This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2006 Putnam, B2
Tags: inequalities , pigeonhole principle
Prove that, for every set $X=\{x_{1},x_{2},\dots,x_{n}\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that \[\left|m+\sum_{s\in S}s\right|\le\frac1{n+1}\]

2018 HMNT, 9
Tags: probability , expected value
$20$ players are playing in a Super Mario Smash Bros. Melee tournament. They are ranked $1-20$, and player $n$ will always beat player $m$ if $n<m$. Out of all possible tournaments where each player plays $18$ distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players that win the same number of games.

2023 Azerbaijan IZhO TST, 4
Tags: number theory
A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.

May Olympiad L1 - geometry, 2005.4
Tags: rectangle , equilateral triangle , paper , geometry
There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.

2025 Kosovo National Mathematical Olympiad`, P2
Tags: inequality , algebra , inequalities
Let $x$ and $y$ be real numbers where at least one of them is bigger than $2$ and $xy+4 > 2(x+y)$ holds. Show that $xy>x+y$.

2024 UMD Math Competition Part I, #25
Tags: geometry
An equilateral triangle $T$ and a circle $C$ are on the same plane. Suppose each side length of $T$ is $6\sqrt3$ and the radius of $C$ is $2.$ The distance between the centers of $T$ and $C$ is $15.$ For every two points $X$ on $T$ and $Y$ on $C,$ let $M(X, Y)$ be the midpoint of segment $\overline{XY}.$ The points $M(X, Y)$ as $X$ varies on $T$ and $Y$ varies on $C$ create a region whose area is $A.$ Find $A.$ \[\mathrm a. ~\pi + 14\sqrt3 \qquad \mathrm b. ~3\pi + 10\sqrt3 \qquad \mathrm c. ~4\pi+9\sqrt3 \qquad\mathrm d. ~\pi + 15\sqrt3 \qquad\mathrm e. ~4\pi+6\sqrt3\]

2014 France Team Selection Test, 3
Tags: number theory , prime divisor , polynomial
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

1986 Tournament Of Towns, (113) 7
Tags: combinatorics
Thirty pupils from the same class decided to exchange visits. Any pupil may make several visits during one evening, but must stay home if he is receiving guests that evening. Prove that in order that each pupil visit each of his classmates (a) four evenings are not enough (b) five evenings are not enough (c) ten evenings are enough (d) even seven evenings are enough

2020 USMCA, 6
Tags:
Let $P$ be a non-constant polynomial with integer coefficients such that if $n$ is a perfect power, so is $P(n)$. Prove that $P(x) = x$ or $P$ is a perfect power of a polynomial with integer coefficients. A perfect power is an integer $n^k$, where $n \in \mathbb Z$ and $k \ge 2$. A perfect power of a polynomial is a polynomial $P(x)^k$, where $P$ has integer coefficients and $k \ge 2$.

Kyiv City MO Seniors 2003+ geometry, 2014.10.4.1
Tags: geometry , angle bisector , angle
In the triangle $ABC$ the side $AC = \tfrac {1} {2} (AB + BC) $, $BL$ is the bisector $\angle ABC$, $K, \, \, M $ - the midpoints of the sides $AB$ and $BC$, respectively. Find the value $\angle KLM$ if $\angle ABC = \beta$

1998 Harvard-MIT Mathematics Tournament, 3
Tags: calculus , geometry , asymptote , integration
Find the area of the region bounded by the graphs $y=x^2$, $y=x$, and $x=2$.

1997 IMO Shortlist, 3
Tags: vector , combinatorics , combinatorial geometry , extremal combinatorics , geometry
For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$

2022 District Olympiad, P4
Tags:
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following relationship for all $x,y\in\mathbb{R}:$\[f(f(y-x)-xf(y))+f(x)=y\cdot(1-f(x)).\]

2012 Online Math Open Problems, 17
Tags:
Find the number of integers $a$ with $1\le a\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation \[x^2(x^2+2z) - y^2(y^2+2z)=a.\] [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]$x,y,z$ are not necessarily distinct.[/list][/hide]

