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 Estonia Math Open Junior Contests, 6
Tags: geometry , 3d geometry , quadratic , algebra
Find all real numbers with the following property: the difference of its cube and its square is equal to the square of the difference of its square and the number itself.

2011 Postal Coaching, 3
Tags: function , functional equation , algebra
Suppose $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that \[2f (f (x)) = (x^2 - x)f (x) + 4 - 2x\] for all real $x$. Find $f (2)$ and all possible values of $f (1)$. For each value of $f (1)$, construct a function achieving it and satisfying the given equation.

1931 Eotvos Mathematical Competition, 1
Tags: number theory
Let $p$ be a prime greater than $2$. Prove that $\frac{2}{p}$ can be expressed in exactly one way in the form $$\frac{1}{x}+\frac{1}{y}$$ where $x$ and $y$ are positive integers with $x > y$.

2011 Tournament of Towns, 2
Tags: geometry , right triangle
On side $AB$ of triangle $ABC$ a point $P$ is taken such that $AP = 2PB$. It is known that $CP = 2PQ$ where $Q$ is the midpoint of $AC$. Prove that $ABC$ is a right triangle.

2013 Israel National Olympiad, 2
Tags: combinatorics , number theory , maximum , set
Let $A=\{n\in\mathbb{Z}\mid 0<n<2013\}$. A subset $B\subseteq A$ is called [b]reduced[/b] if for any two numbers $x,y\in B$, we must have $x\cdot y \notin B$. For example, any subset containing the numbers $3,5,15$ cannot be reduced, and same for a subset containing $4,16$. [list=a] [*] Find the maximal size of a reduced subset of $A$. [*] How many reduced subsets are there with that maximal size? [/list]

2021 Israel TST, 2
Tags:
Find all functions $f:\mathbb{R}\to \mathbb{R}$ so that for any reals $x,y$ the following holds: \[f(x\cdot f(x+y))+f(f(y)\cdot f(x+y))=(x+y)^2\]

2013 Stanford Mathematics Tournament, 3
Tags:
Queen Jack likes a 5-card hand if and only if the hand contains only queens and jacks. Considering all possible 5-card hands that can come from a standard 52-card deck, how many hands does Queen Jack like?

2017 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , diophantine , diophantine equation
Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.

1935 Eotvos Mathematical Competition, 2
Tags: geometry , symmetry
Prove that a finite point set cannot have more than one center of symmetry.

2025 Bulgarian Spring Mathematical Competition, 10.3
Tags: table , perfect square , algebra
In the cell $(i,j)$ of a table $n\times n$ is written the number $(i-1)n + j$. Determine all positive integers $n$ such that there are exactly $2025$ rows not containing a perfect square.

1973 AMC 12/AHSME, 33
Tags:
When one ounce of water is added to a mixture of acid and water, the new mixture is $ 20\%$ acid. When one ounce of acid is added to the new mixture, the result is $ 33\frac13\%$ acid. The percentage of acid in the original mixture is $ \textbf{(A)}\ 22\% \qquad \textbf{(B)}\ 24\% \qquad \textbf{(C)}\ 25\% \qquad \textbf{(D)}\ 30\% \qquad \textbf{(E)}\ 33\frac13 \%$

2014 Grand Duchy of Lithuania, 3
Tags: combinatorics , table , coloring
In a table $n\times n$ some unit squares are coloured black and the other unit squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $n$?

2020 Jozsef Wildt International Math Competition, W59
Tags: inequalities
If $a_k>0~(k=1,2,\ldots,n)$ then prove that $$\sum_{\text{cyc}}\left(\frac{(a_1+a_2+\ldots+a_{n-1})^2}{a_n}+\frac{a_n^2}{a_1}\right)\ge\frac{n^2}2\sum_{k=1}^na_k$$ [i]Proposed by Mihály Bencze[/i]

2019 Cono Sur Olympiad, 2
Tags: number theory , digit
We say that a positive integer $M$ with $2n$ digits is [i]hypersquared[/i] if the following three conditions are met: [list] [*]$M$ is a perfect square. [*]The number formed by the first $n$ digits of $M$ is a perfect square. [*]The number formed by the last $n$ digits of $M$ is a perfect square and has exactly $n$ digits (its first digit is not zero). [/list] Find a hypersquared number with $2000$ digits.

2010 Contests, 1
Tags: modular arithmetic , number theory
Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$

2021 Science ON all problems, 4
Tags: geometry , angle chasing , circles , projective
$ABCD$ is a cyclic convex quadrilateral whose diagonals meet at $X$. The circle $(AXD)$ cuts $CD$ again at $V$ and the circle $(BXC)$ cuts $AB$ again at $U$, such that $D$ lies strictly between $C$ and $V$ and $B$ lies strictly between $A$ and $U$. Let $P\in AB\cap CD$.\\ \\ If $M$ is the intersection point of the tangents to $U$ and $V$ at $(UPV)$ and $T$ is the second intersection of circles $(UPV)$ and $(PAC)$, prove that $\angle PTM=90^o$.\\ \\ [i](Vlad Robu)[/i]

2005 China Team Selection Test, 3
Tags: modular arithmetic , number theory , binomial expansion , fermat number , power of 2
$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.

2012 Stanford Mathematics Tournament, 3
Tags: geometry , 3d geometry
Express $\frac{2^3-1}{2^3+1}\times\frac{3^3-1}{3^3+1}\times\frac{4^3-1}{4^3+1}\times\dots\times\frac{16^3-1}{16^3+1}$ as a fraction in lowest terms.

2013 AMC 12/AHSME, 23
Tags: search , modular arithmetic
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? ${ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 $

2014 Middle European Mathematical Olympiad, 8
Tags: modular arithmetic , quadratic , inequalities , geometry , 3d geometry , number theory
Determine all quadruples $(x,y,z,t)$ of positive integers such that \[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]

2015 Nordic, 2
Tags: prime number , number theory
Find the primes ${p, q, r}$, given that one of the numbers ${pqr}$ and ${p + q + r}$ is ${101}$ times the other.

2021 Kyiv City MO Round 1, 8.2
Tags: number theory , digit
Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada? [i]Proposed by Oleksii Masalitin[/i]

2024 Chile TST IMO, 1
Tags: combinatorics
Consider a set of \( n \geq 3 \) points in the plane where no three are collinear. Prove that the points can be labeled as \( P_1, P_2, \dots, P_n \) so that the angles \( \angle P_i P_{i+1} P_{i+2} \) are less than \( 90^\circ \) for all \( i \).

2002 Estonia National Olympiad, 2
Tags: acute , obtuse , orthocenter , angle , geometry
Let $ABC$ be a non-right triangle with its altitudes intersecting in point $H$. Prove that $ABH$ is an acute triangle if and only if $\angle ACB$ is obtuse.

2022 Grand Duchy of Lithuania, 4
Tags: number theory , perfect square
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.