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

2009 Indonesia TST, 1
Tags: modular arithmetic , induction , number theory
Prove that for all odd $ n > 1$, we have $ 8n \plus{} 4|C^{4n}_{2n}$.

2001 Brazil National Olympiad, 4
Tags: inequalities , algebra , function , trigonometry
A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)

MOAA Team Rounds, 2018.1
Tags: geometry , team
In $\vartriangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\vartriangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a prime. Determine the value of $m + n + p$.

2017 NIMO Problems, 6
Tags:
Let $n$ be a positive integer, and let $S_n = \{1, 2, \ldots, n\}$. For a permutation $\sigma$ of $S_n$ and an integer $a \in S_n$, let $d(a)$ be the least positive integer $d$ for which \[\underbrace{\sigma(\sigma(\ldots \sigma(a) \ldots))}_{d \text{ applications of } \sigma} = a\](or $-1$ if no such integer exists). Compute the value of $n$ for which there exists a permutation $\sigma$ of $S_n$ satisfying the equations \[\begin{aligned} d(1) + d(2) + \ldots + d(n) &= 2017, \\ \frac{1}{d(1)} + \frac{1}{d(2)} + \ldots + \frac{1}{d(n)} &= 2. \end{aligned}\] [i]Proposed by Michael Tang[/i]

2005 AMC 10, 9
Tags: probability
One fair die has faces $ 1$, $ 1$, $ 2$, $ 2$, $ 3$, $ 3$ and another has faces $ 4$, $ 4$, $ 5$, $ 5$, $ 6$, $ 6$. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd? $ \textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{4}{9}\qquad \textbf{(C)}\ \frac{1}{2}\qquad \textbf{(D)}\ \frac{5}{9}\qquad \textbf{(E)}\ \frac{2}{3}$

2007 China Team Selection Test, 2
Tags: modular arithmetic , algebra
Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$

2012 Hanoi Open Mathematics Competitions, 5
Tags: functional equation , algebra
Let $f(x)$ be a function such that $f(x)+2f\left(\frac{x+2010}{x-1}\right)=4020 - x$ for all $x \ne 1$. Then the value of $f(2012)$ is (A) $2010$, (B) $2011$, (C) $2012$, (D) $2014$, (E) None of the above.

2017 ISI Entrance Examination, 2
Tags: geometry
Consider a circle of radius $6$. Let $B,C,D$ and $E$ be points on the circle such that $BD$ and $CE$, when extended, intersect at $A$. If $AD$ and $AE$ have length $5$ and $4$ respectively, and $DBC$ is a right angle, then show that the length of $BC$ is $\frac{12+9\sqrt{15}}{5}$.

1991 Greece National Olympiad, 1
Tags: algebra , polynomial
Find all polynomials $P(x)$ , such that $$P(x^3+1)=\left(P (x+1)\right)^3$$

Today's calculation of integrals, 859
Tags: geometry , calculus , derivative , integration , calculus computations
In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

2010 AMC 10, 19
Tags: perpendicular bisector , trigonometry , pythagorean theorem , quadratic , trig identities , law of cosines , geometry
A circle with center $ O$ has area $ 156\pi$. Triangle $ ABC$ is equilateral, $ \overline{BC}$ is a chord on the circle, $ OA \equal{} 4\sqrt3$, and point $ O$ is outside $ \triangle ABC$. What is the side length of $ \triangle ABC$? $ \textbf{(A)}\ 2\sqrt3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 4\sqrt3 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18$

2008 Princeton University Math Competition, A9
Tags: number theory
Find the number of positive integer solutions of $(x^2 + 2)(y^2 + 3)(z^2 + 4) = 60xyz$.

2008 Bulgaria Team Selection Test, 1
Tags: graph theory , induction , hamiltonian path , chessboard , combinatorics
Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

2011 Junior Macedonian Mathematical Olympiad, 3
Tags:
All numbers from $1$ to $32$ a written on the stars from the picture below, such that each number is written once. Can all sums of the numbers written in each square that is not divided in smaller squares be equal?

1980 All Soviet Union Mathematical Olympiad, 293
Tags: geometry , combinatorial geometry , plane , vector
Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

2010 Saint Petersburg Mathematical Olympiad, 7
Tags: geometry
Incircle of $ABC$ tangent $AB,AC,BC$ in $C_1,B_1,A_1$. $AA_1$ intersect incircle in $E$. $N$ is midpoint $B_1A_1$. $M$ is symmetric to $N$ relatively $AA_1$. Prove that $\angle EMC= 90$

2013 Dutch IMO TST, 3
Tags: prime number , number theory
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.

2014 Turkey Junior National Olympiad, 3
Tags: pigeonhole principle , combinatorics
There are $2014$ balls with $106$ different colors, $19$ of each color. Determine the least possible value of $n$ so that no matter how these balls are arranged around a circle, one can choose $n$ consecutive balls so that amongst them, there are $53$ balls with different colors.

1986 IMO Longlists, 56
Tags: geometry
Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed into a circle with center $O$. Consider the circular arc with endpoints $A_1,A_6$ not containing $A_2$. For any point $M$ of that arc denote by $h_i$ the distance from $M$ to the line $A_iA_{i+1} \ (1 \leq i \leq 5)$. Construct $M$ such that the sum $h_1 + \cdots + h_5$ is maximal.

2024 Oral Moscow Geometry Olympiad, 6
Tags: geometry
An unequal acute-angled triangle $ABC$ with an orthocenter $H$ is given, $M$ is the midpoint of side $BC$. Points $K$ and $L$ lie on a line passing through $H$ and perpendicular to $AM$ such a $KB$ and $LC$ perpendicular to $BC$. Point $N$ lies on the line $HM$, and the lines $AN$ and $AH$ are symmetric with respect to the line $AM$. Prove that a circle with a diameter $AN$ touches two circles: centered at $K$ and with a radius $KB$ and with a center $L$ and radius $LC$.

2015 Putnam, B4
Tags: algebra , summation , geometric series , rip kent
Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c.$ Express \[\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}\] as a rational number in lowest terms.

2025 Azerbaijan IZhO TST, 4
Tags: functional equation , algebra
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that $$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$ for any $x;y\in\mathbb{Q}$

2008 iTest Tournament of Champions, 4
Tags:
Find the maximum of $x+y$ given that $x$ and $y$ are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\]

2009 Princeton University Math Competition, 2
Tags:
Find the number of ordered pairs $(a, b)$ of positive integers that are solutions of the following equation: \[a^2 + b^2 = ab(a+b).\]

2021 Malaysia IMONST 1, 15
Tags: perfect square , number theory
Find the sum of all integers $n$ with this property: both $n$ and $n + 2021$ are perfect squares.