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

2014 Saudi Arabia GMO TST, 2
Tags: number theory , greatest common divisor , gcd
Let $p \ge 2$ be a prime number and $\frac{a_p}{b_p}= 1 +\frac12+ .. +\frac{1}{p^2 -1}$, where $a_p$ and $b_p$ are two relatively prime positive integers. Compute gcd $(p, b_p)$.

2023-IMOC, C4
Tags: combinatorics
A ghost leg is a game with some vertical lines and some horizontal lines. A player starts at the top of the vertical line and go downwards, and always walkthrough a horizontal line if he encounters one. We define a layer is some horizontal line with the same height and has no duplicated endpoints. Find the smallest number of layers needed to grant that you can walk from $(1, 2, \ldots , n)$ on the top to any permutation $(\sigma_1, \sigma_2, \ldots, \sigma_n)$ on the bottom.

2006 Princeton University Math Competition, 8
Tags: algebra
Evaluate the sum $$\sum_{n=0}^{\infty}\frac{5n+7}{6^n}$$

2015 Junior Balkan Team Selection Tests - Moldova, 8
Tags: least common multiple , lcm , number theory
Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities: $$[a, b] =1000, [b, c] = 2000, [c, a] =2000.$$ ([x, y]represents the least common multiple of positive integers x,y)

2016 India PRMO, 1
Tags: algebra , diophantine equation , diophantine
Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$.

2001 Manhattan Mathematical Olympiad, 5
Tags:
Factorize the expression $a^3 + b^3 + c^3 - 3abc$.

2009 Moldova Team Selection Test, 3
Tags: function , floor function , combinatorics
[color=darkblue]Weightlifter Ruslan has just finished the exercise with a weight, which has $ n$ small weights on one side and $ n$ on the another. At each stage he takes some weights from one of the sides, such that at any moment the difference of the numbers of weights on the sides does not exceed $ k$. What is the minimal number of stages (in function if $ n$ and $ k$), which Ruslan need to take off all weights..[/color]

1990 Romania Team Selection Test, 5
Tags: circumcircle , circles , area , geometric inequalities , inequalities , geometry
Let $O$ be the circumcenter of an acute triangle $ABC$ and $R$ be its circumcenter. Consider the disks having $OA,OB,OC$ as diameters, and let $\Delta$ be the set of points in the plane belonging to at least two of the disks. Prove that the area of $\Delta$ is greater than $R^2/8$.

2010 Middle European Mathematical Olympiad, 4
Tags: inequalities , number theory
Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]

2011 USAMO, 2
Tags: invariant , algorithm , rotation , geometry
An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.

1992 IMO Shortlist, 9
Tags: algebra , polynomial , functional equation , iteration
Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.

2011 Today's Calculation Of Integral, 719
Tags: calculus , integration , trigonometry , calculus computations
Compute $\int_0^x \sin t\cos t\sin (2\pi\cos t)\ dt$.

2000 Argentina National Olympiad, 2
Tags: geometry , angle bisector , median , parallel , perpendicular
Given a triangle $ABC$ with side $AB$ greater than $BC$, let $M$ be the midpoint of $AC$ and $L$ be the point at which the bisector of angle $\angle B$ intersects side $AC$. The line parallel to $AB$, which intersects the bisector $BL$ at $D$, is drawn by $M$, and the line parallel to the side $BC$ that intersects the median $BM$ at $E$ is drawn by $L$. Show that $ED$ is perpendicular to $BL$.

2024 ELMO Shortlist, N6
Tags: number theory
Given a positive integer whose base-$10$ representation is $\overline{d_k\ldots d_0}$ for some integer $k \geq 0$, where $d_k \neq 0$, a move consists of selecting some integers $0 \leq i \leq j \leq k$, such that the digits $d_j,\ldots,d_i$ are not all $0$, erasing them from $n$, and replacing them with a divisor of $\overline{d_j\ldots d_i}$ (this divisor need not have the same number of digits as $\overline{d_j\ldots d_i}$). Prove that for all sufficiently large even integers $n$, we may apply some sequence of moves to $n$ to transform it into $2024$. [i]Allen Wang[/i]

2025 CMIMC Geometry, 3
Tags: geometry
Let $AB$ be a segment of length $1.$ Let $\odot A, \odot B$ be circles with radius $\overline{AB}$ centered at $A, B.$ Denote their intersection points $C, D.$ Draw circles $\odot C, \odot D$ with radius $\overline{CD}.$ Denote their intersection points $C, D.$ Draw circles $\odot C, \odot D$ with radius $\overline{CD}.$ Denote the intersection points of $\odot C$ and $\odot D$ by $E, F.$ Draw circles $\odot E, \odot F$ with radius $\overline{EF}$ and denote their intersection points $G, H.$ Compute the area of the pentagon $ACFHE.$

2022 Balkan MO Shortlist, A4
Tags: functional equation , algebra
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

1998 Romania National Olympiad, 1
Tags: calculus , integration , function , real analysis
Suppose that $a,b\in\mathbb{R}^+$ which $a+b<1$ and $f:[0,+\infty) \rightarrow [0,+\infty) $ be the increasing function s.t. $\forall x\geq 0 ,\int _0^x f(t)dt=\int _0^{ax} f(t)dt+\int _0^{bx} f(t)dt$. Prove that $\forall x\geq 0 , f(x)=0$

2014 SDMO (Middle School), 2
Tags: probability , expected value
A dog has three trainers: [list] [*]The first trainer gives him a treat right away. [*]The second trainer makes him jump five times, then gives him a treat. [*]The third trainer makes him jump three times, then gives him no treat. [/list] The dog will keep picking trainers with equal probability until he gets a treat. (The dog's memory isn't so good, so he might pick the third trainer repeatedly!) What is the expected number of times the dog will jump before getting a treat?

2017 Math Prize for Girls Problems, 16
Tags:
Samantha is about to celebrate her sweet 16th birthday. To celebrate, she chooses a five-digit positive integer of the form SWEET, in which the two E's represent the same digit but otherwise the digits are distinct. (The leading digit S can't be 0.) How many such integers are divisible by 16?

1986 Tournament Of Towns, (116) 4
Tags: algebra , continuous , analysis , function , functional equation , functional
The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous. (A.I. Plotkin, Leningrad)

2021 USAMO, 5
Tags: algebra
Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations: \begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\ &\vdots & &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*}

Novosibirsk Oral Geo Oly IX, 2020.4
Tags: equal angles , square
Points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ of square $ABCD$, respectively. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

1996 Greece Junior Math Olympiad, 2
Tags: midpoint , geometry , area
In a triangle $ABC$ let $D,E,Z,H,G$ be the midpoints of $BC,AD,BD,ED,EZ$ respectively. Let $I$ be the intersection of $BE,AC$ and let $K$ be the intersection of $HG,AC$. Prove that: a) $AK=3CK$ b) $HK=3HG$ c) $BE=3EI$ d) $(EGH)=\frac{1}{32}(ABC)$ Notation $(...)$ stands for area of $....$

1966 AMC 12/AHSME, 27
Tags:
At his usual rate a man rows $15$ miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream's current is: $\text{(A)} \ 2 \qquad \text{(B)} \ \frac52 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ \frac72 \qquad \text{(E)} \ 4$

2014 Contests, 2
Tags: tangent , geometry , concurrency , tangential quadrilateral
A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.