</math> is the midpoint of <math>\overline{IS}</math>. The circumcircles of <math>\triangle KID</math> and <math>\triangle MAN</math> intersect at points <math>L_1</math> and <math>L_2</math>. Prove that <math>\Omega</math> passes through the midpoint of either <math>\overline{IL_1}</math> or <math>\overline{IL_2}</math>. (2017 USAMO, Problem 3)
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'''9.5:''' An [i]anti-Pascal triangle[/i] is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from <math>1</math> to <math>10</math>.
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<cmath>\begin{array}{
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c@{\hspace{4pt}}c@{\hspace{4pt}}
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c@{\hspace{4pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c
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} \vspace{4pt}
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& & & 4 & & &  \\\vspace{4pt}
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& & 2 & & 6 & &  \\\vspace{4pt}
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& 5 & & 7 & & 1 & \\\vspace{4pt}
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8 & & 3 & & 10 & & 9 \\\vspace{4pt}
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\end{array}</cmath>Does there exist an anti-Pascal triangle with <math>2018</math> rows which contains every integer from <math>1</math> to <math>1 + 2 + 3 + \dots + 2018</math>? (2018 IMO, Problem 3)
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'''10:''' Prove that there exists a positive constant <math>c</math> such that the following statement is true:
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Consider an integer <math>n > 1</math>, and a set <math>\mathcal S</math> of <math>n</math> points in the plane such that the distance between any two different points in <math>\mathcal S</math> is at least 1. It follows that there is a line <math>\ell</math> separating <math>\mathcal S</math> such that the distance from any point of <math>\mathcal S</math> to <math>\ell</math> is at least <math>cn^{-1/3}</math>.
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(A line <math>\ell</math> separates a set of points S if some segment joining two points in <math>\mathcal S</math> crosses <math>\ell</math>.) (2020 IMO, Problem 6)
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'''>10:''' Let <math>P_1P_2\dotsb P_{100}</math> be a cyclic <math>100</math>-gon and let <math>P_i = P_{i+100}</math> for all <math>i</math>. Define <math>Q_i</math> as the intersection of diagonals <math>\overline{P_{i-2}P_{i+1}}</math> and <math>\overline{P_{i-1}P_{i+2}}</math> for all integers <math>i</math>.
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Suppose there exists a point <math>P</math> satisfying <math>\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}</math> for all integers <math>i</math>. Prove that the points <math>Q_1,Q_2,\dots, Q_{100}</math> are concyclic. (2020 USA TST, Problem 6)
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'''>10:''' Given positive integer <math>n</math>. Prove that for any integers <math>a_1,a_2,\cdots,a_n,</math> at least <math>\lceil \tfrac{n(n-6)}{19} \rceil</math> numbers from the set <math>\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \}</math> cannot be represented as <math>a_i-a_j (1 \le i, j \le n)</math>. (2021 China TST, Day 1 Problem 3)
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= Competitions =
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==Introductory Competitions==
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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].
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=== [[MOEMS]] ===
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*Division E: '''1'''
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*: ''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])
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*Division M: '''1'''
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*: ''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])
  
 
=== [[AMC 8]] ===
 
=== [[AMC 8]] ===
  
 
* Problem 1 - Problem 12: '''1'''  
 
* Problem 1 - Problem 12: '''1'''  
*: ''What is the number of degrees in the smaller angle between the hour hand and the minute hand on a clock that reads seven o'clock?'' ([[1989 AJHSME Problems/Problem 10|Solution]])
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*: ''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]])
* Problem 13 - Problem 25: '''2'''
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* Problem 13 - Problem 25: '''1.5-2'''
*: ''A fifth number, <math>n</math>, is added to the set <math>\{ 3,6,9,10 \}</math> to make the mean of the set of five numbers equal to its median.  What is the number of possible values of <math>n</math>? '' ([[1988 AJHSME Problems/Problem 21|Solution]])
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*: ''How many positive factors does <math>23,232</math> have?'' ([[2018 AMC 8 Problems/Problem 18|Solution]])
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=== [[Mathcounts]] ===
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* Countdown: '''1-2.'''
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* Sprint: '''1-1.5''' (school/chapter), '''1.5-2''' (State), '''2-2.5''' (National)
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* Target: '''1-2''' (school/chapter), '''1.5-2.5''' (State), '''2.5-3.5''' (National)
  
 
=== [[AMC 10]] ===
 
=== [[AMC 10]] ===
  
* Problem 1 - 5: '''1'''
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* Problem 1 - 10: '''1-2'''
*: ''The larger of two consecutive odd integers is three times the smaller. What is their sum?'' ([[2007 AMC 10A Problems/Problem 4|Solution]])
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*: ''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]])
* Problem 6 - 20: '''2'''
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* Problem 11 - 20: '''2-3'''
*: ''How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?'' ([[2006 AMC 10A Problems/Problem 19|Solution]])
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*: ''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]])
* Problem 21 - 25: '''3'''
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* Problem 21 - 25: '''3.5-4.5'''
*: ''Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?'' ([[2006 AMC 10B Problems/Problem 25|Solution]])
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*: ''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]])
  
=== [[AMC 12]] ===
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===[[CEMC|CEMC Multiple Choice Tests]]===
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This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.
  
