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

2020 Malaysia IMONST 1, 5
Tags: last digit , number theory , modulo
Determine the last digit of $5^5+6^6+7^7+8^8+9^9$.

2000 AMC 8, 17
Tags:
The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b = \dfrac{a^2}{b}$. Determine $[(1\otimes 2)\otimes 3] - [1\otimes (2\otimes 3)]$. $\text{(A)}\ -\dfrac{2}{3} \qquad \text{(B)}\ -\dfrac{1}{4} \qquad \text{(C)}\ 0 \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{2}{3}$

2011 Kyiv Mathematical Festival, 2
Tags: number theory , sum of digits
Is it possible to represent number $2011... 2011$, where number $2011$ is written $20112011$ times, as a product of some number and sum of its digits?

2022 Stanford Mathematics Tournament, 9
Tags:
Let $f(x,y)=(\cos x+y\sin x)^2$. We may express $\text{max}_xf(x,y)$, the maximum value of $f(x,y)$ over all values of $x$ for a given fixed value of $y$, as a function of $y$, call it $g(y)$. Let the smallest positive value $x$ which achieves this maximum value of $f(x,y)$ for a given $y$ be $h(y)$. Compute \[\int_1^{2+\sqrt{3}}\frac{h(y)}{g(y)}\text{d}y.\]

2013 Balkan MO Shortlist, G3
Tags: geometry
Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $M,N$. A line $\ell$ is tangent to $\Gamma_1 ,\Gamma_2$ at $A$ and $B$, respectively. The lines passing through $A$ and $B$ and perpendicular to $\ell$ intersects $MN$ at $C$ and $D$ respectively. Prove that $ABCD$ is a parallelogram.

2007 Romania National Olympiad, 3
Tags: inequalities , geometry , 3d geometry , tetrahedron
a) In a triangle $ MNP$, the lenghts of the sides are less than $ 2$. Prove that the lenght of the altitude corresponding to the side $ MN$ is less than $ \sqrt {4 \minus{} \frac {MN^2}{4}}$. b) In a tetrahedron $ ABCD$, at least $ 5$ edges have their lenghts less than $ 2$.Prove that the volume of the tetrahedron is less than $ 1$.

1963 Swedish Mathematical Competition., 2
Tags: geometry , chessboard , max , square
The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

1987 Swedish Mathematical Competition, 2
Tags: arc , geometric inequalities , equal area , geometry
A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.

2010 Olympic Revenge, 3
Tags: geometric transformation , rotation , geometry
Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions: $i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$. $ii)$ There are no two lines of $S$ which are parallel.

2002 China Team Selection Test, 3
Tags: algebra , polynomial , number theory
The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

2023 ELMO Shortlist, G3
Tags: geometry
Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\). [i]Proposed by Karthik Vedula[/i]

2006 Grigore Moisil Urziceni, 3
Tags: number theory , numerical analysis , arithmetic sequence , integer sequence , sequence
Let be a sequence $ \left( b_n \right)_{n\ge 1} $ of integers, having the following properties: $ \text{(i)} $ the sequence $ \left( \frac{b_n}{n} \right)_{n\ge 1} $ is convergent. $ \text{(ii)} m-n|b_m-b_n, $ for any natural numbers $ m>n. $ Prove that there exists an index from which the sequence $ \left( b_n \right)_{n\ge 1} $ is an arithmetic one. [i]Cristinel Mortici[/i]

2016 Mathematical Talent Reward Programme, MCQ: P 11
Tags: geometry , rectangle , perimeter
In rectangle $ABCD$, $AD=1$, $P$ is on $AB$ and $DB$ and $DP$ trisect $\angle ADC$. What is the perimeter $\triangle BDP$ [list=1] [*] $3+\frac{\sqrt{3}}{3}$ [*] $2+\frac{4\sqrt{3}}{3}$ [*] $2+2\sqrt{2}$ [*] $\frac{3+3\sqrt{5}}{2}$ [/list]

2025 Belarusian National Olympiad, 11.6
Tags: geometry
Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$. Prove that $ZA=ZH$. [i]Vadzim Kamianetski[/i]

2020 Durer Math Competition Finals, 11
Tags: geometry , angle
The convex quadrilateral $ABCD$ has $|AB| = 8$, $|BC| = 29$, $|CD| = 24$ and $|DA| = 53$. What is the area of the quadrilateral if $\angle ABC + \angle BCD = 270^o$?

