Difference between revisions of "AoPS Wiki:Competition ratings"

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.

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.

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. early AMC problems and 10 is hardest level, i.e. China IMO Team Selection Test. When considering problem difficulty put more emphasis on problem-solving aspects and less so on technical skill requirements.[1]

Competitions

AMC 8

• 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? (Solution)
• Problem 13 - Problem 25: 2
A fifth number, $n$, is added to the set $\{ 3,6,9,10 \}$ to make the mean of the set of five numbers equal to its median. What is the number of possible values of $n$? (Solution)

AMC 10

• Problem 1 - 5: 1
The larger of two consecutive odd integers is three times the smaller. What is their sum? (Solution)
• Problem 6 - 20: 2
How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression? (Solution)
• Problem 21 - 25: 3
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? (Solution)

AMC 12

• Problem 1-5: 2
A solid box is $15$ cm by $10$ cm by $8$ cm. A new solid is formed by removing a cube $3$ cm on a side from each corner of this box. What percent of the original volume is removed? (Solution)
• Problem 3-18: 3
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 22-25: 4
Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. The value of $x_4 - x_1$ is $m + n\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. What is $m + n + p$? (Solution)

AIME

• Problem 1 - 5: 3
If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$? (Solution)
• Problem 5 - 10: 4
Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. (Solution)
• Problem 11 - 15: 5
A right cone|right circular cone has a base with radius $600$ and height $200\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $125$, 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 $375\sqrt{2}.$ Find the least distance that the fly could have crawled. (Solution)
• Problem 14 - 15: 6
Find the largest integer $n$ satisfying the following conditions: (i) $n^2$ can be expressed as the difference of two consecutive cubes; (ii) $2n + 79$ is a perfect square. (Solution)

APMO

Problem 1: 6.5 Problem 2: 7

Austrian MO

• Gebietswettbewerb Für Fortgeschrittene, Problems 1-4: 2
• Bundeswettbewerb Für Fortgeschrittene, Teil 1. Problems 1-4: 3
• Bundeswettbewerb Für Fortgeschrittene, Teil 2, Problems 1-6: 4

Indonesian MO

• Problem 1/5: 2.5
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? <url>viewtopic.php?t=294065 (Solution)</url>
• Problem 2/6: 3.5
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 $x$.<url>viewtopic.php?t=294067 (Solution)</url>

• Problem 3/7: 4.5
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 $b\nmid m$.

• Problem 4/8: 5.5
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.

ARML

• Individuals, Problem 1-5,7,9: 3
• Individuals, Problem 6,8: 4
• Individuals, Problem 10: 6.5
• Team/power, Problem 1-5: 3.5
• Team/power, Problem 6-10: 5

Balkan MO

• Problem 1: 6
Solve the equation $3^x - 5^y = z^2$ in positive integers.
• Problem 2: x
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: x
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

$(i)$ the distances $C_1C_2,...C_{95}C_{96}, C_{96}C_{1}$ are all equal to $\sqrt {13}$

$(ii)$ the closed broken line $C_1C_2...C_{96}C_1$ has a centre of symmetry?

• Problem 4: x
Denote by $S$ the set of all positive integers. Find all functions $f: S \rightarrow S$ such that

$f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2$ for all $m,n \in S$.

China TST

• Problem 1/4:
• Problem 2/5:
• Problem 3/6: 9.5
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.

Germany Bundeswettbewerb Mathematik

• Round 1, Problem 1: x
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 $2008$ 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 $ABC$ angle bisector $w_{\alpha},$ median $s_b$ and the altitude $h_c$ intersect in one point if $w_{\alpha},$ side $BC$ and the circle around foot of the altitude $h_c$ have vertex $A$ 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 $\{(0,0),(0,1),(1,0),(1,1)\}$. Given a finite number of moves can you yield the configuration $\{(0,0), (1,1), (3,0), (2, - 1)\}$ ? <url>viewtopic.php?p=1194636#1194636 (Solution)</url>
• Round 2, Problem 1: x
Determine all real $x$ satisfying the equation

$$\sqrt [5]{x^3 + 2x} = \sqrt [3]{x^5 - 2x}.$$ 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 $a,b,c$ chosen such that the quotients $\frac {bc}{b + c},$ $\frac {ca}{c + a}$ and $\frac {ab}{a + b}$ are integers. Prove that $a,b,c$ 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
On a bookcase there are $n \geq 3$ 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>

HMMT

• Individuals, Problem 1-5: 4
• Individuals, Problem 6-10: 7

IMO

• Problem 1/4: 6.5
Find all functions $f: (0, \infty) \mapsto (0, \infty)$ (so $f$ is a function from the positive real numbers) such that
$\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}$

for all positive real numbers $w,x,y,z,$ satisfying $wx = yz.$ (Solution)

• Problem 2/5: 7.5
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: 9.5
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 $P$. (<url>viewtopic.php?p=572824#572824 Solution</url>)

IMO Shortlist

• Problem 1-2: 5
• Problem 3-4: 7
• Problem 5+: 8.5+

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: 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 $n$.

Miklós Schweitzer

• Problem 1-3:
• Problem 4-6:
• Problem 7-9:
• Problem 10-12:

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)

Putnam

• Problem A/B,1-2: 6.5
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy = 1$ and both branches of the hyperbola $xy = - 1.$ (A set $S$ in the plane is called convex if for any two points in $S$ the line segment connecting them is contained in $S.$) (<url>viewtopic.php?p=978383#p978383 Solution</url>)
• Problem A/B,3-4: 7.5
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 $ab\le n$. (<url>viewtopic.php?p=383280#383280 Solution</url>)
• 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 $S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \phi,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}$. (<url>viewtopic.php?t=127810 Solution</url>)

USAMO

• Problem 1/4: 6.5
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
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: 8.5
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)

USAMTS

• Problem 1-2: 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: 5
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)

USA TST

(seems to vary more than other contests; estimates based on 08 and 09)

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

Scale

[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.

1. 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
2. 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).
3. 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.
4. Intermediate-leveled problem solvers, the most difficult questions on AMC 12s (#22-25s), more difficult AIME-styled questions #6-10
5. Difficult AIME problems (#10-13), others, simple proof-based problems (JBMO etc)
6. High-leveled AIME-styled questions, not requiring proofs (#12-15). Introductory-leveled Olympiad-level questions (#1-4s).
7. Intermediate-leveled Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.
8. High-level difficult Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc.