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

2013 Brazil Team Selection Test, 2
Tags: parallelogram , reflection , moving points , ptolemy sinus lemma , geometry
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.

1970 IMO, 1
Tags: geometry , ratio , incircle , radius
$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

2016 Costa Rica - Final Round, A1
Tags: system of equations , algebra
Find all solutions of the system $\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0 $ $\sqrt[3]{\frac{xz^4}{y}}+5wy=0 $ $\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0$ $x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16$ where $x, y, z \ge 0$ and $w \in R$ [hide=PS] I attached the system, in case I have any typos[/hide]

2017 Macedonia JBMO TST, 3
Tags: inequalities
Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen?

Russian TST 2017, P1
Tags: geometry , nine point center
The diagonals of a convex quadrilateral divide it into four triangles. Prove that the nine point centers of these four triangles either lie on one straight line, or are the vertices of a parallelogram.

CIME II 2018, 12
Tags:
Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$. [I]Proposed by [b]FedeX333X[/b][/I]

1997 Brazil Team Selection Test, Problem 5
Tags: geometry , geometric inequality , inequalities , geometric inequalities , triangle , area
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

2009 Tournament Of Towns, 3
Tags: 3d geometry , tetrahedron , sphere , geometry
Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent. [i](7 points)[/i]

2015 Postal Coaching, 1
Tags: polynomial , algebra
Find all real polynomials $P(x)$ that satisfy $$P(x^3-2)=P(x)^3-2$$

1980 AMC 12/AHSME, 29
Tags: algebra , system of equations
How many ordered triples $(x,y,z)$ of integers satisfy the system of equations below? \[ \begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} \] $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{a finite number greater than 2} \qquad \text{(E)} \ \text{infinately many}$

JOM 2023, 4
Tags: algebra
Given $n$ positive real numbers $x_1,x_2,x_3,...,x_n$ such that $$\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n$$ Determine the minimum value of $x_1+x_2+x_3+...+x_n$. [i]Proposed by Loh Kwong Weng[/i]

2021 Baltic Way, 19
Tags: polynomial , number theory
Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.

2024 Mozambican National MO Selection Test, P3
Tags: positive integer , nt , number theory , diophantine equation , factorization , prime number
Find all triples of positive integers $(a,b,c)$ such that: $a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$

2007 ITest, 39
Tags: number theory , relatively prime
Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the sum of the real solutions to the equation \[\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}.\] Find $a+b$.

2015 South East Mathematical Olympiad, 8
Tags: algebra , number theory
Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit.

Ukraine Correspondence MO - geometry, 2016.11
Tags: geometry , trapezoid , square , angle
Inside the square $ABCD$ mark the point $P$, for which $\angle BAP = 30^o$ and $\angle BCP = 15^o$. The point $Q$ was chosen so that $APCQ$ is an isosceles trapezoid ($PC\parallel AQ$). Find the angles of the triangle $CAM$, where $M$ is the midpoint of $PQ$.

2001 All-Russian Olympiad Regional Round, 8.4
Tags: rectangle , tiles , combinatorics
An angle of size $n \times m$, where $m, n \ge 2$, is called a figure, resulting from a rectangle of size $n \times m$ cells by removing the rectangle size $(n - 1) \times (m - 1)$ cells. Two players take turns making moves consisting in painting in a corner an arbitrary non-zero number of cells forming a rectangle or square.

VMEO III 2006 Shortlist, N11
Tags: power , number theory
Prove that the composition of the sets of one of the following two forms is finite: (a) $2^{2^n}+1$ (b) $6^{2^n}+1$

2014 BMT Spring, 2
Tags: probability , combinatorics
If I roll three fair $4$-sided dice, what is the probability that the sum of the resulting numbers is relatively prime to the product of the resulting numbers?

2002 AIME Problems, 8
Tags: function , floor function , parameterization , algebra
Find the least positive integer $k$ for which the equation $\lfloor \frac{2002}{n}\rfloor = k$ has no integer solutions for $n.$ (The notation $\lfloor x \rfloor$ means the greatest integer less than or equal to $x.$)

2012 Online Math Open Problems, 3
Tags:
Darwin takes an $11\times 11$ grid of lattice points and connects every pair of points that are 1 unit apart, creating a $10\times 10$ grid of unit squares. If he never retraced any segment, what is the total length of all segments that he drew? [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]The problem asks for the total length of all *unit* segments (with two lattice points in the grid as endpoints) he drew.[/list][/hide]

1986 IMO Longlists, 76
Tags: number theory , diophantine equation , geometry , pythagorean theorem , algebra
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

May Olympiad L2 - geometry, 2003.2
Tags: geometry , rectangle , perpendicular
Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the segment $MN$.

2008 Gheorghe Vranceanu, 4
Tags: number theory , modular arithmetic
Find the largest natural number $ k $ which has the property that there is a partition of the natural numbers $ \bigcup_{1\le j\le k} V_j, $ an index $ i\in\{ 1,\ldots ,k \} $ and three natural numbers $ a,b,c\in V_i, $ satisfying $ a+2b=4c. $

2015 ASDAN Math Tournament, 8
Tags: team test
Let $f(x)=\tfrac{x+a}{x+b}$ for real numbers $x$ such that $x\neq -b$. Compute all pairs of real numbers $(a,b)$ such that $f(f(x))=-\tfrac{1}{x}$ for $x\neq0$.