2023 MMATHS, 12
Tags:
Let $ABC$ be a triangle with incenter $I,$ circumcenter $O,$ and $A$-excenter $J_A.$ The incircle of $\triangle{ABC}$ touches side $BC$ at a point $D.$ Lines $OI$ and $J_AD$ meet at a point $K.$ Line $AK$ meets the circumcircle of $\triangle{ABC}$ again at a point $L \neq A.$ If $BD=11, CD=5,$ and $AO=10,$ the length of $DL$ can be expressed as $\tfrac{m\sqrt{p}}{n},$ where $m,n,p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p.$

2022 China Team Selection Test, 6
Tags: number theory , divisibility , divisor
Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$, it holds that $a \nmid b$ and $b \nmid a$. Show that \[ \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}. \]

2001 Romania National Olympiad, 4
Tags: 3d geometry , geometry
In the cube $ABCDA'B'C'D'$, with side $a$, the plane $(AB'D')$ intersects the planes $(A'BC),(A'CD),(A'DB)$ after the lines $d_1,d_2$ and $d_3$ respectively. a) Show that the lines $d_1,d_2,d_3$ intersect pairwise. b) Determine the area of the triangle formed by these three lines.

2008 Mathcenter Contest, 10
Tags: combinatorics , this problem is a nightmare
One test is a multiple choice test with $5$ questions, each with $4$ options, $2000$ candidates, each choosing only one answer for each item.Find the smallest possible integer $n$ that gives a student's answer sheet the following properties: In the student's answer sheet $n$, there are four sheets in it. Any two of the four tiles have exactly the same three answers. [i](tatari/nightmare)[/i]

2014 Iran MO (3rd Round), 2
Tags: geometric transformation , reflection , circumcircle , angle bisector , geometry
$\triangle{ABC}$ is isosceles$(AB=AC)$. Points $P$ and $Q$ exist inside the triangle such that $Q$ lies inside $\widehat{PAC}$ and $\widehat{PAQ} = \frac{\widehat{BAC}}{2}$. We also have $BP=PQ=CQ$.Let $X$ and $Y$ be the intersection points of $(AP,BQ)$ and $(AQ,CP)$ respectively. Prove that quadrilateral $PQYX$ is cyclic. [i](20 Points)[/i]

2015 Tournament of Towns, 2
Tags: geometry , geometric transformation , reflection , circumcircle
A point $X$ is marked on the base $BC$ of an isosceles $\triangle ABC$, and points $P$ and $Q$ are marked on the sides $AB$ and $AC$ so that $APXQ$ is a parallelogram. Prove that the point $Y$ symmetrical to $X$ with respect to line $PQ$ lies on the circumcircle of the $\triangle ABC$. [i]($5$ points)[/i]

Durer Math Competition CD Finals - geometry, 2015.C4
Tags: geometry , perimeter , arc , max
On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible?

Ukrainian TYM Qualifying - geometry, I.7
Tags: ratio , geometry , polygon , inscribed , area
Given a natural number $n> 3$. On the plane are considered convex $n$ - gons $F_1$ and $F_2$ such that on each side of $F_1$ lies one vertex of $F_2$ and no two vertices $F_1$ and $F_2$ coincide. For each $n$, determine the limits of the ratio of the areas of the polygons $F_1$ and $F_2$. For each $n$, determine the range of the areas of the polygons $F_1$ and $F_2$, if $F_1$ is a regular $n$-gon. Determine the set of values of this value for other partial cases of the polygon $F_1$.

MBMT Guts Rounds, 2015.15
Tags:
Two (not necessarily different) numbers are chosen independently and at random from $\{1, 2, 3, \dots, 10\}$. On average, what is the product of the two integers? (Compute the expected product. That is, if you do this over and over again, what will the product of the integers be on average?)

2007 Tournament Of Towns, 3
Tags: combinatorics , combinatorial geometry
Michael is at the centre of a circle of radius $100$ metres. Each minute, he will announce the direction in which he will be moving. Catherine can leave it as is, or change it to the opposite direction. Then Michael moves exactly $1$ metre in the direction determined by Catherine. Does Michael have a strategy which guarantees that he can get out of the circle, even though Catherine will try to stop him?

1984 IMO Longlists, 12
Tags: number theory , equation , polynomial , diophantine equation
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]