Revision as of 17:21, 31 May 2024

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 - IMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this.

Notes:

Multiple choice tests like AMC are rated as though they are free-response. Test-takers can use the answer choices as hints, and so correctly answer more AMC questions than Mathcounts or AIME problems of similar difficulty.
Some Olympiads are taken in 2 sessions, with 2 similarly difficult sets of questions, numbered as one set. For these the first half of the test (questions 1-3) is similar difficulty to the second half (questions 4-6).

Scale

1: Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, MATHCOUNTS Chapter, AMC 8 1-20, AMC 10 1-10, AMC 12 1-5, 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 (AMC 8 21-25, harder MATHCOUNTS States questions, AMC 10 11-20, AMC 12 5-15, AIME 1-3), traditional middle/high school word problems with extremely complex problem solving.

3: Advanced Beginner problems that require more creative thinking (harder MATHCOUNTS National questions, AMC 10 21-25, AMC 12 15-20, AIME 4-6).

4: Intermediate-level problems (AMC 12 21-25, AIME 7-9).

5: More difficult AIME problems (10-12), simple proof-based Olympiad-style problems (early JBMO questions, easiest USAJMO 1/4).

6: High-leveled AIME-styled questions (13-15). Introductory-leveled Olympiad-level questions (harder USAJMO 1/4 and easier USAJMO 2/5, easier USAMO and IMO 1/4).

7: Tougher Olympiad-level questions, may require more technical knowledge (harder USAJMO 2/5 and most USAJMO 3/6, extremely hard USAMO and IMO 1/4, easy-medium USAMO and IMO 2/5).

8: High-level Olympiad-level questions (medium-hard USAMO and IMO 2/5, easiest USAMO and IMO 3/6).

9: Expert Olympiad-level questions (average USAMO and IMO 3/6).

9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve (hard USAMO and IMO 3/6).

10: Historically hard problems, generally unsuitable for very hard competitions (such as the IMO) due to being exceedingly tedious, long, and difficult (e.g. very few students are capable of solving on a worldwide basis).

Examples

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? (2003 AMC 8, Problem 1)

1: How many integer values of $x$ satisfy $|x| < 3\pi$ ? (2021 Spring 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: A fair $6$ -sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? (2021 Spring AMC 10B, Problem 18)

2.5: $A$ , $B$ , $C$ 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: 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)

3.75: 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 2011/1)

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

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. (USAMO 2021/1, USAJMO 2021/2)

7: We say that a finite set $\mathcal{S}$ in the plane is balanced if, for any two different points $A$ , $B$ in $\mathcal{S}$ , there is a point $C$ in $\mathcal{S}$ such that $AC=BC$ . We say that $\mathcal{S}$ is centre-free if for any three points $A$ , $B$ , $C$ in $\mathcal{S}$ , there is no point $P$ in $\mathcal{S}$ such that $PA=PB=PC$ .

Show that for all integers $n\geq 3$ , there exists a balanced set consisting of $n$ points.
Determine all integers $n\geq 3$ for which there exists a balanced centre-free set consisting of $n$ points.

(IMO 2015/1)

7.5: 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 $x \neq 0$ . (USAMO 2014/2)

8: For each positive integer $n$ , the Bank of Cape Town issues coins of denomination $\frac1n$ . Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac{1}{2}$ , prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$ . (IMO 2014/5)

8.5: Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$ . The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$ , and $AB$ at $D, E,$ and $F$ , respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$ . Line $AR$ meets $\omega$ again at $P$ . The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$ .

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$ . (IMO 2019/6)

9: Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$ . (IMO 2022/3)

9.5: An anti-Pascal triangle 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 $1 + 2 + 3 + \dots + 2018$ ? (IMO 2018/3)

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 $\ell$ .) (IMO 2020/6)

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: 1-1.25
The coordinates of $\triangle ABC$ are $A(5,7)$ , $B(11,7)$ , and $C(3,y)$ , with $y>7$ . The area of $\triangle ABC$ is 12. What is the value of $y$ ? (Solution)
Problem 13 - Problem 25: 1.5-2
A small airplane has $4$ rows of seats with $3$ seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be $2$ adjacent seats in the same row for the couple? (Solution)

Mathcounts

Countdown: 0.5-2.
Sprint: 1-1.5 (school/chapter), 1.5-2 (State), 2-2.5 (National)
Target: 1-2 (school/chapter), 1.5-2.5 (State), 2-2.5 (National)
Team: 1-2.5 (school/chapter), 1-3 (State), 1.5-3.5 (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)

Isosceles trapezoid ABCD has parallel sides $AD$ and $BC$ , with $BC<AD$ and $AB=CD$ . There is a point P on the plane such that $PA=1$ , $PB=2$ , $PC=3$ , and $PD=4$ . What is $BC/AD$ ? (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

0.5-2

Introduction to Algebra by AoPS

1-3.5 Extremely basic book for AMC 10

Introduction to Counting and Probability by AoPS

1-3.5

Introduction to Number Theory by AoPS

1-3.5

Introduction to Geometry by AoPS

1-4.25

===105 Algebra by Awesome Math === 2-6

====== 112 y Awesome Math ===1-5 ===111 Algebra by Awesome Math ===1-6

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 (Easy):3-4
Consider polynomials $P(x)$ of degree at most $3$ , each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ . How many such polynomials satisfy $P(-1) = -9$ ? (Solution)

Problem 21-25(Harder): 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
An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, the ant does not return to its previous vertex, but chooses to move to one of the other two adjacent vertices. All choices are selected at random so that each of the possible moves is equally likely. The probability that after exactly $8$ moves that ant is at a vertex of the top face on the cube is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ (Solution)
Problem 10 - 12: 5-5.5
Triangle $ABC$ has side lengths $AB=7,BC=8,$ and $CA=9.$ Circle $\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Then $AK=\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ (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 11-17 (MS): 3-4.5
Problems 18-20 (MS): 4-4.75
Problems 1-10 (HS): 1.5-3.5
Problems 11-20 (HS): 3.5-4.75
Problems 21-30 (HS): 4.5-6

LMT

Easy Problems: 0.5
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

===[108 /Awesome Math ]]===2.5-8

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

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: 5
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

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$ , $CB_1=B_1A$ , $AC_1=C_1B$ , and $\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$ Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$

Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ all pass through two common points.

(Note: a scalene triangle is one where no two sides have equal length.)

https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]

IMO Shortlist

Problem 1-2: 5.5-7
Problem 3-4: 7-8
Problem 5+: 9-10

@@ Line 357: / Line 357: @@
 * 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]] ===

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