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 District Olympiad, 3
Tags: probability , real analysis
Let $\{x_n\}_{n\geq 0}$ be a sequence of real numbers which satisfy \[ (x_{n+1} - x_n)(x_{n+1}+x_n+1) \leq 0, \quad n\geq 0. \] a) Prove that the sequence is bounded; b) Is it possible that the sequence is not convergent?

1996 Polish MO Finals, 2
Tags: circumcircle , geometry
Let $P$ be a point inside a triangle $ABC$ such that $\angle PBC = \angle PCA < \angle PAB$. The line $PB$ meets the circumcircle of triangle $ABC$ at a point $E$ (apart from $B$). The line $CE$ meets the circumcircle of triangle $APE$ at a point $F$ (apart from $E$). Show that the ratio $\frac{\left|APEF\right|}{\left|ABP\right|}$ does not depend on the point $P$, where the notation $\left|P_1P_2...P_n\right|$ stands for the area of an arbitrary polygon $P_1P_2...P_n$.

2025 Kyiv City MO Round 1, Problem 5
Tags: number theory
Find all quadruples of positive integers \( (a, p, q, r) \), where \( p, q, r \) are prime numbers, such that the following equation holds: \[ p^2q^2 + q^2r^2 + r^2p^2 + 3 = 4 \cdot 13^a. \] [i]Proposed by Oleksii Masalitin[/i]

2004 Purple Comet Problems, 12
Tags: function
If $f(x, y) = xy + 2x + y + 1$, find $f(f(2, f(3, 4)), 5)$.

2023 Romania Team Selection Test, P2
Tags: number theory
Find all positive integers, such that there exist positive integers $a, b, c$, satisfying $\gcd(a, b, c)=1$ and $n=\gcd(ab+c, ac-b)=a+b+c$.

PEN S Problems, 23
Tags:
Observe that \[\frac{1}{1}+\frac{1}{3}=\frac{4}{3}, \;\; 4^{2}+3^{2}=5^{2},\] \[\frac{1}{3}+\frac{1}{5}=\frac{8}{15}, \;\; 8^{2}+{15}^{2}={17}^{2},\] \[\frac{1}{5}+\frac{1}{7}=\frac{12}{35}, \;\;{12}^{2}+{35}^{2}={37}^{2}.\] State and prove a generalization suggested by these examples.

2004 Romania Team Selection Test, 9
Tags: pigeonhole principle , linear algebra , matrix , inequalities , combinatorics
Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements. Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.

2015 China Second Round Olympiad, 1
Tags: inequalities , probabilistic method
Let $a_1, a_2, \ldots, a_n$ be real numbers.Prove that you can select $\varepsilon _1, \varepsilon _2, \ldots, \varepsilon _n\in\{-1,1\}$ such that$$\left( \sum_{i=1}^{n}a_{i}\right)^2 +\left( \sum_{i=1}^{n}\varepsilon _ia_{i}\right)^2 \leq(n+1)\left( \sum_{i=1}^{n}a^2_{i}\right).$$

2013 Online Math Open Problems, 34
Tags: number theory , relatively prime
For positive integers $n$, let $s(n)$ denote the sum of the squares of the positive integers less than or equal to $n$ that are relatively prime to $n$. Find the greatest integer less than or equal to \[ \sum_{n\mid 2013} \frac{s(n)}{n^2}, \] where the summation runs over all positive integers $n$ dividing $2013$. [i]Ray Li[/i]

2014 Balkan MO Shortlist, N4
Tags: number theory
A [i]special number[/i] is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with \[ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. \] Prove that i) there are infinitely many special numbers; ii) $2014$ is not a special number. [i]Romania[/i]

2004 Purple Comet Problems, 13
Tags:
How many three digit numbers are made up of three distinct digits?

2012 Flanders Math Olympiad, 4
Tags: geometry , angle bisector , angle chasing , angle
In $\vartriangle ABC, \angle A = 66^o$ and $| AB | <| AC |$. The outer bisector in $A$ intersects $BC$ in $D$ and $| BD | = | AB | + | AC |$. Determine the angles of $\vartriangle ABC$.

2019 Regional Competition For Advanced Students, 2
Tags: cyclic , parallel , equal segments , geometry
The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$.

2016 Costa Rica - Final Round, F3
Tags: digit , functional equation , functional , algebra
Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$, b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$, c) $f (10) = 0$. Find $f (2016).$

2011 AMC 12/AHSME, 3
Tags:
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$. How many dollars must LeRoy give to Bernardo so that they share the costs equally? $ \textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B $

1979 Romania Team Selection Tests, 1.
Tags: geometry , locus
Let $\triangle ABC$ be a triangle with $\angle BAC=60^\circ$, $M$ be a point in its interior and $A',\, B',\, C'$ be the orthogonal projections of $M$ on the sides $BC,\, CA,\, AB$. Determine the locus of $M$ when the sum $A'B+B'C+C'A$ is constant. [i]Horea Călin Pop[/i]

2025 Thailand Mathematical Olympiad, 2
Tags: combinatorics
A school sent students to compete in an academic olympiad in $11$ differents subjects, each consist of $5$ students. Given that for any $2$ different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least $4$ different subjects.

1973 Poland - Second Round, 6
Tags: number theory , greatest common divisor
Prove that for every non-negative integer $m$ there exists a polynomial w with integer coefficients such that $2^m$ is the greatest common divisor of the numbers $$ a_n = 3^n + w(n), n = 0, 1, 2, ....$$

2020 March Advanced Contest, 2
Tags: geometry , concyclic
An acute triangle \(ABC\) has circumcircle \(\Gamma\) and circumcentre \(O\). The incentres of \(AOB\) and \(AOC\) are \(I_b\) and \(I_c\) respectively. Let \(M\) be the the point on \(\Gamma\) such that \(MB = MC\) and \(M\) lies on the same side of \(BC\) as \(A\). Prove that the points \(M\), \(A\), \(I_b\), and \(I_c\) are concyclic.

2004 Federal Math Competition of S&M, 4
Tags: sequence
The sequence $(a_n)$ is given by $a_1 = x \in \mathbb{R}$ and $3a_{n+1} = a_n+1$ for $n \geq 1$. Set $A = \sum_{n=1}^\infty \Big[ a_n - \frac{1}{6}\Big]$, $B = \sum_{n=1}^\infty \Big[ a_n + \frac{1}{6}\Big]$. Compute the sum $A+B$ in terms of $x$.

2023 Durer Math Competition Finals, 2
Tags: number theory , diophantine equation
[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\ [b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.

2021 AIME Problems, 1
Tags:
Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$.)

2003 Balkan MO, 1
Tags: pigeonhole principle , algebra , polynomial , induction , number theory , greatest common divisor
Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

2003 Cuba MO, 1
Tags: algebra , polynomial , trinomial
The roots of the equation $x^2 + (3a + b)x + a^2 + 2b^2 = 0$ are $x_1$ and $x_2$ with $x_1 \ne x_2$. Determine the values of $a$ and $b$ so that the roots of the equation $ x^2 - 2a(3a + 2b)x + 5a^2b^2 + 4b^4 = 0$ let $x^2_1$ and $x^2_2$.

2016 Belarus Team Selection Test, 3
Tags: number theory , divisibility
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.