Olympiad problems

1992 Bundeswettbewerb Mathematik, 2

A positive integer $n$ is called [i]good [/i] if they sum up in one and only one way at least of two positive integers whose product also has the value $n$. Here representations that differ only in the order of the summands are considered the same viewed. Find all good positive integers.

2001 Hungary-Israel Binational, 2

Tags: graph theory , combinatorics

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. If $n \geq 5$ and $e(G_{n}) \geq \frac{n^{2}}{4}+2$, prove that $G_{n}$ contains two triangles that share exactly one vertex.

1986 IMO Longlists, 33

Tags: locus , locus problems , rotation , polygon , geometry

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

2014 Romania Team Selection Test, 3

Tags: inequalities

Determine the smallest real constant $c$ such that \[\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2\] for all positive integers $n$ and all positive real numbers $x_1,\cdots ,x_n$.

2020 LMT Spring, 4

Tags:

Suppose there are $n$ ordered pairs of positive integers $(a_i,b_i)$ such that $a_i+b_i=2020$ and $a_ib_i$ is a multiple of $2020$, where $1\le i \le n$. Compute the sum \[\sum_{i=1}^{n} a_i+b_i.\]

2015 Iran Geometry Olympiad, 1

Tags: geometry

Given a circle and Points $P,B,A$ on it.Point $Q$ is Interior of this circle such that: $1)$ $\angle PAQ=90$. $ 2)PQ=BQ$. Prove that $\angle AQB - \angle PQA=\stackrel{\frown}{AB}$. proposed by Davoud Vakili, Iran.

2003 Moldova Team Selection Test, 1

Tags: algebra , vieta , polynomial

Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form $ P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1$, if it is known that all the roots of them are positive reals. [i]Proposer[/i]: [b]Baltag Valeriu[/b]

2015 Thailand TSTST, 2

Tags: algebra , functional equation

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ and let $f : \mathbb{N}\to\mathbb{R}$. Prove that there is an infinite subset $A$ of $\mathbb{N}$ such that $f$ is increasing on $A$ or $f$ is decreasing on $A$.

1996 All-Russian Olympiad Regional Round, 11.7

Tags: angle , geometry , similar triangles

In triangle $ABC$, a point $O$ is taken such that $\angle COA = \angle B + 60^o$, $\angle COB = \angle A + 60^o$, $\angle AOB = \angle C + 60^o$.Prove that if a triangle can be formed from the segments $AO$, $BO$, $CO$, then a triangle can also be formed from the altitudes of triangle $ABC$ and these triangles are similar.

2024 JBMO TST - Turkey, 8

Tags: combinatorics , game strategy

There is $207$ boxes on the table which numbered $1,2, \dots , 207$ respectively. Firstly Aslı puts a red ball in each of the $100$ boxes that she chooses and puts a white ball in each of the remaining ones. After that Zehra, writes a pair $(i,j)$ on the blackboard such that $1\leq i \leq j \leq 207$. Finally, Aslı tells Zehra that for every pair; whether the color of the balls which is inside the box which numbered by these numbers are the same or not. Find the least possible value of $N$ such that Zehra can guarantee finding all colors that has been painted to balls in each of the boxes with writing $N$ pairs on the blackboard.

2013 Iran MO (3rd Round), 2

Tags: combinatorics , modular arithmetic

How many rooks can be placed in an $n\times n$ chessboard such that each rook is threatened by at most $2k$ rooks? (15 points) [i]Proposed by Mostafa Einollah zadeh[/i]

2006 Sharygin Geometry Olympiad, 8.5

Tags: convex , convex polygon , diagonal , side , geometry

Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

2017 Iran Team Selection Test, 1

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

2010 Spain Mathematical Olympiad, 3

Tags: number theory

Let $p$ be a prime number and $A$ an infinite subset of the natural numbers. Let $f_A(n)$ be the number of different solutions of $x_1+x_2+\ldots +x_p=n$, with $x_1,x_2,\ldots x_p\in A$. Does there exist a number $N$ for which $f_A(n)$ is constant for all $n<N$?

2024 Mongolian Mathematical Olympiad, 1

Tags: diophantine equation , number theory , factorial

Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\] [i]Proposed by Otgonbayar Uuye.[/i]

2018 PUMaC Team Round, 6

Tags:

Let $\tau(n)$ be the number of distinct positive divisors of $n$ (including $1$ and itself). Find the sum of all positive integers $n$ satisfying $n=\tau(n)^3.$

2010 Putnam, A4

Tags: putnam number theory , logarithm , greatest common divisor , modular arithmetic , number theory

Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.

2015 Tuymaada Olympiad, 3

Tags: polynomial , algebra

$P(x,y)$ is polynomial with real coefficients and $P(x+2y,x+y)=P(x,y)$. Prove that exists polynomial $Q(t)$ such that $P(x,y)=Q((x^2-2y^2)^2)$ [i]A. Golovanov[/i]

2022 Kyiv City MO Round 2, Problem 4

Tags: geometry

Let $\omega$ denote the circumscribed circle of triangle $ABC$, $I$ be its incenter, and $K$ be any point on arc $AC$ of $\omega$ not containing $B$. Point $P$ is symmetric to $I$ with respect to point $K$. Point $T$ on arc $AC$ of $\omega$ containing point $B$ is such that $\angle KCT = \angle PCI$. Show that the bisectors of angles $AKC$ and $ATC$ meet on line $CI$. [i](Proposed by Anton Trygub)[/i]

2010 IFYM, Sozopol, 7

Tags: trigonometric equation , algebra

Prove the following equality: $4 sin\frac{2\pi }{7}-tg \frac{\pi }{7}=\sqrt{7}$

2012 Thailand Mathematical Olympiad, 4

Tags: geometric inequality , right triangle , incenter , geometry , area

Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.

2005 Today's Calculation Of Integral, 18

Tags: integration , trigonometry , logarithm , calculus , calculus computations

Calculate the following indefinite integrals. [1] $\int (\sin x+\cos x)^4 dx$ [2] $\int \frac{e^{2x}}{e^x+1}dx$ [3] $\int \sin ^ 4 xdx$ [4] $\int \sin 6x\cos 2xdx$ [5] $\int \frac{x^2}{\sqrt{(x+1)^3}}dx$

1998 Harvard-MIT Mathematics Tournament, 6

Tags: calculus

Edward, the author of this test, had to escape from prison to work in the grading room today. He stopped to rest at a place $1,875$ feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of $100 \dfrac{\text{ft}}{\text{s}}$. At the same time, Edward started running away with an acceleration of $1 \dfrac{\text{ft}}{\text{s}^2}$. Assuming that air resistance causes the arrow to decelerate at $1 \dfrac{\text{ft}}{\text{s}^2}$, and that it does hit Edward, how fast was the arrow moving at the moment of impact (in $\dfrac{\text{ft}}{\text{s}}$)?

2010 China Team Selection Test, 3

Tags: pigeonhole principle , ceiling function , number theory

Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers $a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.

2016 Regional Olympiad of Mexico West, 1

Tags: algebra

Indra has a bag for bringing flowers for her grandmother. The first day she brings $n$ flowers. From the second day Indra tries to bring three times plus one with respect to the number of flowers of the previous day. However, if this number is greater or equal to $40$, Indra substracts multiples of $40$ until the remainder is less than this number, since her bag cannot containt so many flowers. For which value of $n$ Indra will bring $30$ flowers the day $2016$?