* Problem 1-5: '''2'''
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* Part A: '''1-1.5'''
*: ''A solid box is <math>15</math> cm by <math>10</math> cm by <math>8</math> cm. A new solid is formed by removing a cube <math>3</math> cm on a side from each corner of this box. What percent of the original volume is removed?'' ([[2003 AMC 12A Problems/Problem 3|Solution]])
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*: ''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)
* Problem 6-20: '''3'''
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* Part B: '''1-2'''
*: ''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]])
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*: ''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)
* Problem 21-25: '''4'''
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* Part C (Gauss/Pascal): '''2-2.5'''
*: ''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]])
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*: ''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)
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* Part C (Cayley/Fermat): '''2.5-3'''
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*: ''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)
  
=== [[AIME]] ===
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===[[CEMC|CEMC Fryer/Galois/Hypatia]]===
  
* Problem 1 - 5: '''3'''
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* Problem 1-2: '''1-2'''
*: ''If <math>\tan x+\tan y=25</math> and <math>\cot x + \cot y=30</math>, what is <math>\tan(x+y)</math>?'' ([[1986 AIME Problems/Problem 3|Solution]])
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* Problem 3-4 (early parts): '''2-3'''
* Problem 5 - 10: '''4'''  
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* Problem 3-4 (later parts): '''3-5'''
*: ''Triangle <math>ABC</math> has <math>AB=21</math>, <math>AC=22</math> and <math>BC=20</math>. Points <math>D</math> and <math>E</math> are located on <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, such that <math>\overline{DE}</math> is [[parallel]] to <math>\overline{BC}</math> and contains the center of the inscribed circle of triangle <math>ABC</math>. Then <math>DE=m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.'' ([[2001 AIME I Problems/Problem 7|Solution]])
 
* Problem 11 - 15: '''5.5'''
 
*: ''A right cone|right circular cone has a base with radius <math>600</math> and height <math> 200\sqrt{7}. </math> A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is <math>125</math>, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is <math>375\sqrt{2}.</math> Find the least distance that the fly could have crawled.'' ([[2004 AIME II Problems/Problem 11|Solution]])
 
  
=== [[APMO]] ===
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===Problem Solving Books for Introductory Students===
Problem 1: '''6.5'''
 
Problem 2: '''7'''
 
  
=== [[Austrian MO]] ===
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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.
  
* Gebietswettbewerb Für Fortgeschrittene, Problems 1-4: '''2'''  
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===[[Prealgebra by AoPS]]===
* Bundeswettbewerb Für Fortgeschrittene, Teil 1. Problems 1-4: '''3'''  
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'''1-2'''
* Bundeswettbewerb Für Fortgeschrittene, Teil 2, Problems 1-6: '''4'''
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===[[Introduction to Algebra by AoPS]]===
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'''1-3.5'''
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===[[Introduction to Counting and Probability by AoPS]]===
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'''1-3.5'''
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===[[Introduction to Number Theory by AoPS]]===
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'''1-3'''
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===[[Introduction to Geometry by AoPS]]===
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'''1-4'''
  
=== [[Indonesian MO]] ===
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==Intermediate Competitions==
* Problem 1/5: '''3.5'''
+
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].
*: '' 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>
 
* Problem 2/6: '''4.5'''
 
*: ''Find the lowest possible values from the function
 
<math>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</math>
 
  
for any real numbers <math>x</math>.''<url>viewtopic.php?t=294067 (Solution)</url>
+
=== [[AMC 12]] ===
* Problem 3/7: '''5'''
 
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if
 
<math>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</math>
 
  
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>.''
+
* Problem 1-10: '''1.5-2'''
* Problem 4/8: '''6'''
+
*: ''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]])
*: ''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>.
+
* Problem 11-20: '''2.5-3.5'''
 +
*: ''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]])
 +
* Problem 21-25: '''4.5-6'''
 +
*: ''Semicircle <math>\Gamma</math> has diameter <math>\overline{AB}</math> of length <math>14</math>. Circle <math>\omega</math> lies tangent to <math>\overline{AB}</math> at a point <math>P</math> and intersects <math>\Gamma</math> at points <math>Q</math> and <math>R</math>. If <math>QR=3\sqrt{3}</math> and <math>\angle QPR=60^{\circ},</math> then the area of <math>\triangle PQR</math> equals <math>\tfrac{a\sqrt{b}}{c},</math> where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer not divisible by the square of any prime. What is <math>a+b+c</math>?'' ([[2021 AMC 12A Problems/Problem 24|Solution]])
  
(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.
+
=== [[AIME]] ===
  
(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.''
+
* Problem 1 - 5: '''3-3.5'''
 +
*: ''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]])
 +
* Problem 6 - 9: '''4-4.5'''
 +
*: ''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]])
 +
* Problem 10 - 12: '''5-5.5'''
 +
*: 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]])
 +
* Problem 13 - 15: '''6-7'''
 +
*: ''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]])
  
 
=== [[ARML]] ===
 
=== [[ARML]] ===
  
* Individuals, Problem 1-5,7,9: '''3'''
+
* Individuals, Problem 1: '''2'''
* Individuals, Problem 6,8: '''4'''
+
 
* Individuals, Problem 10: '''6.5'''
+
* Individuals, Problems 2, 3, 4, 5, 7, and 9: '''3'''
* Team/power, Problem 1-5: '''3.5'''
+
 
 +
* Individuals, Problems 6 and 8: '''4'''  
 +
 
 +
* Individuals, Problem 10: '''5.5'''
 +
 
 +
* Team/power, Problem 1-5: '''3.5'''  
 +
 
 
* Team/power, Problem 6-10: '''5'''
 
* Team/power, Problem 6-10: '''5'''
  
=== [[Balkan MO]] ===
+
===[[HMMT|HMMT (November)]]===
 +
* Individual Round, Problem 6-8: '''4'''
 +
* Individual Round, Problem 10: '''4.5'''
 +
* Team Round: '''4-5'''
 +
* Guts: '''3.5-5.25'''
  
