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

2017 HMNT, 2
Tags: geometry
[b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$.

2006 Hanoi Open Mathematics Competitions, 8
Tags: algebra , polynomial
Find all polynomials P(x) such that P(x)+P(1/x)=x+1/x

2001 Romania Team Selection Test, 3
Tags: modular arithmetic , number theory , combinatorics , additive number theory , additive combinatorics , frobenius
Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$

2019 Israel National Olympiad, 5
Tags: number theory
Guy has 17 cards. Each of them has an integer written on it (the numbers are not necessarily positive, and not necessarily different from each other). Guy noticed that for each card, the square of the number written on it equals the sum of the numbers on the 16 other cards. What are the numbers on Guy's cards? Find all of the options.

2023 Malaysian IMO Training Camp, 8
Tags: combinatorics
Given two positive integers $m$ and $n$, find the largest $k$ in terms of $m$ and $n$ such that the following condition holds: Any tree graph $G$ with $k$ vertices has two (possibly equal) vertices $u$ and $v$ such that for any other vertex $w$ in $G$, either there is a path of length at most $m$ from $u$ to $w$, or there is a path of length at most $n$ from $v$ to $w$. [i]Proposed by Ivan Chan Kai Chin[/i]

1995 Bundeswettbewerb Mathematik, 4
Tags: multiple , digit , number theory
Prove that every integer $k > 1$ has a multiple less than $k^4$ whose decimal expension has at most four distinct digits.

1960 Miklós Schweitzer, 2
Tags: complex number
[b]2.[/b] Construct a sequence $(a_n)_{n=1}^{\infty}$ of complex numbers such that, for every $l>0$, the series $\sum_{n=1}^{\infty} \mid a_n \mid ^{l}$ be divergent, but for almost all $\theta$ in $(0,2\pi)$, $\prod_{n=1}^{\infty} (1+a_n e^{i\theta})$ be convergent. [b](S. 11)[/b]

1990 China National Olympiad, 1
Tags: geometry
Given a convex quadrilateral $ABCD$, side $AB$ is not parallel to side $CD$. The circle $O_1$ passing through $A$ and $B$ is tangent to side $CD$ at $P$. The circle $O_2$ passing through $C$ and $D$ is tangent to side $AB$ at $Q$. Circle $O_1$ and circle $O_2$ meet at $E$ and $F$. Prove that $EF$ bisects segment $PQ$ if and only if $BC\parallel AD$.

2023 BMT, 4
Tags: geometry
Let $\omega$ be a circle with center $O$ and radius $8$, and let $A$ be a point such that $AO = 17$. Let $P$ and $Q$ be points on $\omega$ such that line segments $\overline{AP}$ and $\overline{AQ}$ are tangent to $\omega$ . Let $B$ and $C$ be points chosen on $\overline{AP}$ and $\overline{AQ}$, respectively, such that $\overline{BC}$ is also tangent to $\omega$ . Compute the perimeter of triangle $\vartriangle ABC$.

2000 Romania Team Selection Test, 1
Tags: function , algebra
Let $n\ge 2$ be a positive integer. Find the number of functions $f:\{1,2,\ldots ,n\}\rightarrow\{1,2,3,4,5 \}$ which have the following property: $|f(k+1)-f(k)|\ge 3$, for any $k=1,2,\ldots n-1$. [i]Vasile Pop[/i]

2013 IMO Shortlist, N1
Tags: number theory , algebra , functional equation
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

2021 Bangladesh Mathematical Olympiad, Problem 12
Tags: algebra , functional equation
A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$.

