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

2019 Jozsef Wildt International Math Competition, W. 59
Tags: tetrahedron , 3d geometry , circumradius , inradius , geometry , inequalities
In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).

1981 Canada National Olympiad, 4
Tags: polynomial , function , algebra
$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.

2024 JHMT HS, 9
Tags: geometry
Compute the smallest positive integer $k$ such that the area of the region bounded by \[ k\min(x,y)+x^2+y^2=0 \] exceeds $100$.

1986 Traian Lălescu, 2.4
Tags: geometry , rectangle , 3d geometry
Prove that $ ABCD $ is a rectangle if and only if $ MA^2+MC^2=MB^2+MD^2, $ for all spatial points $ M. $

2019 CMIMC, 7
Tags: algebra , number theory
For all positive integers $n$, let \[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute $f(2019) - f(2018)$. Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.

2022 Macedonian Team Selection Test, Problem 3
Tags: algebra , function , functional equation
We consider all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(f(n)+n)=n$ and $f(a+b-1) \leq f(a)+f(b)$ for all positive integers $a, b, n$. Prove that there are at most two values for $f(2022)$. $\textit {Proposed by Ilija Jovcheski}$

2008 Costa Rica - Final Round, 4
Tags: inequalities
Let $ x$, $ y$ and $ z$ be non negative reals, such that there are not two simultaneously equal to $ 0$. Show that $ \frac {x \plus{} y}{y \plus{} z} \plus{} \frac {y \plus{} z}{x \plus{} y} \plus{} \frac {y \plus{} z}{z \plus{} x} \plus{} \frac {z \plus{} x}{y \plus{} z} \plus{} \frac {z \plus{} x}{x \plus{} y} \plus{} \frac {x \plus{} y}{z \plus{} x}\geq\ 5 \plus{} \frac {x^{2} \plus{} y^{2} \plus{} z^{2}}{xy \plus{} yz \plus{} zx}$ and determine the equality cases.

1999 Greece JBMO TST, 1
Tags: combinatorics , coloring , number theory
A circle is divided in $100$ equal parts and the points of this division are colored green or yellow, such that when between two points of division $A,B$ there are exactly $4$ division points and the point $A$ is green, then the point $B$ shall be yellow. Which points are more, the green or the yellow ones?

2007 Mongolian Mathematical Olympiad, Problem 1
Tags: combinatorics
Find the number of subsets of the set $\{1,2,3,...,5n\}$ such that the sum of the elements in each subset are divisible by $5$.

2021 Moldova EGMO TST, 9
Tags: geometry
Let $ABCD$ be a square and $E$ a on point diagonal $(AC)$, different from its midpoint. $H$ and $K$ are the orthoceneters of triangles $ABE$ and $ADE$. Prove that $AH$ and $CK$ are parallel.

2021 Belarusian National Olympiad, 11.1
Tags: algebra , functional equation
Find all functions $f: \mathbb{R} \to \mathbb{R}$, such that for all real $x,y$ the following equation holds:$$f(x-0.25)+f(y-0.25)=f(x+\lfloor y+0.25 \rfloor - 0.25)$$

2015 Peru Cono Sur TST, P7
Tags: combinatorics , geometry
In the plan $6$ points were located such that the distance between two damages of them is greater than or equal to $1$. Prove that it is possible to choose two of those points such that their distance is greater than or equal to $2 \cos{18}$ Observation: It might help you to know that $\cos{18} = 0.95105\ldots$ and $\cos{24} = 0.91354\ldots$

1983 IMO Longlists, 26
Tags: number theory
Let $a, b, c$ be positive integers satisfying $\gcd (a, b) = \gcd (b, c) = \gcd (c, a) = 1$. Show that $2abc-ab-bc-ca$ cannot be represented as $bcx+cay +abz$ with nonnegative integers $x, y, z.$

2016 Brazil Team Selection Test, 3
Tags: combinatorics , game
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

2017 ASDAN Math Tournament, 2
Tags:
Let $5$ and $13$ be lengths of two sides of a right triangle. Compute the sum of all possible lengths of the third side.

2006 China Second Round Olympiad, 3
Tags: inequalities
Suppose $A = {x|5x-a \le 0}$, $B = {x|6x-b > 0}$, $a,b \in \mathbb{N}$, and $A \cap B \cap \mathbb{N} = {2,3,4}$. The number of such pairs $(a,b)$ is ${ \textbf{(A)}\ 20\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}} 42\qquad $

2013 Korea Junior Math Olympiad, 3
Tags: algebra , sequence , recurrence relation , sum , positive integer , diophantine equation
$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$). For positive integer $n$, de fine as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$ Prove that $b_n$ is positive integer.

1984 Brazil National Olympiad, 3
Tags: geometry , rectangle , dodecahedron , regular polygon , 3d geometry
Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.

2002 China Western Mathematical Olympiad, 2
Tags: modular arithmetic , inequalities , number theory
Given a positive integer $ n$, find all integers $ (a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions: $ (1): a_{1}\plus{}a_{2}\plus{}\cdots\plus{}a_{n}\ge n^2;$ $ (2): a_{1}^2\plus{}a_{2}^2\plus{}\cdots\plus{}a_{n}^2\le n^3\plus{}1.$

2023 Polish Junior Math Olympiad First Round, 3.
Tags: geometry
Let $ABCD$ be a rectangle. Point $E$ lies on side $AB$, and point $F$ lies on segment $CE$. Prove that if triangles $ADE$ and $CDF$ have equal areas, then triangles $BCE$ and $DEF$ also have equal areas.

2015 Saudi Arabia IMO TST, 1
Tags: geometry , incenter
Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$, $H$ the foot of the altitude of $ABC$ at $A$ and $P$ a point inside $ABC$ lying on the bisector of $\angle BAC$. The circle of diameter $AP$ cuts $(O)$ again at $G$. Let $L$ be the projection of $P$ on $AH$. Prove that if $GL$ bisects $HP$ then $P$ is the incenter of the triangle $ABC$. Lê Phúc Lữ

2023 UMD Math Competition Part I, #9
Tags: geometry
The Amazing Prime company ships its products in boxes whose length, width, and height (in inches) are prime numbers. If the volume of one of their boxes is $105$ cubic inches, what is its surface area (that is, the sum of the areas of the 6 sides of the box) in square inches? $$ \mathrm a. ~ 21\qquad \mathrm b.~71\qquad \mathrm c. ~77 \qquad \mathrm d. ~05 \qquad \mathrm e. ~142 $$

2016 India Regional Mathematical Olympiad, 2
Tags: algebra , inequalities
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.

1999 Singapore Senior Math Olympiad, 2
Tags: geometry , trigonometry , triangle area , area
In $\vartriangle ABC$ with edges $a, b$ and $c$, suppose $b + c = 6$ and the area $S$ is $a^2 - (b -c)^2$. Find the value of $\cos A$ and the largest possible value of $S$.

2025 Kosovo National Mathematical Olympiad`, P4
Tags: geometry
Let $D$ be a point on the side $AC$ of triangle $\triangle ABC$ such that $AB=AD=DC$ and let $E$ be a point on the side $BC$ such that $BE=2CE$. Prove that $\angle BDE = 90 ^{\circ}$.