* Problem 1: '''6'''
+
===[[CEMC|CEMC Euclid]]===
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. ''
 
* Problem 2: '''6.5'''
 
*: '' 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>. ''
 
* Problem 3: '''7.5'''
 
*: '' 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
 
  
<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>
+
* Problem 1-6: '''1-3'''
 +
* Problem 7-10: '''3-6'''
  
<math>(ii)</math> the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry? ''
+
===[[Purple Comet! Math Meet|Purple Comet]]===
* Problem 4: '''8'''
 
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that
 
  
<math>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2</math> for all <math>m,n \in S</math>. ''
+
* Problems 1-10 (MS): '''1.5-3'''
 +
* Problems 11-20 (MS): '''3-4.5'''
 +
* Problems 1-10 (HS): '''1.5-3.5'''
 +
* Problems 11-20 (HS): '''3.5-4.75'''
 +
* Problems 21-30 (HS): '''4.5-6'''
 +
* Problems 18-20 (MS): '''4-4.75'''
  
=== [[China TST]] ===
+
===[[Lexington Math Tournament|LMT]]===
  
* Problem 1/4:
+
* Easy Problems: '''1-2'''
* Problem 2/5:  
+
*: ''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>. ''
* Problem 3/6: '''9.5'''  
+
* Medium Problems: '''2-4'''
*: ''Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form [copy error] or not where [copy error] are nonnegative integers.''
+
*: ''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>. ''
 +
* Hard Problems: '''5-7'''
 +
*: ''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.''
  
=== Germany Bundeswettbewerb Mathematik ===
+
==Problem Solving Books for Intermediate Students==
  
* Round 1, Problem 1: '''x'''
+
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.
*: '' Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter? '' <url>viewtopic.php?p=1194585#1194585 (Solution)</url>
 
* Round 1, Problem 2: '''x'''
 
*: '' Represent the number <math>2008</math> as a sum of natural number such that the addition of the reciprocals of the summands yield 1. '' <url>viewtopic.php?p=1194595#1194595 (Solution)</url>
 
* Round 1, Problem 3: '''x'''
 
*: '' Prove: In an acute triangle <math>ABC</math> angle bisector <math>w_{\alpha},</math> median <math>s_b</math> and the altitude <math>h_c</math> intersect in one point if <math>w_{\alpha},</math> side <math>BC</math> and the circle around foot of the altitude <math>h_c</math> have vertex <math>A</math> as a common point. '' <url>viewtopic.php?p=1194631#1194631 (Solution)</url>
 
* Round 1, Problem 4: '''x'''
 
*: '' In a planar coordinate system we got four pieces on positions with coordinates. You can make a move according to the following rule: You can move a piece to a new position if there is one of the other pieces in the middle of the old and new position. Initially the four pieces have positions <math>\{(0,0),(0,1),(1,0),(1,1)\}</math>. Given a finite number of moves can you yield the configuration <math>\{(0,0), (1,1), (3,0), (2, - 1)\}</math> ? '' <url>viewtopic.php?p=1194636#1194636 (Solution)</url>
 
* Round 2, Problem 1: '''x'''
 
*: '' Determine all real <math>x</math> satisfying the equation
 
<cmath>\sqrt [5]{x^3 + 2x} = \sqrt [3]{x^5 - 2x}.</cmath>
 
Odd roots for negative radicands shall be included in the discussion. '' <url>viewtopic.php?p=1249364#1249364 (Solution)</url>
 
* Round 2, Problem 2: '''x'''
 
*: '' Let the positive integers <math>a,b,c</math> chosen such that the quotients <math>\frac {bc}{b + c},</math> <math>\frac {ca}{c + a}</math> and <math>\frac {ab}{a + b}</math> are integers. Prove that <math>a,b,c</math> have a common divisor greater than 1. '' <url>viewtopic.php?p=1249366#1249366 (Solution)</url>
 
* Round 2, Problem 3: '''x'''
 
*: '' Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black. '' <url>viewtopic.php?p=1249370#1249370 (Solution)</url>
 
  
* Round 2, Problem 4: '''x'''
+
===[[Intermediate Algebra by AoPS]]===
*: '' On a bookcase there are <math>n \geq 3</math> books side by side by different authors. A librarian considers the first and second book from left and exchanges them iff they are not alphabetically sorted. Then he is doing the same operation with the second and third book from left etc. Using this procedure he iterates through the bookcase three times from left to right. Considering all possible initial book configurations how many of them will then be alphabetically sorted? '' <url>viewtopic.php?p=1249370#1249370 (Solution)</url>
+
'''2.5-6.5/7''', may vary across chapters
  
=== [[HMMT]] ===
+
===[[Intermediate Counting & Probability by AoPS]]===
 +
'''3.5-7.5/8''', may vary across chapters
  
* Individuals, Problem 1-5: '''4'''
+
===[[Precalculus by AoPS]]===
* Individuals, Problem 6-10: '''7'''
+
'''2-8''', may vary across chapters
  
=== [[IMO]] ===
+
==Beginner Olympiad Competitions==
 +
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].
  
