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: 66

2000 District Olympiad (Hunedoara), 4
Tags: geometry , contests , Olympiad
Let be a circle centeted at $ O, $ and $ A,B,C, $ points situated on this circle. Show that if $$ \left|\overrightarrow{OA} +\overrightarrow{OB}\right| = \left|\overrightarrow{OB} +\overrightarrow{OC}\right| = \left|\overrightarrow{OC} +\overrightarrow{OA}\right| , $$ then $ A=B=C, $ or $ ABC $ is an equilateral triangle.

2016 Bangladesh Mathematical Olympiad, 2
Tags: number theory , modular arithmetic , Divisibility , Bdmo , contests
(a) How many positive integer factors does $6000$ have? (b) How many positive integer factors of $6000$ are not perfect squares?

2000 District Olympiad (Hunedoara), 3
Tags: geometry , contests , Olympiad , Locus problems
Let $ \alpha $ be a plane and let $ ABC $ be an equilateral triangle situated on a parallel plane whose distance from $ \alpha $ is $ h. $ Find the locus of the points $ M\in\alpha $ for which $$ \left|MA\right| ^2 +h^2 = \left|MB\right|^2 +\left|MC\right|^2. $$

2014 Polish MO Finals, 1
Tags: contests , number theory , relatively prime
Let $k,n\ge 1$ be relatively prime integers. All positive integers not greater than $k+n$ are written in some order on the blackboard. We can swap two numbers that differ by $k$ or $n$ as many times as we want. Prove that it is possible to obtain the order $1,2,\dots,k+n-1, k+n$.

2013 Bangladesh Mathematical Olympiad, 4
Tags: algebra , contests
Higher Secondary P4 If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.

2000 District Olympiad (Hunedoara), 3
Tags: limits , Sequences , calculus , real analysis , contests , romania
Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

2021 Argentina National Olympiad, 5
Tags: number theory , Perfect Squares , contests
Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.

2014 Polish MO Finals, 2
Tags: contests , algebra
Let $k\ge 2$, $n\ge 1$, $a_1, a_2,\dots, a_k$ and $b_1, b_2, \dots, b_n$ be integers such that $1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n$. Prove that if $a_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n$, then $a_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n$.

2013 Bangladesh Mathematical Olympiad, 1
Tags: geometry , contests
Higher Secondary P1 A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$, $AB=AE$, $BC=DE$, $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$, $BD$ is perpendicular to both $AB$ and $DE$. Prove that $ABCDE$ is a degenerate pentagon.

2000 District Olympiad (Hunedoara), 4
Tags: functions , limits , real analysis , contests , romania , calculus , integration
Let $ f:[0,1]\longrightarrow\mathbb{R}_+^* $ be a Riemann-integrable function. Calculate $ \lim_{n\to\infty}\left(-n+\sum_{i=1}^ne^{\frac{1}{n}\cdot f\left(\frac{i}{n}\right)}\right) . $

2021 Argentina National Olympiad, 5
Tags: number theory , prime numbers , contests , Argentina
The sequence $a_n (n\geq 1)$ of natural numbers is defined as $a_{n+1}=a_n+b_n,$ where $b_n$ is the number that has the same digits as $a_n$ but in the opposite order ($b_n$ can start with $0$). For example, if $a_1=180,$ then $a_2=261, a_3=423.$ a) Decide if $a_1$ can be chosen so that $a_7$ is prime. b) Decide if $a_1$ can be chosen so that $a_5$ is prime.

2015 Polish MO Finals, 1
Tags: contests , system of equations
Solve the system $$\begin{cases} x+y+z=1\\ x^5+y^5+z^5=1\end{cases}$$ in real numbers.

2015 Polish MO Finals, 2
Tags: contests , polynomial , algebra , calculus , integration
Let $P$ be a polynomial with real coefficients. Prove that if for some integer $k$ $P(k)$ isn't integral, then there exist infinitely many integers $m$, for which $P(m)$ isn't integral.

