Olympiad problems

2016 Tournament Of Towns, 3

Given a square with side $10$. Cut it into $100$ congruent quadrilaterals such that each of them is inscribed into a circle with diameter $\sqrt{3}$. [i](5 points)[/i] [i]Ilya Bogdanov[/i]

2013 Mediterranean Mathematics Olympiad, 3

Tags: inequalities

Let $x,y,z$ be positive reals for which: $\sum (xy)^{2}=6xyz$ Prove that: $\sum \sqrt{\frac{x}{x+yz}}\geq \sqrt{3}$.

2009 Balkan MO Shortlist, C1

Tags: symmetry , geometry , rectangle , combinatorics

A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$, ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry? [i]Bulgaria[/i]

1966 IMO Longlists, 56

Tags: geometry , 3d geometry , tetrahedron , circumcircle , sphere

In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.

2013 NIMO Problems, 1

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Let $a$, $b$, $c$, $d$, $e$ be positive reals satisfying \begin{align*} a + b &= c \\ a + b + c &= d \\ a + b + c + d &= e.\end{align*} If $c=5$, compute $a+b+c+d+e$. [i]Proposed by Evan Chen[/i]

1999 National Olympiad First Round, 19

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$ k$ black pieces are placed on $ k$ consecutive squares of top row starting from upper left of a $ 2\times 5$ board. We are placing white pieces on empty squares one by one in arbitrary order. Two squares is said to adjacent if they have common vertex. When a white piece is placed on a square, the pieces on adjacent squares change their color. For which $ k$, when all the squares are filled, it is possible that color of every piece is white? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

2021 Romania National Olympiad, 2

Tags: vector , geometry , trigonometry

Let $P_0, P_1,\ldots, P_{2021}$ points on the unit circle of centre $O$ such that for each $n\in \{1,2,\ldots, 2021\}$ the length of the arc from $P_{n-1}$ to $P_n$ (in anti-clockwise direction) is in the interval $\left[\frac{\pi}2,\pi\right]$. Determine the maximum possible length of the vector: \[\overrightarrow{OP_0}+\overrightarrow{OP_1}+\ldots+\overrightarrow{OP_{2021}}.\] [i]Mihai Iancu[/i]

2019 CCA Math Bonanza, I13

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Convex quadrilateral $ABCD$ has $AB=20$, $BC=CD=26$, and $\angle{ABC}=90^\circ$. Point $P$ is on $DA$ such that $\angle{PBA}=\angle{ADB}$. If $PB=20$, compute the area of $ABCD$. [i]2019 CCA Math Bonanza Individual Round #13[/i]

2023 Iranian Geometry Olympiad, 2

Tags: geometry

A convex hexagon $ABCDEF$ with an interior point $P$ is given. Assume that $BCEF$ is a square and both $ABP$ and $PCD$ are right isosceles triangles with right angles at $B$ and $C$, respectively. Lines $AF$ and $DE$ intersect at $G$. Prove that $GP$ is perpendicular to $BC$. [i]Proposed by Patrik Bak - Slovakia[/i]

2011 Saudi Arabia IMO TST, 1

Tags: algebra , inequalities

Let $a, b, c$ be real numbers such that $ab + bc + ca = 1$. Prove that $$\frac{(a + b)^2 + 1}{c^2+2}+\frac{(b + c)^2 + 1}{a^2+2}+ \frac{(c + a)^2 + 1}{b^2+2} \ge 3$$

2025 CMIMC Algebra/NT, 10

Tags: algebra , number theory

Let $a_n$ be a recursively defined sequence with $a_0=2024$ and $a_{n+1}=a_n^3+5a_n^2+10a_n+6$ for $n\ge 0.$ Determine the value of $$\sum_{n=0}^{\infty} \frac{2^n(a_n+1)}{a_n^2+3a_n+4}.$$

2021 Science ON Seniors, 2

Tags: number theory , divisibility , order

Find all pairs $(p,q)$ of prime numbers such that $$p^q-4~|~q^p-1.$$ [i](Vlad Robu)[/i]

2023 Brazil National Olympiad, 4

Tags: number theory , diophantine equation , minimum value , algebra

Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$

2014 Moldova Team Selection Test, 4

Tags: combinatorics

On a circle $n \geq 1$ real numbers are written, their sum is $n-1$. Prove that one can denote these numbers as $x_1, x_2, ..., x_n$ consecutively, starting from a number and moving clockwise, such that for any $k$ ($1\leq k \leq n$) $ x_1 + x_2+...+x_k \geq k-1$.