* Problem 1/4: '''6.5'''
+
=== [[USAMTS]] ===
*: ''Find all functions <math>f: (0, \infty) \mapsto (0, \infty)</math> (so <math>f</math> is a function from the positive real numbers) such that
+
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:
<center><math>\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}</math></center> for all positive real numbers <math>w,x,y,z,</math> satisfying <math>wx = yz.</math> ([[2008 IMO Problems/Problem 4|Solution]])
+
* Problem 1-2: '''3-4'''
''
+
*: ''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])
* Problem 2/5: '''7.5'''
+
* Problem 3-5: '''4-6'''
*: ''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]])
+
*: ''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])
* Problem 3/6: '''9.5'''
 
*: ''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>)
 
  
=== [[IMO Shortlist]] ===
+
=== [[Indonesia Mathematical Olympiad|Indonesia MO]] ===
 +
* Problem 1/5: '''3.5'''
 +
*: '' 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])
 +
* Problem 2/6: '''4.5'''
 +
*: ''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])
 +
* Problem 3/7: '''5'''
 +
*: ''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])
 +
* Problem 4/8: '''6'''
 +
*: ''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>.''
  
* Problem 1-2: '''5'''
+
::''(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.''
* Problem 3-4: '''7'''
 
* Problem 5+: '''8.5+'''
 
  
=== [[Iran NMO]] ===
+
::''(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' ([https://artofproblemsolving.com/community/c6h294069 Solution])
  
=== [[Iran TST]] ===
+
=== [[Central American Olympiad]] ===
 +
* Problem 1: '''4'''
 +
*: ''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])
 +
* Problem 2,4,5: '''5-6'''
 +
*: ''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])
 +
* Problem 3/6: '''6.5'''
 +
*: ''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])
  
 
=== [[JBMO]] ===
 
=== [[JBMO]] ===
Line 159: Line 262:
 
*: ''Find all real numbers <math>a,b,c,d</math> such that  
 
*: ''Find all real numbers <math>a,b,c,d</math> such that  
 
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''
 
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''
* Problem 2: '''5'''
+
* Problem 2: '''4.5-5'''
 
*: ''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>.''
 
*: ''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>.''
 
* Problem 3: '''5'''
 
* Problem 3: '''5'''
Line 166: Line 269:
 
*: ''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>.''
 
*: ''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>.''
  
=== [[Mathcounts]] ===
+
==Olympiad Competitions==
 +
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].
 +
 
 +
=== [[USAJMO]] ===
 +
* Problem 1/4: '''5'''
 +
*: ''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.''
 +
 
 +
::''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]])
 +
 
 +
* Problem 2/5: '''6-6.5'''
 +
*: ''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]])
 +
 
 +
* Problem 3/6: '''7'''
 +
*: ''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]])
 +
 
 +
===[[HMMT|HMMT (February)]]===
 +
* Individual Round, Problem 1-5: '''5'''
 +
* Individual Round, Problem 6-10: '''5.5-6'''
 +
* Team Round: '''7.5'''
 +
* HMIC: '''8'''
 +
 
 +
=== [[Canadian MO]] ===
 +
 
 +
* Problem 1: '''5.5'''
 +
* Problem 2: '''6'''
 +
* Problem 3: '''6.5'''
 +
* Problem 4: '''7-7.5'''
 +
* Problem 5: '''7.5-8'''
 +
 
 +
=== Austrian MO ===
 +
 
 +
* Regional Competition for Advanced Students, Problems 1-4: '''5'''
 +
* Federal Competition for Advanced Students, Part 1. Problems 1-4: '''6'''
 +
* Federal Competition for Advanced Students, Part 2, Problems 1-6: '''7'''
 +
 
 +
=== [[Iberoamerican Math Olympiad]] ===
 +
 
 +
* Problem 1/4: '''5.5'''
 +
* Problem 2/5: '''6.5'''
 +
* Problem 3/6: '''7.5'''
  
=== [[Miklós Schweitzer]] ===
+
=== [[APMO]] ===
 +
*Problem 1: '''6'''
 +
*Problem 2: '''7'''
 +
*Problem 3: '''7'''
 +
*Problem 4: '''7.5'''
 +
*Problem 5: '''8.5'''
  
* Problem 1-3:
+
=== Balkan MO ===
* Problem 4-6:
 
* Problem 7-9:
 
* Problem 10-12:
 
  
=== [[MOEMS]] ===
+
* Problem 1: '''6'''
*Division E: '''1'''
+
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. ''
*: ''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])
+
* Problem 2: '''6.5'''
*Division M: '''1'''
+
*: '' 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>. ''
*: ''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])
+
* Problem 3: '''7.5'''
 +
*: '' <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''
 +
 
 +
::<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>''
  
=== [[Putnam]] ===
+
::<math>(\rm ii)</math> ''the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry?''
 +
* Problem 4: '''8'''
 +
*: '' 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>
  
* Problem A/B,1-2: '''6.5'''
+
==Hard Olympiad Competitions==
*: ''Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola <math>xy = 1</math> and both branches of the hyperbola <math>xy = - 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>)'' (<url>viewtopic.php?p=978383#p978383 Solution</url>)
+
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].
* Problem A/B,3-4: '''7.5'''
 
*: ''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>.'' (<url>viewtopic.php?p=383280#383280 Solution</url>)
 
* Problem A/B,5-6: '''9'''
 
*: ''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) = \phi,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' (<url>viewtopic.php?t=127810 Solution</url>)
 
  
 
=== [[USAMO]] ===
 
=== [[USAMO]] ===
 
+
* Problem 1/4: '''6-7'''
* Problem 1/4: '''6.5'''
 
 
*: ''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]])  
 
*: ''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]])  
* Problem 2/5: '''7'''
+
* Problem 2/5: '''7-8'''
 
*: ''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]])
 
*: ''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]])
* Problem 3/6: '''8.5'''
+
* Problem 3/6: '''8-9'''
*: ''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]])  
+
*: ''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]])
 +
 
 +
=== [[USA TST]] ===
 +
 
 +
 
  
=== [[USAMTS]] ===
+
* Problem 1/4/7: '''6.5-7'''
 +
* Problem 2/5/8: '''7.5-8'''
 +
* Problem 3/6/9: '''8.5-9'''
 +
 
 +
=== [[Putnam]] ===
  
* Problem 1-2: '''4'''
+
* Problem A/B,1-2: '''7'''
*: ''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])
+
*: ''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])
* Problem 3-5: '''5'''
+
* Problem A/B,3-4: '''8'''
*: ''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])
+
*: ''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])
 +
* Problem A/B,5-6: '''9'''
 +
*: ''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])
  
=== [[USA TST]] ===
+
=== [[China TST]] (hardest problems)===
  
(seems to vary more than other contests; estimates based on 08 and 09)
+
* Problem 1/4: '''8-8.5'''
 +
*: ''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>''
 +
* Problem 2/5: '''9'''
 +
*: ''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>''
 +
* Problem 3/6: '''9.5-10'''
 +
*: ''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>''
  
* Problem 1/4/7: '''7.5'''
+
=== [[IMO]] ===
* Problem 2/5/8: '''8'''
 
* Problem 3/6/9: '''8.5'''
 
  
 +
* Problem 1/4: '''5.5-7'''
 +
*: ''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]])
  
 +
* Problem 2/5: '''7-8'''
 +
*: ''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]])
  
== Scale ==
+
* Problem 3/6: '''9-10'''
{{ref|1}} All levels 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, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this.  
+
*: ''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])
  
# Problems strictly for beginners, on the easiest elementary school or middle school levels. Examples would be MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, very easy AMC 10/12 questions, and others that involve standard techniques introduced up to the middle school level
+
=== [[IMO Shortlist]] ===
# For motivated beginners, harder questions from the previous categories (hardest middle-school level questions, #5-20 on AMC 10, #5-10 on AMC 12, easiest AIME questions, etc).
 
# For those not too familiar with standard techniques, #21-25 on AMC 10, #11-20ish on AMC 12, #1-5 on AIMEs, and analogous contests.
 
# Intermediate-leveled problem solvers, the most difficult questions on AMC 12s (#22-25s), more difficult AIME-styled questions #6-10
 
# Difficult AIME problems (#10-13), others, simple proof-based problems (JBMO etc)
 
# High-leveled AIME-styled questions, not requiring proofs (#12-15). Introductory-leveled Olympiad-level questions (#1-4s).
 
# Intermediate-leveled Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.
 
# High-level difficult Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.
 
# Difficult Olympiad-level questions, eg #3,6s on difficult Olympiad contests.
 
# Problems occasionally even unsuitable for normal grade school level competitions due to being exceedingly tedious/long/difficult (eg very few students are capable of solving, even on a worldwide basis), or involving techniques beyond high school level mathematics.
 
  
== See also ==
+
* Problem 1-2: '''5.5-7'''
* <url>viewtopic.php?p=1565063#1565063 Forum discussion of wiki entry </url>
+
* Problem 3-4: '''7-8'''
 +
* Problem 5+: '''8-10'''
  
 
[[Category:Mathematics competitions]]
 
[[Category:Mathematics competitions]]

Latest revision as of 20:44, 1 October 2021

This page contains an approximate estimation of the difficulty level of various 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.

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.

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. early AMC problems and 10 is hardest level, e.g. China IMO Team Selection Test. When considering problem difficulty put more emphasis on problem-solving aspects and less so on technical skill requirements.

Scale

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.

  1. 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
  2. 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
  3. Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20 on AMC 12, easier #1-5 on AIMEs, etc.)
  4. Intermediate-level problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.
  5. More difficult AIME problems (#10-12), simple proof-based problems (JBMO)
  6. High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1-4).
  7. Tougher Olympiad-level questions, #1-4 that require more technical knowledge than new students to Olympiad-type questions have, easier #2-5, etc.
  8. High-level Olympiad-level questions, #2-5s on difficult Olympiad contest and easier #3,6s, etc.
  9. Expert Olympiad-level questions, #3-6 on difficult Olympiad contests.
  10. Super Expert problems, problems occasionally even unsuitable for very hard competitions (such as the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).

For reference, here are problems from each of the difficulty levels 1-10: <1: Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? (2015 AMC 8, Problem 1)

1: How many integer values of $x$ satisfy $|x| < 3\pi$? (2021 AMC 10B, Problem 1)

1.5: A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$ (2020 AMC 8, Problem 19)

2: For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Let $N=f(11)+f(13)+f(14)$. Which of the following relations is true? (2002 AMC 12A, Problem 14)

2.5: Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? (2018 AMC 10A, Problem 24)

3: $A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$? (2013 AMC 12A, Problem 16)

3.5: Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\log(kx)=2\log(x+2)$ has exactly one real solution. (2017 AIME II, Problem 7)

4: Define a sequence recursively by $x_0=5$ and\[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\]for all nonnegative integers $n.$ Let $m$ be the least positive integer such that\[x_m\leq 4+\frac{1}{2^{20}}.\]In which of the following intervals does $m$ lie?

