Revision as of 11:30, 27 May 2020

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 Olympiads such as our very own AoPS!

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.

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

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
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), 2-3 (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: 0.5-1.5
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
Points $A$ and $B$ lie on a circle centered at $O$ , and $\angle AOB = 60^\circ$ . A second circle is internally tangent to the first and tangent to both $\overline{OA}$ and $\overline{OB}$ . What is the ratio of the area of the smaller circle to that of the larger circle? (Solution)
Problem 21-25: 4-5.5
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: 3-3.5
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
Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$ . (Solution)
Problem 10 - 12: 5-5.5
Let $R$ be the set of all possible remainders when a number $2^n$ , $n$ 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: 6-6.5
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: 5
Guts: 3.5-5.25

CEMC Euclid

Problem 1-6: 1-3
Problem 7-10: 3-5

Purple Comet

Problems 1-10 (MS): 2-3
Problems 11-20 (MS): 3-4
Problems 1-10 (HS): 1-2
Problems 11-20 (HS): 2-3
Problems 21-30 (HS): 3-4

Philippine Mathematical Olympiad Qualifying Round

Problem 1-15: 2
Problem 16-25: 3
Problem 26-30: 4

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? <url>viewtopic.php?t=294065 (Solution)</url>
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 $x$ .<url>viewtopic.php?t=294067 (Solution)</url>

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 $b\nmid m$ . <url>viewtopic.php?t=294068 (Solution)</url>

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. <url>viewtopic.php?t=294069 (Solution)</url>

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. (<url>viewtopic.php?p=903856#903856 Solution</url>)
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. (<url>viewtopic.php?p=291301#291301 Solution</url>)
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 $\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}$ . (<url>viewtopic.php?p=828841#p828841 Solution</url>

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
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

Ibero American 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

$(i)$ 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: 8
Denote by $S$ the set of all positive integers. Find all functions $f: S \rightarrow S$ such that

$f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2$ 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: 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: 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: 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

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

Problem 1/4/7: 7
Problem 2/5/8: 8
Problem 3/6/9: 9.5

Putnam

Problem A/B,1-2: 7
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 $S.$ ) (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 $S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}$ . (<url>viewtopic.php?t=127810 Solution</url>)

China TST

Problem 1/4: 7-8
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: 8.5-9
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 $P$ . (<url>viewtopic.php?p=572824#572824 Solution</url>)

IMO Shortlist

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

@@ Line 205: / Line 205: @@
 ==Olympiad Competitions==
-This category consists of standard Olympiad competitions, usually ones from national Olympiads.  Average difficulty is from 5.5 to 8.  A full list is available [http://artofproblemsolving.com/wiki/index.php?title=Special%3ASearch&search=Category%3AOlympiad+mathematics+competitions here].
+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]] ===

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