Olympiad problems

1975 Bulgaria National Olympiad, Problem 6

Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$. [i](H. Lesov)[/i]

2013 IberoAmerican, 6

Tags: rotation , geometry , geometric transformation , combinatorics proposed , combinatorics , combinatorial geometry

A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.

2004 Uzbekistan National Olympiad, 2

Tags: geometry , quadratics , number theory unsolved , number theory

Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.

2006 Taiwan TST Round 1, 2

Tags: analytic geometry , trigonometry , geometry , geometry proposed

Let $P$ be a point on the plane. Three nonoverlapping equilateral triangles $PA_1A_2$, $PA_3A_4$, $PA_5A_6$ are constructed in a clockwise manner. The midpoints of $A_2A_3$, $A_4A_5$, $A_6A_1$ are $L$, $M$, $N$, respectively. Prove that triangle $LMN$ is equilateral.

2012 India PRMO, 5

Tags: number theory

Let $S_n = n^2 + 20n + 12$, $n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square?

2014 ELMO Shortlist, 9

Tags: logarithms , number theory proposed , number theory

Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]

2017 Polish MO Finals, 3

Tags: number theory , Poland

Integers $a_1, a_2, \ldots, a_n$ satisfy $$1<a_1<a_2<\ldots < a_n < 2a_1.$$ If $m$ is the number of distinct prime factors of $a_1a_2\cdots a_n$, then prove that $$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$

PEN K Problems, 26

Tags: function , calculus , integration , induction , Functional Equations

The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$.

2009 Junior Balkan Team Selection Test, 2

Tags: combinatorics proposed , combinatorics

From the set $ \{1,2,3,\ldots,2009\}$ we choose $ 1005$ numbers, such that sum of any $ 2$ numbers isn't neither $ 2009$ nor $ 2010$. Find all ways on we can choose these $ 1005$ numbers.

1970 Swedish Mathematical Competition, 4

Tags: Cubic , analysis , algebra , polynomial

Let $p(x) = (x- x_1)(x- x_2)(x- x_3)$, where $x_1, x_2$ and $x_3$ are real. Show that $p(x) p''(x) \le p'(x)^2$ for all $x$.

1993 Poland - First Round, 10

Tags: inequalities

Given positive real numbers $p,q$ with $p+q=1$. Prove that for all positive integers $m,n$ the following inequality holds $(1-p^m)^n+(1-q^n)^m \geq 1$.

2024 Belarusian National Olympiad, 9.6

Tags: algebra

Given pairs $(a_1,b_1)$, $(a_2,b_2),\ldots, (a_n,b_n)$ of non-negative real numbers such that for any real $x$ and $y$ the equality $$\sqrt{a_1x^2+b_1y^2}+\sqrt{a_2x^2+b_2y^2}+\ldots+\sqrt{a_nx^2+b_ny^2}=\sqrt{x^2+y^2}$$ Prove that $a_1=b_1,a_2=b_2,\ldots$,$a_n=b_n$ [i]A. Vaidzelevich[/i]

2021 JHMT HS, 4

Tags: combinatorics , 2021

For positive integers $n,$ let $f(n)$ equal the number of subsets of the first $13$ positive integers whose members sum to $n.$ Compute \[ \sum_{n=46}^{86} f(n). \]

2021 Cono Sur Olympiad, 3

Tags: combinatorics

In a tennis club, each member has exactly $k > 0$ friends, and a tournament is organized in rounds such that each pair of friends faces each other in matches exactly once. Rounds are played in simultaneous matches, choosing pairs until they cannot choose any more (that is, among the unchosen people, there is not a pair of friends which has its match pending). Determine the maximum number of rounds the tournament can have, depending on $k$.

2016 CCA Math Bonanza, L2.3

Tags: CCA Math Bonanza , inequalities

Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$. $D$ is a point on $AB$ such that $CD\perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $$AC^2+\left(3\cdot CD\right)^2+BC^2?$$ [i]2016 CCA Math Bonanza Lightning #2.3[/i]

2009 ELMO Problems, 1

Tags: number theory , greatest common divisor , number theory unsolved

Let $a,b,c$ be positive integers such that $a^2 - bc$ is a square. Prove that $2a + b + c$ is not prime. [i]Evan o'Dorney[/i]

2006 Federal Competition For Advanced Students, Part 2, 2

Tags: inequalities unsolved , inequalities

Let $ a,b,c$ be positive real numbers. Show that $ 3(a \plus{} b \plus{} c) \ge 8 \sqrt [3]{abc} \plus{} \sqrt [3]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3} }.$

1896 Eotvos Mathematical Competition, 1

Tags: algebra , inequalities

If $k$ is the number of distinct prime divisors of a natural number $n$, prove that log $n \geq k$ log $2$.

LMT Team Rounds 2021+, 13

Tags: combinatorics

Ella lays out $16$ coins heads up in a $4\times 4$ grid as shown. [img]https://cdn.artofproblemsolving.com/attachments/3/3/a728be9c51b27f442109cc8613ef50d61182a0.png[/img] On a move, Ella can flip all the coins in any row, column, or diagonal (including small diagonals such as $H_1$ & $H_4$). If rotations are considered distinct, how many distinct grids of coins can she create in a finite number of moves?

2007 South africa National Olympiad, 1

Tags: algebra proposed , algebra

Determine whether $ \frac{1}{\sqrt{2}} \minus{} \frac{1}{\sqrt{6}}$ is less than or greater than $ \frac{3}{10}$.

2018 Middle European Mathematical Olympiad, 8

Tags: number theory

An integer $n $ is called silesian if there exist positive integers $a,b$ and $c$ such that $$n=\frac{a^2+b^2+c^2}{ab+bc+ca}.$$ $(a)$ prove that there are infinitely many silesian integers. $(b)$ prove that not every positive integer is silesian.

2022 Sharygin Geometry Olympiad, 8

Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometry

Points $P,Q,R$ lie on the sides $AB,BC,CA$ of triangle $ABC$ in such a way that $AP=PR, CQ=QR$. Let $H$ be the orthocenter of triangle $PQR$, and $O$ be the circumcenter of triangle $ABC$. Prove that $$OH||AC$$.

2021 Austrian MO National Competition, 4

Tags: game , combinatorics , game strategy

On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$. Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)

Oliforum Contest I 2008, 2

Tags: algebra

Let $ a_1,a_2,...,a_n$ with arithmetic mean equals zero; what is the value of: $ \sum_{j=1}^n{\frac{1}{a_j(a_j+a_{j+1})(a_j+a_{j+1}+a_{j+2})...(a_j+a_{j+1}+...+a_{j+n-2})}}$ , where $ a_{n+k}=a_k$ ?

2021 Sharygin Geometry Olympiad, 10-11.4

Tags: 3D geometry , combinatorial geometry , geometry , pyramid

Can a triangle be a development of a quadrangular pyramid?