$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty)$ (2019 AMC 10B, Problem 24 and 2019 AMC 12B, Problem 22)

4.5: Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square. (USAJMO 2011, Problem 1)

5: Find all triples $(a,b,c)$ of real numbers such that the following system holds: \[\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}\] (JBMO 2020, Problem 1)

5.5: Triangle $ABC$ has $\angle BAC = 60^{\circ}$, $\angle CBA \leq 90^{\circ}$, $BC=1$, and $AC \geq AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? (2011 AMC 12A, Problem 25)

6: Let $\triangle ABC$ be an acute triangle with circumcircle $\omega,$ and let $H$ be the intersection of the altitudes of $\triangle ABC.$ Suppose the tangent to the circumcircle of $\triangle HBC$ at $H$ intersects $\omega$ at points $X$ and $Y$ with $HA=3,HX=2,$ and $HY=6.$ The area of $\triangle ABC$ can be written in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n.$ (2020 AIME I, Problem 15)

6.5: Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that\[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\]Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent. (2021 USAMO Problem 1, 2021 USAJMO Problem 2)

7: Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear. (2013 IMO, Problem 1)

7.5: A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other. Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges $\overrightarrow{uv}$ and $\overrightarrow{vw}$, those two edges are in different colors. Note that it is permissible for $\overrightarrow{uv}$ and $\overrightarrow{uw}$ to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each $n$, determine the minimum directed-edge-chromatic-number over all tournaments on $n$ vertices. (2015 USA TST, Problem 5)

8: Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that\[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\]for all $x, y \in \mathbb{Z}$ with $x \neq 0$. (2014 USAMO, Problem 2)

8.5: Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying\[\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)\]for all $x,y>0$. (2018 IMO Shortlist, Problem A5)

9: Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$. Ray $AI$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$; the circle with diameter $\overline{DM}$ cuts $\Omega$ again at $K$. Lines $MK$ and $BC$ meet at $S$, and $N$ is the midpoint of $\overline{IS}$. The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L_2$. Prove that $\Omega$ passes through the midpoint of either $\overline{IL_1}$ or $\overline{IL_2}$. (2017 USAMO, Problem 3)

9.5: An [i]anti-Pascal triangle[/i] is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$. \[\begin{array}{ c@{\hspace{4pt}}c@{\hspace{4pt}} c@{\hspace{4pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c } \vspace{4pt}  & & & 4 & & &  \\\vspace{4pt}  & & 2 & & 6 & &  \\\vspace{4pt}  & 5 & & 7 & & 1 & \\\vspace{4pt}  8 & & 3 & & 10 & & 9 \\\vspace{4pt} \end{array}\]Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1 + 2 + 3 + \dots + 2018$? (2018 IMO, Problem 3)

10: Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$.

(A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) (2020 IMO, Problem 6)

>10: Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$.

Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic. (2020 USA TST, Problem 6)

>10: Given positive integer $n$. Prove that for any integers $a_1,a_2,\cdots,a_n,$ at least $\lceil \tfrac{n(n-6)}{19} \rceil$ numbers from the set $\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \}$ cannot be represented as $a_i-a_j (1 \le i, j \le n)$. (2021 China TST, Day 1 Problem 3)

Competitions

Introductory Competitions

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 here.

MOEMS

  • Division E: 1
    The whole number $N$ is divisible by $7$. $N$ leaves a remainder of $1$ when divided by $2,3,4,$ or $5$. What is the smallest value that $N$ can be? (Solution)
  • Division M: 1
    The value of a two-digit number is $10$ times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number. (Solution)

AMC 8

  • Problem 1 - Problem 12: 1
    The $\emph{harmonic mean}$ 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? (Solution)
  • Problem 13 - Problem 25: 1.5-2
    How many positive factors does $23,232$ have? (Solution)

Mathcounts

  • Countdown: 1-2.
  • Sprint: 1-1.5 (school/chapter), 1.5-2 (State), 2-2.5 (National)
  • Target: 1-2 (school/chapter), 1.5-2.5 (State), 2.5-3.5 (National)

AMC 10

  • Problem 1 - 10: 1-2
    A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box? (Solution)
  • Problem 11 - 20: 2-3
    For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? (Solution)
  • Problem 21 - 25: 3.5-4.5
    The vertices of an equilateral triangle lie on the hyperbola $xy=1$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? (Solution)

CEMC Multiple Choice Tests

This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.

  • Part A: 1-1.5
    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)
  • Part B: 1-2
    Two lines with slopes $\tfrac14$ and $\tfrac54$ intersect at $(1,1)$. What is the area of the triangle formed by these two lines and the vertical line $x = 5$? (2017 Cayley Problem 19)
  • Part C (Gauss/Pascal): 2-2.5
    Suppose that $\tfrac{2009}{2014} + \tfrac{2019}{n} = \tfrac{a}{b}$, where $a$, $b$, and $n$ are positive integers with $\tfrac{a}{b}$ in lowest terms. What is the sum of the digits of the smallest positive integer $n$ for which $a$ is a multiple of 1004? (2014 Pascal Problem 25)
  • Part C (Cayley/Fermat): 2.5-3
    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)

CEMC Fryer/Galois/Hypatia

  • Problem 1-2: 1-2
  • Problem 3-4 (early parts): 2-3
  • Problem 3-4 (later parts): 3-5

Problem Solving Books for Introductory Students

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.

Prealgebra by AoPS

1-2

Introduction to Algebra by AoPS

1-3.5

Introduction to Counting and Probability by AoPS

1-3.5

Introduction to Number Theory by AoPS

1-3

Introduction to Geometry by AoPS

1-4

Intermediate Competitions

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 here.