2021 Argentina National Olympiad Level 2, 4
Tags: combinatorics , contests
The sum of several positive integers, not necessarily different, all of them less than or equal to $10$, is equal to $S$. We want to distribute all these numbers into two groups such that the sum of the numbers in each group is less than or equal to $80.$ Determine all values ​​of $S$ for which this is possible.

2021 Argentina National Olympiad, 4
Tags: contests , combinatorics
Martu wants to build a set of cards with the following properties: • Each card has a positive integer on it. • The number on each card is equal to one of $5$ possible numbers. • If any two cards are taken and added together, it is always possible to find two other cards in the set such that the sum is the same. Determine the fewest number of cards Martu's set can have and give an example for that number.

2021 Argentina National Olympiad, 1
Tags: contests , number theory , prime numbers
Determine all pairs of prime numbers $p$ and $q$ greater than $1$ and less than $100$, such that the following five numbers: $$p+6,p+10,q+4,q+10,p+q+1,$$ are all prime numbers.

2021 Argentina National Olympiad, 3
Tags: geometry , right triangle , Angle Chasing , contests
Let $ABC$ be an isosceles right triangle at $A$ with $AB=AC$. Let $M$ and $N$ be on side $BC$, with $M$ between $B$ and $N,$ such that $$BM^2+ NC^2= MN^2.$$ Determine the measure of the angle $\angle MAN.$

2018 Malaysia National Olympiad, B2
Tags: Proof , number theory , contests
Prove that the number $ 9^{(a_1 + a_2)(a_2 + a_3)(a_3 + a_4)...(a_{98} + a_{99})(a_{99} + a_1)}$ − $1$ is divisible by $10$, for any choice of positive integers $a_1, a_2, a_3, . . . , a_{99}$.

2021 Argentina National Olympiad, 3
Tags: geometry , arcs , lengths , contests
A circle is divided into $2n$ equal arcs by $2n$ points. Find all $n>1$ such that these points can be joined in pairs using $n$ segments, all of different lengths and such that each point is the endpoint of exactly one segment.

2016 Bangladesh Mathematical Olympiad, 1
Tags: Bdmo , contests , contest problem , number theory , modular arithmetic
(a) Show that $n(n + 1)(n + 2)$ is divisible by $6$. (b) Show that $1^{2015} + 2^{2015} + 3^{2015} + 4^{2015} + 5^{2015} + 6^{2015}$ is divisible by $7$.

2021 Argentina National Olympiad, 4
Tags: algebra , Argentina , contests
Find the real numbers $x, y, z$ such that, $$\frac{1}{x}+\frac{1}{y+z}=\frac{1}{2}, \frac{1}{y}+\frac{1}{z+x}=\frac{1}{3}, \frac{1}{z}+\frac{1}{x+y}=\frac{1}{4}.$$

2016 IberoAmerican, 2
Tags: contests , Iberoamerican , algebra , Iberoamerican 2016
Find all positive real numbers $(x,y,z)$ such that: $$x = \frac{1}{y^2+y-1}$$ $$y = \frac{1}{z^2+z-1}$$ $$z = \frac{1}{x^2+x-1}$$

2018 Malaysia National Olympiad, A5
Tags: number theory , contests , Perfect Squares , difference of squares
Determine the value of $(101 \times 99)$ - $(102 \times 98)$ + $(103 \times 97)$ − $(104 \times 96)$ + ... ... + $(149 \times 51)$ − $(150 \times 50)$.

2014 Contests, 3
Tags: contests , geometry , 3D geometry , tetrahedron , sphere
A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.

2015 Polish MO Finals, 2
Tags: contests , geometry
Prove that diagonals of a convex quadrilateral are perpendicular if and only if inside of the quadrilateral there is a point, whose orthogonal projections on sides of the quadrilateral are vertices of a rectangle.