2015 Thailand TSTST, 2
Tags: inequalities
Let $a, b, c\in (0, 1)$ with $a + b + c = 1$. Prove that $$\frac{a^5+b^5}{a^3+b^3}+\frac{b^5+c^5}{b^3+c^3}+\frac{c^5+a^5}{c^3+a^3}\geq\frac{a}{8+b^3+c^3}+\frac{b}{8+c^3+a^3}+\frac{c}{8+a^3+b^3}.$$

2022 Nordic, 3
Tags: combinatorial game theory , number theory
Anton and Britta play a game with the set $M=\left \{ 1,2,\dots,n-1 \right \}$ where $n \geq 5$ is an odd integer. In each step Anton removes a number from $M$ and puts it in his set $A$, and Britta removes a number from $M$ and puts it in her set $B$ (both $A$ and $B$ are empty to begin with). When $M$ is empty, Anton picks two distinct numbers $x_1, x_2$ from $A$ and shows them to Britta. Britta then picks two distinct numbers $y_1, y_2$ from $B$. Britta wins if $(x_1x_2(x_1-y_1)(x_2-y_2))^{\frac{n-1}{2}}\equiv 1\mod n$ otherwise Anton wins. Find all $n$ for which Britta has a winning strategy.

1996 Czech and Slovak Match, 4
Tags: number theory , integer equation , function equation , algebra
Decide whether there exists a function $f : Z \rightarrow Z$ such that for each $k =0,1, ...,1996$ and for any integer $m$ the equation $f (x)+kx = m$ has at least one integral solution $x$.

2012 Tournament of Towns, 1
Tags: combinatorics
It is possible to place an even number of pears in a row such that the masses of any two neighbouring pears differ by at most $1$ gram. Prove that it is then possible to put the pears two in a bag and place the bags in a row such that the masses of any two neighbouring bags differ by at most $1$ gram.

2017 Switzerland - Final Round, 7
Tags: number theory , perfect square , diophantine , diophantine equation
Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$

2006 AIME Problems, 2
Tags:
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$.

Kharkiv City MO Seniors - geometry, 2015.11.3
Tags: geometry , rectangle , right angle , midpoint
In the rectangle $ABCD$, point $M$ is the midpoint of the side $BC$. The points $P$ and $Q$ lie on the diagonal $AC$ such that $\angle DPC = \angle DQM = 90^o$. Prove that $Q$ is the midpoint of the segment $AP$.

2022 Thailand Online MO, 10
Tags: function , algebra , number theory , functional equation
Let $\mathbb{Q}$ be the set of rational numbers. Determine all functions $f : \mathbb{Q}\to\mathbb{Q}$ satisfying both of the following conditions. [list=disc] [*] $f(a)$ is not an integer for some rational number $a$. [*] For any rational numbers $x$ and $y$, both $f(x + y) - f(x) - f(y)$ and $f(xy) - f(x)f(y)$ are integers. [/list]

2025 Belarusian National Olympiad, 9.5
Tags: combinatorics
Polina and Yan have $n$ cards, on the first card on one side $1$ is written, on the other side $n+1$, on the second card on one side $2$ is written, on the other side $n+2$, etc. Polina laid all cards in a circle in some order. Yan wants to turn some cards such that the numbers on the top sides of adjacent cards were not coprime. For every positive integer $n \geq 3$ determine can Yan accomplish that regardless of the actions of Polina. [i]M. Shutro[/i]

2002 Singapore Senior Math Olympiad, 1
Tags: function , functional equation , number theory
Let $f: N \to N$ be a function satisfying the following: $\bullet$ $f(ab) = f(a)f(b)$, whenever the greatest common divisor of $a$ and $b$ is $1$. $\bullet$ $f(p + q) = f(p)+ f(q)$ whenever $p$ and $q$ are primes. Determine all possible values of $f(2002)$. Justify your answers.