AMC 12

  • Problem 1-10: 1.5-2
    What is the value of \[\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?\] (Solution)
  • Problem 11-20: 2.5-3.5
    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? (Solution)
  • Problem 21-25: 4.5-6
    Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt{3}$ and $\angle QPR=60^{\circ},$ then the area of $\triangle PQR$ equals $\tfrac{a\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$? (Solution)

AIME

  • Problem 1 - 5: 3-3.5
    Consider the integer \[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\]Find the sum of the digits of $N$. (Solution)
  • Problem 6 - 9: 4-4.5
    How many positive integers $N$ less than $1000$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$? (Solution)
  • Problem 10 - 12: 5-5.5
    Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$.Let $S$ be the sum of all elements in $R$. Find the remainder when $S$ is divided by $1000$ (Solution)
  • Problem 13 - 15: 6-7
    Let \[P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).\] Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2} = a_{k} + b_{k}i$ for $k = 1,2,\ldots,r,$ where $i = \sqrt { - 1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let \[\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},\] where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m + n + p.$. (Solution)

ARML

  • Individuals, Problem 1: 2
  • Individuals, Problems 2, 3, 4, 5, 7, and 9: 3
  • Individuals, Problems 6 and 8: 4
  • Individuals, Problem 10: 5.5
  • Team/power, Problem 1-5: 3.5
  • Team/power, Problem 6-10: 5

HMMT (November)

  • Individual Round, Problem 6-8: 4
  • Individual Round, Problem 10: 4.5
  • Team Round: 4-5
  • Guts: 3.5-5.25

CEMC Euclid

  • Problem 1-6: 1-3
  • Problem 7-10: 3-6

Purple Comet

  • Problems 1-10 (MS): 1.5-3
  • Problems 11-20 (MS): 3-4.5
  • Problems 1-10 (HS): 1.5-3.5
  • Problems 11-20 (HS): 3.5-4.75
  • Problems 21-30 (HS): 4.5-6
  • Problems 18-20 (MS): 4-4.75

LMT

  • Easy Problems: 1-2
    Let trapezoid $ABCD$ be such that $AB||CD$. Additionally, $AC = AD = 5$, $CD = 6$, and $AB = 3$. Find $BC$.
  • Medium Problems: 2-4
    Let $\triangle LMN$ have side lengths $LM = 15$, $MN = 14$, and $NL = 13$. Let the angle bisector of $\angle MLN$ meet the circumcircle of $\triangle LMN$ at a point $T \ne L$. Determine the area of $\triangle LMT$.
  • Hard Problems: 5-7
    A magic $3 \times 5$ 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 $15$ cells (so there are $2^{15}$ 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 $3$ 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.

Problem Solving Books for Intermediate Students

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.

Intermediate Algebra by AoPS

2.5-6.5/7, may vary across chapters

Intermediate Counting & Probability by AoPS

3.5-7.5/8, may vary across chapters

Precalculus by AoPS

2-8, may vary across chapters

Beginner Olympiad Competitions

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 here.

USAMTS

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:

  • Problem 1-2: 3-4
    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. (Solution)
  • Problem 3-5: 4-6
    Call a positive real number groovy if it can be written in the form $\sqrt{n} + \sqrt{n + 1}$ for some positive integer $n$. Show that if $x$ is groovy, then for any positive integer $r$, the number $x^r$ is groovy as well. (Solution)

Indonesia MO

  • Problem 1/5: 3.5
    In a drawer, there are at most $2009$ 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 $\frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true? (Solution)
  • Problem 2/6: 4.5
    Find the lowest possible values from the function \[f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009\] for any real numbers $x$. (Solution)
  • Problem 3/7: 5
    A pair of integers $(m,n)$ is called good if \[m\mid n^2 + n \ \text{and} \ n\mid m^2 + m\] Given 2 positive integers $a,b > 1$ which are relatively prime, prove that there exists a good pair $(m,n)$ with $a\mid m$ and $b\mid n$, but $a\nmid n$ and $b\nmid m$. (Solution)
  • Problem 4/8: 6
    Given an acute triangle $ABC$. The incircle of triangle $ABC$ touches $BC,CA,AB$ respectively at $D,E,F$. The angle bisector of $\angle A$ cuts $DE$ and $DF$ respectively at $K$ and $L$. Suppose $AA_1$ is one of the altitudes of triangle $ABC$, and $M$ be the midpoint of $BC$.
(a) Prove that $BK$ and $CL$ are perpendicular with the angle bisector of $\angle BAC$.
(b) Show that $A_1KML$ is a cyclic quadrilateral. (Solution)

Central American Olympiad

  • Problem 1: 4
    Find all three-digit numbers $abc$ (with $a \neq 0$) such that $a^{2} + b^{2} + c^{2}$ is a divisor of 26. (Solution)
  • Problem 2,4,5: 5-6
    Show that the equation $a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005$ has no integer solutions. (Solution)
  • Problem 3/6: 6.5
    Let $ABCD$ be a convex quadrilateral. $I = AC\cap BD$, and $E$, $H$, $F$ and $G$ are points on $AB$, $BC$, $CD$ and $DA$ respectively, such that $EF \cap GH = I$. If $M = EG \cap AC$, $N = HF \cap AC$, show that $\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}$. (Solution)

JBMO

  • Problem 1: 4
    Find all real numbers $a,b,c,d$ such that

\[\left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right.\]

  • Problem 2: 4.5-5
    Let $ABCD$ be a convex quadrilateral with $\angle DAC=\angle BDC=36^\circ$, $\angle CBD=18^\circ$ and $\angle BAC=72^\circ$. The diagonals intersect at point $P$. Determine the measure of $\angle APD$.
  • Problem 3: 5
    Find all prime numbers $p,q,r$, such that $\frac pq-\frac4{r+1}=1$.
  • Problem 4: 6
    A $4\times4$ table is divided into $16$ 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 $n$ moves all the $16$ cells were black. Find all possible values of $n$.

Olympiad Competitions

This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available here.