1952 Moscow Mathematical Olympiad, 213

Tags: geometric sequence , integer , sum , algebra

Given a geometric progression whose denominator $q$ is an integer not equal to $0$ or $-1$, prove that the sum of two or more terms in this progression cannot equal any other term in it.

2023 HMNT, 5

Tags: number theory

Compute the unique positive integer $n$ such that $\frac{n^3-1989}{n}$ is a perfect square.

2010 Contests, 2

Tags: number theory

Determine the number of positive integers $n$ for which $(n+15)(n+2010)$ is a perfect square.

2024 Australian Mathematical Olympiad, P2

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral. Point $P$ is on line $CB$ such that $CP=CA$and $B$ lies between $C$ and $P$. Point $Q$ is on line $CD$ such that $CQ=CA$ and $D$ lies between $C$ and $Q$. Prove that the incentre of triangle $ABD$ lies on line $PQ.$

1968 Miklós Schweitzer, 11

Tags: probability , probability and stats

Let $ A_1,...,A_n$ be arbitrary events in a probability field. Denote by $ C_k$ the event that at least $ k$ of $ A_1,...,A_n$ occur. Prove that \[ \prod_{k=1}^n P(C_k) \leq \prod_{k=1}^n P(A_k).\] [i]A. Renyi[/i]

1996 USAMO, 4

Tags: induction , function , modular arithmetic , strong induction , combinatorics

An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a [i]binary sequence of length [/i]$n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.

2010 Contests, 4

Tags: inequalities

If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that $\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$

1987 IMO Longlists, 49

Tags: analytic geometry , ceiling function , combinatorics

In the coordinate system in the plane we consider a convex polygon $W$ and lines given by equations $x = k, y = m$, where $k$ and $m$ are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of $W$ intersects a square if the boundary contains an interior point of the square. Prove that the boundary of $W$ intersects at most $4 \lceil d \rceil $ unit squares, where $d$ is the maximal distance of points belonging to $W$ (i.e., the diameter of $W$) and $\lceil d \rceil$ is the least integer not less than $d.$

MathLinks Contest 1st, 1

Tags: number theory

Let $a, m$ be two positive integers, $a \ne 10^k$, for all non-negative integers $k$ and $d_1, d_2, ... , d_m$ random decimal$^1$ digits with $d_1 > 0$. Prove that there exists some positive integer $n$ for which the representation in the decimal base of the number $a^n$ begins with the digits $d_1, d_2, ... , d_m$ in this order. $^1$ lesser or equal with $9$

2005 Tournament of Towns, 5

Tags: geometry , 3d geometry , invariance

A cube lies on the plane. After being rolled a few times (over its edges), it is brought back to its initial location with the same face up. Could the top face have been rotated by 90 degrees? [i](5 points)[/i]

2008 Stars Of Mathematics, 2

Tags: combinatorics

The $ 2^N$ vertices of the $ N$-dimensional hypercube $ \{0,1\}^N$ are labelled with integers from $ 0$ to $ 2^N \minus{} 1$, by, for $ x \equal{} (x_1,x_2,\ldots ,x_N)\in \{0,1\}^N$, \[v(x) \equal{} \sum_{k \equal{} 1}^{N}x_k2^{k \minus{} 1}.\] For which values $ n$, $ 2\leq n \leq 2^n$ can the vertices with labels in the set $ \{v|0\leq v \leq n \minus{} 1\}$ be connected through a Hamiltonian circuit, using edges of the hypercube only? [i]E. Bazavan & C. Talau[/i]