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

1954 Moscow Mathematical Olympiad, 284
Tags: geometry , 3d geometry , symmetry , pyramid
How many planes of symmetry can a triangular pyramid have?

2005 Junior Balkan MO, 1
Tags: quadratic
Find all positive integers $x,y$ satisfying the equation \[ 9(x^2+y^2+1) + 2(3xy+2) = 2005 . \]

2022 Bolivia IMO TST, P1
Tags: algebra , gauss identity
Find all possible values of $\frac{1}{x}+\frac{1}{y}$, if $x,y$ are real numbers not equal to $0$ that satisfy $$x^3+y^3+3x^2y^2=x^3y^3$$

2021 Korea Winter Program Practice Test, 8
Tags: number theory , polynomial , algebra
$P$ is an monic integer coefficient polynomial which has no integer roots. deg$P=n$ and define $A$ $:=${$v_2(P(m))|m\in Z, v_2(P(m)) \ge 1$}. If $|A|=n$, show that all of the elements of $A$ is smaller than $\frac{3}{2}n^2$.

Estonia Open Senior - geometry, 2012.1.3
Tags: right triangle , median , circumcenter , geometry
Let $ABC$ be a triangle with median AK. Let $O$ be the circumcenter of the triangle $ABK$. a) Prove that if $O$ lies on a midline of the triangle $ABC$, but does not coincide with its endpoints, then $ABC$ is a right triangle. b) Is the statement still true if $O$ can coincide with an endpoint of the midsegment?

2005 Croatia National Olympiad, 1
Tags: algebra , equation , polynomial
Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.

2005 Turkey MO (2nd round), 6
Tags: number theory
Suppose that a sequence $(a_n)_{n=1}^{\infty}$ of integers has the following property: For all $n$ large enough (i.e. $n \ge N$ for some $N$ ), $a_n$ equals the number of indices $i$, $1 \le i < n$, such that $a_i + i \ge n$. Find the maximum possible number of integers which occur infinitely many times in the sequence.

1992 China National Olympiad, 1
Tags: inequalities , triangle inequality , algebra
Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$. Suppose that $\lambda$ ($|\lambda|>1$) is a complex root of the equation, prove that $\lambda^{n+1}=1$.

2016 NIMO Problems, 1
Tags: arithmetic sequence
Suppose $a_1$, $a_2$, $a_3$, $\dots$ is an arithmetic sequence such that \[a_1+a_2+a_3+\cdots+a_{48}+a_{49}=1421.\] Find the value of $a_1+a_4+a_7+a_{10}+\cdots+a_{49}$. [i]Proposed by Tony Kim[/i]

LMT Accuracy Rounds, 2022 S3
Tags: number theory
Find the difference between the greatest and least values of $lcm (a,b,c)$, where $a$, $b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive.

2023 Bulgaria JBMO TST, 3
Tags: geometry , incircle , excircle , similarity , circumcircle , incenter
Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.

2016 Switzerland - Final Round, 3
Tags: algebra , number theory
Find all primes $p, q$ and natural numbers $n$ such that: $p(p+1)+q(q+1)=n(n+1)$

2005 Turkey Team Selection Test, 1
Tags: search , function , number theory
Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .

2007 IMO Shortlist, 3
Tags: combinatorics , modular arithmetic , counting , triplets
Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

2017 Estonia Team Selection Test, 8
Tags: inequalities , three variable inequality , ineqstd
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$ [i]Proposed by Tigran Margaryan, Armenia[/i]

2008 F = Ma, 8
Tags:
Riders in a carnival ride stand with their backs against the wall of a circular room of diameter $\text{8.0 m}$. The room is spinning horizontally about an axis through its center at a rate of $\text{45 rev/min}$ when the floor drops so that it no longer provides any support for the riders. What is the minimum coefficient of static friction between the wall and the rider required so that the rider does not slide down the wall? (a) $\text{0.0012}$ (b) $\text{0.056}$ (c) $\text{0.11}$ (d) $\text{0.53}$ (e) $\text{8.9}$

2009 Harvard-MIT Mathematics Tournament, 7
Tags:
Simplify the product \[ \prod_{m=1}^{100}\prod_{n=1}^{100}\frac{x^{n+m}+x^{n+m+2}+x^{2n+1}+x^{2m+1}}{x^{2n}+2x^{n+m}+x^{2m}} \] Express your answer in terms of $x$.

2019 USMCA, 24
Tags:
Let $ABC$ be a triangle with $\angle A = 60^\circ$, $AB = 12$, $AC = 14$. Point $D$ is on $BC$ such that $\angle BAD = \angle CAD$. Extend $AD$ to meet the circumcircle at $M$. The circumcircle of $BDM$ intersects $AB$ at $K \neq B$, and line $KM$ intersects the circumcircle of $CDM$ at $L \neq M$. Find $\frac{KM}{LM}$.

2007 Korea Junior Math Olympiad, 4
Tags: geometry , perpendicular bisector , concyclic , collinear
Let $P$ be a point inside $\triangle ABC$. Let the perpendicular bisectors of $PA,PB,PC$ be $\ell_1,\ell_2,\ell_3$. Let $D =\ell_1 \cap \ell_2$ , $E=\ell_2 \cap \ell_3$, $F=\ell_3 \cap \ell_1$. If $A,B,C,D,E,F$ lie on a circle, prove that $C, P,D$ are collinear.

2002 China Team Selection Test, 2
Tags: 3d geometry , sphere , vector , induction , geometry
There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.

1968 Yugoslav Team Selection Test, Problem 6
Tags: geometry , 3d geometry , tetrahedron
Prove that the incenter coincides with the circumcenter of a tetrahedron if and only if each pair of opposite edges are of equal length.

2013 Greece JBMO TST, 1
Tags: algebra , inequalities
If x,y<0 prove that $\left(x+\frac{2}{y} \right) \left(\frac{y}{x}+2 \right)\geq 8$. When do we have equality?

2014 VJIMC, Problem 3
Tags: inequalities
Let $n\ge2$ be an integer and let $x>0$ be a real number. Prove that $$\left(1-\sqrt{\tanh x}\right)^n+\sqrt{\tanh(nx)}<1.$$

2009 Harvard-MIT Mathematics Tournament, 1
Tags:
If $a$ and $b$ are positive integers such that $a^2-b^4= 2009$, find $a+b$.

VMEO III 2006, 10.3
Tags: algebra , functional
Find all functions $f : R \to R$ that satisfy $f(x^2 + f(y) - y) = (f(x))^2$ for all $x,y \in R$.