USAJMO

  • Problem 1/4: 5
    There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear.
A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even. (Solution)
  • Problem 2/5: 6-6.5
    Let $a,b,c$ be positive real numbers such that $a+b+c=4\sqrt[3]{abc}$. Prove that \[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\] (Solution)
  • Problem 3/6: 7
    Two rational numbers $\tfrac{m}{n}$ and $\tfrac{n}{m}$ are written on a blackboard, where $m$ and $n$ are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ and $y$ written on the board and write either their arithmetic mean $\tfrac{x+y}{2}$ or their harmonic mean $\tfrac{2xy}{x+y}$ on the board as well. Find all pairs $(m,n)$ such that Evan can write $1$ on the board in finitely many steps. (Solution)

HMMT (February)

  • Individual Round, Problem 1-5: 5
  • Individual Round, Problem 6-10: 5.5-6
  • Team Round: 7.5
  • HMIC: 8

Canadian MO

  • Problem 1: 5.5
  • Problem 2: 6
  • Problem 3: 6.5
  • Problem 4: 7-7.5
  • Problem 5: 7.5-8

Austrian MO

  • Regional Competition for Advanced Students, Problems 1-4: 5
  • Federal Competition for Advanced Students, Part 1. Problems 1-4: 6
  • Federal Competition for Advanced Students, Part 2, Problems 1-6: 7

Iberoamerican Math Olympiad

  • Problem 1/4: 5.5
  • Problem 2/5: 6.5
  • Problem 3/6: 7.5

APMO

  • Problem 1: 6
  • Problem 2: 7
  • Problem 3: 7
  • Problem 4: 7.5
  • Problem 5: 8.5

Balkan MO

  • Problem 1: 6
    Solve the equation $3^x - 5^y = z^2$ in positive integers.
  • Problem 2: 6.5
    Let $MN$ be a line parallel to the side $BC$ of a triangle $ABC$, with $M$ on the side $AB$ and $N$ on the side $AC$. The lines $BN$ and $CM$ meet at point $P$. The circumcircles of triangles $BMP$ and $CNP$ meet at two distinct points $P$ and $Q$. Prove that $\angle BAQ = \angle CAP$.
  • Problem 3: 7.5
    A $9 \times 12$ 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 $C_1,C_2...,C_{96}$ in such way that the following to conditions are both fulfilled
$(\rm i)$ the distances $C_1C_2,...C_{95}C_{96}, C_{96}C_{1}$ are all equal to $\sqrt {13}$
$(\rm ii)$ the closed broken line $C_1C_2...C_{96}C_1$ has a centre of symmetry?
  • Problem 4: 8
    Denote by $S$ the set of all positive integers. Find all functions $f: S \rightarrow S$ such that \[f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2\text{ for all }m,n \in S.\]

Hard Olympiad Competitions

This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available here.

USAMO

  • Problem 1/4: 6-7
    Let $\mathcal{P}$ be a convex polygon with $n$ sides, $n\ge3$. Any set of $n - 3$ diagonals of $\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\mathcal{P}$ into $n - 2$ triangles. If $\mathcal{P}$ is regular and there is a triangulation of $\mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $n$. (Solution)
  • Problem 2/5: 7-8
    Three nonnegative real numbers $r_1$, $r_2$, $r_3$ are written on a blackboard. These numbers have the property that there exist integers $a_1$, $a_2$, $a_3$, not all zero, satisfying $a_1r_1 + a_2r_2 + a_3r_3 = 0$. We are permitted to perform the following operation: find two numbers $x$, $y$ on the blackboard with $x \le y$, then erase $y$ and write $y - x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $0$ on the blackboard. (Solution)
  • Problem 3/6: 8-9
    Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots. (Solution)

USA TST

  • Problem 1/4/7: 6.5-7
  • Problem 2/5/8: 7.5-8
  • Problem 3/6/9: 8.5-9

Putnam

  • Problem A/B,1-2: 7
    Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola $xyz = 1$ and both branches of the hyperbola $xwy = - 1.$ (A set $S$ in the plane is called convex if for any two points in $S$ the line segment connecting them is contained in $S.$) (Solution)
  • Problem A/B,3-4: 8
    Let $H$ be an $n\times n$ matrix all of whose entries are $\pm1$ and whose rows are mutually orthogonal. Suppose $H$ has an $a\times b$ submatrix whose entries are all $1.$ Show that $ab\le n$. (Solution)
  • Problem A/B,5-6: 9
    For any $a > 0$, define the set $S(a) = \{[an]|n = 1,2,3,...\}$. Show that there are no three positive reals $a,b,c$ such that $S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}$. (Solution)

China TST (hardest problems)

  • Problem 1/4: 8-8.5
    Given an integer $m,$ prove that there exist odd integers $a,b$ and a positive integer $k$ such that \[2m=a^{19}+b^{99}+k*2^{1000}.\]
  • Problem 2/5: 9
    Given a positive integer $n=1$ and real numbers $a_1 < a_2 < \ldots < a_n,$ such that $\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,$ prove that for any positive real number $x,$ \[\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.\]
  • Problem 3/6: 9.5-10
    Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Define $S_k=\sum_{i=0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]

IMO

  • Problem 1/4: 5.5-7
    Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line. (Solution)
  • Problem 2/5: 7-8
    Let $P(x)$ be a polynomial of degree $n>1$ with integer coefficients, and let $k$ be a positive integer. Consider the polynomial $Q(x) = P( P ( \ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t)=t$. (Solution)
  • Problem 3/6: 9-10
    Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at least twice the area of $P$. (Solution)

IMO Shortlist

  • Problem 1-2: 5.5-7
  • Problem 3-4: 7-8
  • Problem 5+: 8-10
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