2006 AMC 12/AHSME, 22
Tags: probability , trigonometry , quadratic , geometry , rectangle , similar triangles , trig identities
A circle of radius $ r$ is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is 1/2. What is $ r$? $ \textbf{(A) } 2\sqrt {2} \plus{} 2\sqrt {3} \qquad \textbf{(B) } 3\sqrt {3} \plus{} \sqrt {2} \qquad \textbf{(C) } 2\sqrt {6} \plus{} \sqrt {3} \qquad \textbf{(D) } 3\sqrt {2} \plus{} \sqrt {6}\\ \textbf{(E) } 6\sqrt {2} \minus{} \sqrt {3}$

1992 National High School Mathematics League, 1
Tags: geometry , cyclic quadrilateral
$A_1A_2A_3A_4$ is cyclic quadrilateral of $\odot O$. $H_1,H_2,H_3,H_4$ are orthocentres of $\triangle A_2A_3A_4,\triangle A_3A_4A_1,\triangle A_4A_1A_2,\triangle A_1A_2A_3$. Prove that $H_1,H_2,H_3,H_4$ are concyclic, and determine its center.

1966 Miklós Schweitzer, 10
Tags: integration , probability and stats
For a real number $ x$ in the interval $ (0,1)$ with decimal representation \[ 0.a_1(x)a_2(x)...a_n(x)...,\] denote by $ n(x)$ the smallest nonnegative integer such that \[ \overline{a_{n(x)\plus{}1}a_{n(x)\plus{}2}a_{n(x)\plus{}3}a_{n(x)\plus{}4}}\equal{}1966 .\] Determine $ \int_0^1n(x)dx$. ($ \overline{abcd}$ denotes the decimal number with digits $ a,b,c,d .$) [i]A. Renyi[/i]

1978 Austrian-Polish Competition, 5
Tags: combinatorics
We are given $1978$ sets of size $40$ each. The size of the intersection of any two sets is exactly $1$. Prove that all the sets have a common element.

2013 Today's Calculation Of Integral, 887
Tags: calculus , integration , function , trigonometry , derivative , calculus computations , logarithm
For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

2024 LMT Fall, B3
Tags: theme
Let $MEW$ and $MOG$ be isosceles right triangles such that $E$, $M$, $O$ are collinear in that order and $G$, $M$, $W$ are collinear in that order. Suppose $ME=MW=\sqrt{6-4\sqrt{2}}$ and $MO=MG=\sqrt{6+2\sqrt{2}}$. Find the least possible area of a circle which contains both triangles $MOG$ and $MEW$.

2018 Mathematical Talent Reward Programme, SAQ: P 1
Tags: inequalities
Let $x, y, z$ are real numbers such that $x<y<z .$ Prove that $$ (x-y)^{3}+(y-z)^{3}+(z-x)^{3}>0 $$

2019 Poland - Second Round, 3
Tags: rational number , algebra , number theory , function
Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and: \begin{align*} \underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)). \end{align*}

2003 Federal Competition For Advanced Students, Part 2, 2
Tags: geometry , geometric transformation , rotation , inequalities
Let $a, b, c$ be nonzero real numbers for which there exist $\alpha, \beta, \gamma \in\{-1, 1\}$ with $\alpha a + \beta b + \gamma c = 0$. What is the smallest possible value of \[\left( \frac{a^3+b^3+c^3}{abc}\right)^2 ?\]

1966 IMO Longlists, 3
Tags: geometry , 3d geometry , prism , analytic geometry
A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.

PEN A Problems, 3
Tags: algebra , polynomial , vieta , function , induction , divisibility theory
Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that \[\frac{a^{2}+b^{2}}{ab+1}\] is the square of an integer.

2016 ASDAN Math Tournament, 9
Tags: algebra test
Let $P(x)$ be a monic cubic polynomial. The line $y=0$ and $y=m$ intersect $P(x)$ at points $A,C,E$ and $B,D<F$ from left to right for a positive real number $m$. If $AB=\sqrt{7}$, $CD=\sqrt{15}$, and $EF=\sqrt{10}$, what is the value of $m$?

2022-IMOC, A6
Tags: algebra , functional equation
Find all functions $f:\mathbb R^+\to \mathbb R^+$ such that $$f(x+y)f(f(x))=f(1+yf(x))$$ for all $x,y\in \mathbb R^+.$ [i]Proposed by Ming Hsiao[/i]