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.

AND:
OR:
NO:

Found problems: 85335

2014 IMO Shortlist, A5
Tags: polynomial , algebra , functional inequality
Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

2005 MOP Homework, 4
Tags: inequalities
Let $x_1$, $x_2$, ..., $x_5$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5=5$. Determine the maximum value of $x_1x_2+x_2x_3+x_3x_4+x_4x_5$.

1988 USAMO, 3
Tags: pigeonhole principle , ceiling function , function , boolean lattice method
A function $f(S)$ assigns to each nine-element subset of $S$ of the set $\{1,2,\ldots, 20\}$ a whole number from $1$ to $20$. Prove that regardless of how the function $f$ is chosen, there will be a ten-element subset $T\subset\{1,2,\ldots, 20\}$ such that $f(T - \{k\})\neq k$ for all $k\in T$.

2008 Baltic Way, 4
Tags: polynomial , algebra
The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$. Show that there is no integer $x$ such that $-6\le P(x)\le 4$ or $6\le P(x)\le 16$.

2020 Estonia Team Selection Test, 3
Tags: number theory , remainder
The prime numbers $p$ and $q$ and the integer $a$ are chosen such that $p> 2$ and $a \not\equiv 1$ (mod $q$), but $a^p \equiv 1$ (mod $q$). Prove that $(1 + a^1)(1 + a^2)...(1 + a^{p - 1})\equiv 1$ (mod $q$) .

2024 Lusophon Mathematical Olympiad, 5
Tags: combinatorics
In a $9\times9$ board, the squares are labeled from 11 to 99, with the first digit indicating the row and the second digit indicating the column. One would like to paint the squares in black or white in a way that each black square is adjacent to at most one other black square and each white square is adjacent to at most one other white square. Two squares are adjacent if they share a common side. How many ways are there to paint the board such that the squares $44$ and $49$ are both black?

2023 CCA Math Bonanza, T4
Tags:
Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Let $E$ be the foot from $B$ to $AC$ and $F$ be the foot from $C$ to $AB.$ Denote $M$ the midpoint of $BC.$ The circumcircles of $\triangle BMF$ and $\triangle CME$ meet at another point $G.$ Compute the length of $GC.$ [i]Team #4[/i]

2011 HMNT, 3
Tags: combinatorics
Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur?

MOAA Team Rounds, 2021.1
Tags: team
The value of \[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2004 ITAMO, 5
Tags: number theory
Decide if the following statement is true or false: For every sequence $\{x_n\}_{n\in \mathbb{N}}$ of non-negative real numbers, there exist sequences $\{a_n\}_{n\in\mathbb{N}}$ and $\{b_n\}_{n\in\mathbb{N}}$ of non-negative real numbers such that: (a) $x_n = a_n + b_n$ for all $n$; (b) $a_1 + \cdots + a_n \le n$ for infinitely many values of $n$; (c) $b_1 + \cdots + b_n \le n$ for infinitely many values of $n$.

Gheorghe Țițeica 2025, P2
Tags: function
Let $n\geq 2$ and consider the functions $f,g:\{1,2,\dots ,n\}\rightarrow\{1,2,\dots ,n\}$ such that $$g(k)=|\{i\mid f(i)\leq f(k)\}|$$ for all $1\leq k\leq n$. [list=a] [*] Show that $f$ is bijective if and only if $g$ is bijective. [*] If $g$ is a given function, find how many functions $f$ (in terms of $g$) satisfy the hypothesis. [/list] [i]Silviu Cristea[/i]

2020 USOMO, 1
Tags: minimize , geometry , olympiad geometry , area of a triangle
Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$. A variable point $X$ is chosen on minor arc $AB$ of $\omega$, and segments $CX$ and $AB$ meet at $D$. Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$, respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized. [i]Proposed by Zuming Feng[/i]

2016 Indonesia TST, 6
Tags: geometry
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

LMT Speed Rounds, 2010.4
Tags:
Determine the largest positive integer that is a divisor of all three of $A=2^{2010}\times3^{2010}, B=3^{2010}\times5^{2010},$ and $C=5^{2010}\times2^{2010}.$

2020 Greece Team Selection Test, 1
Tags: functional , functional equation , algebra
Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

2021 HMNT, 6
Tags: number theory
Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying $$3a + 5b \equiv 19 \,\,\, (mod \,\,\, n + 1)$$ $$4a + 2b \equiv 25 \,\,\, (mod \,\,\, n + 1)$$ Find $ 2a + 6b$.

2001 AMC 12/AHSME, 3
Tags:
The state income tax where Kristin lives is levied at the rate of $ p \%$ of the first $ \$28000$ of annual income plus $ (p \plus{} 2) \%$ of any amount above $ \$28000$. Kristin noticed that the state income tax she paid amounted to $ (p \plus{} 0.25) \%$ of her annual income. What was her annual income? $ \textbf{(A)} \ \$28000 \qquad \textbf{(B)} \ \$32000 \qquad \textbf{(C)} \ \$35000 \qquad \textbf{(D)} \ \$42000 \qquad \textbf{(E)} \ \$56000$

2016 NIMO Problems, 7
Tags:
Suppose $a$, $b$, $c$, and $d$ are positive real numbers which satisfy the system of equations \[\begin{aligned} a^2+b^2+c^2+d^2 &= 762, \\ ab+cd &= 260, \\ ac+bd &= 365, \\ ad+bc &= 244. \end{aligned}\] Compute $abcd.$ [i]Proposed by Michael Tang[/i]

2010 ISI B.Math Entrance Exam, 1
Tags: floor function , modular arithmetic , calendar
Prove that in each year , the $13^{th}$ day of some month occurs on a Friday .

1995 Tournament Of Towns, (461) 6
Tags: geometry , combinatorics , combinatorial geometry
Does there exist a nonconvex polyhedron such that not one of its vertices is visible from a point $M$ outside it? (The polyhedron is made out of an opaque material.) (AY Belov, S Markelov)

1999 Bosnia and Herzegovina Team Selection Test, 5
Tags: set , product , sum , number theory , combinatorics
For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that $a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$ $b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$ where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$

1979 IMO Longlists, 19
Tags: number theory
For $k = 1, 2, \ldots$ consider the $k$-tuples $(a_1, a_2, \ldots, a_k)$ of positive integers such that \[a_1 + 2a_2 + \cdots + ka_k = 1979.\] Show that there are as many such $k$-tuples with odd $k$ as there are with even $k$.

2024 CMIMC Integration Bee, 15
Tags: calculus , integration
\[\int_0^\infty 1+\cos\left(\tfrac 1{\sqrt x}\right)-2\cos\left(\tfrac 1{\sqrt {2x}}\right)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2014 IFYM, Sozopol, 8
Tags: algebra , inequality , triangle inequality , inequalities
Prove that, if $a,b,c$ are sides of a triangle, then we have the following inequality: $3(a^3 b+b^3 c+c^3 a)+2(ab^3+bc^3+ca^3 )\geq 5(a^2 b^2+a^2 c^2+b^2 c^2 )$.

2015 Kyiv Math Festival, P3
Tags: positive integer , number theory , sum
Is it true that every positive integer greater than $50$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?