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

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P3
Tags: number theory , diophantine equation , 3 variables
Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$. [i]Proposed by Viktor Simjanoski[/i]

2003 Junior Balkan MO, 3
Tags: geometry , circumcircle , angle bisector
Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

2022 International Zhautykov Olympiad, 1
Tags: algebra , polynomial
Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities $$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$ Prove that the degrees of the three polynomials are all even.

2014-2015 SDML (High School), 6
Tags: function , real analysis
Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$. $\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$

1985 AMC 12/AHSME, 1
Tags:
If $ 2x \plus{} 1 \equal{} 8$, then $ 4x \plus{} 1 \equal{}$ $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 19$

1998 German National Olympiad, 4
Tags: polynomial , quadratic , algebra
Let $a$ be a positive real number. Then prove that the polynomial \[ p(x)=a^3x^3+a^2x^2+ax+a \] has integer roots if and only if $a=1$ and determine those roots.

2023 Hong Kong Team Selection Test, Problem 5
Tags: geometry
Let $ABCD$ be a convex quadrilateral with $AB=5$, $AD=17$, and $CD=6$. If the angle bisector of $\angle{BAD}$ and $\angle{ADC}$ intersect at the midpoint of $BC$, find the area of $ABCD$.

2019 Novosibirsk Oral Olympiad in Geometry, 7
Tags: geometry , acute , square
Cut a square into eight acute-angled triangles.

2009 AMC 12/AHSME, 17
Tags: geometry , 3d geometry , probability
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? $ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {3}{16}\qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {3}{8}\qquad \textbf{(E)}\ \frac {1}{2}$

2011 Today's Calculation Of Integral, 760
Tags: calculus , integration , trigonometry , calculus computations
Prove that there exists a positive integer $n$ such that $\int_0^1 x\sin\ (x^2-x+1)dx\geq \frac {n}{n+1}\sin \frac{n+2}{n+3}.$

2019 Greece Team Selection Test, 2
Tags: geometry , circumcircle
Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).

2012 USAJMO, 2
Tags: geometry , fibonacci
Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.

2025 Poland - First Round, 2
Tags: geometry , rectangle
Let $ABCD$ be a rectangle inscribed in circle $\omega$ with center $O$. Line $l$ passes trough $O$ and intersects lines $BC$ and $AD$ at points $E$ and $F$ respectively. Points $K$ and $L$ are the intersection points of $l$ and $\omega$ and points $K, E, F, L$ lie in this order on the line $l$. Lines tangent to $w$ in $K$ and $L$ intersect $CD$ at $M$ and $N$ respectively. Prove that $E, F, M, N$ lie on a common circle.

2019 India PRMO, 23
Tags: cyclic quadrilateral , geometry
Let $ABCD$ be a convex cyclic quadilateral. Suppose $P$ is a point in the plane of the quadilateral such that the sum of its distances from the vertices of $ABCD$ is the least. If $$\{PC, PB, PC, PD\} = \{3, 4, 6, 8\}$$, what is the maxumum possible area of $ABCD$?

2023 AMC 12/AHSME, 13
Tags: 3d geometry , diagonal
A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$? $\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$

2018 Cono Sur Olympiad, 3
Tags: algebra , combinatorics
Define the product $P_n=1! \cdot 2!\cdot 3!\cdots (n-1)!\cdot n!$ a) Find all positive integers $m$, such that $\frac {P_{2020}}{m!}$ is a perfect square. b) Prove that there are infinite many value(s) of $n$, such that $\frac {P_{n}}{m!}$ is a perfect square, for at least two positive integers $m$.

2010 Greece National Olympiad, 1
Tags: quadratic , number theory
Solve in the integers the diophantine equation $$x^4-6x^2+1 = 7 \cdot 2^y.$$

2007 Estonia Math Open Junior Contests, 9
Tags: inequalities , combinatorics
In an exam with k questions, n students are taking part. A student fails the exam if he answers correctly less than half of all questions. Call a question easy if more than half of all students answer it correctly. For which pairs (k, n) of positive integers is it possible that (a) all students fail the exam although all questions are easy; (b) no student fails the exam although no question is easy?

2014 Contests, 2
Tags: polynomial , algebra , number theory
find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.

1959 AMC 12/AHSME, 35
Tags:
The symbol $\ge$ means "greater than or equal to"; the symbol $\le$ means "less than or equal to". In the equation $(x-m)^2-(x-n)^2=(m-n)^2$; m is a fixed positive number, and $n$ is a fixed negative number. The set of values $x$ satisfying the equation is: $ \textbf{(A)}\ x\ge 0 \qquad\textbf{(B)}\ x\le n\qquad\textbf{(C)}\ x=0\qquad\textbf{(D)}\ \text{the set of all real numbers}\qquad\textbf{(E)}\ \text{none of these} $

2021 Thailand Mathematical Olympiad, 1
Tags: geometry
Let $\triangle ABC$ be an isosceles triangle such that $AB=AC$. Let $\omega$ be a circle centered at $A$ with a radius strictly less than $AB$. Draw a tangent from $B$ to $\omega$ at $P$, and draw a tangent from $C$ to $\omega$ at $Q$. Suppose that the line $PQ$ intersects the line $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$.

2004 China Team Selection Test, 3
Tags: algebra , polynomial , calculus , integration , number theory
Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

2016 Bosnia and Herzegovina Junior BMO TST, 4
Tags: algebra , inequalities
Let $x$, $y$ and $z$ be positive real numbers such that $\sqrt{xy} + \sqrt{yz} + \sqrt{zx} = 3$. Prove that $\sqrt{x^3+x} + \sqrt{y^3+y} + \sqrt{z^3+z} \geq \sqrt{6(x+y+z)}$

1977 Bundeswettbewerb Mathematik, 4
Tags: algebra
Find all functions $f : \mathbb R \to \mathbb R$ such that \[f(x)+f\left(1-\frac{1}{x}\right)=x,\] holds for all real $x$.

1998 IMC, 4
Tags: function , calculus , derivative , real analysis
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ is twice differentiable and satisfies $f(0)=2,f'(0)=-2,f(1)=1$. Prove that there is a $\xi \in ]0,1[$ for which we have $f(\xi)\cdot f'(\xi)+f''(\xi)=0$.