Olympiad problems

2009 Indonesia MO, 2

Find the lowest possible values from the function \[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\] for any real numbers $ x$.

2001 Croatia National Olympiad, Problem 1

Tags: Diophantine equation , number theory , PreSchool

Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.

2015 Dutch IMO TST, 5

Tags: number theory , divides , min , max , positive integers

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

Oliforum Contest III 2012, 5

Tags: geometry , cyclic quadrilateral , orthogonal

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

2023 Brazil EGMO Team Selection Test, 2

Tags: number theory , Sets , prime numbers

Let $A$ be a finite set made up of prime numbers. Determine if there exists an infinite set $B$ that satisfies the following conditions: $(i)$ the prime factors of any element of $B$ are in $A$; $(ii)$ no term of $B$ divides another element of this set.

2019 ELMO Shortlist, N4

Tags: number theory , Divisibility

A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$. [i]Proposed by Carl Schildkraut and Holden Mui[/i]

1980 Austrian-Polish Competition, 1

Tags: algebra , arithmetic sequence , Arithmetic Progression , Sequence , IMO Shortlist

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

2020-2021 Fall SDPC, 5

Tags: geometry

Let $ABC$ be a triangle with area $1$. Let $D$ be a point on segment $BC$. Let points $E$ and $F$ on $AC$ and $AB$, respectively, satisfy $DE || AB$ and $DF || AC$. Compute, with proof, the area of the quadrilateral with vertices at $E$, $F$, the midpoint of $BD$, and the midpoint of $CD$.

2007 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , modulo , prime

Let $p > 3$ be a prime number and $ x$ an integer, denote by $r ( x )\in \{ 0 , 1 , ... , p - 1 \}$ to the rest of $x$ modulo $p$ . Let $x_1, x_2, ... , x_k$ ( $2 < k < p$) different integers modulo $p$ and not divisible by $p$. We say that a number $a \in \{ 1 , 2 ,..., p -1 \}$ is [i]good [/i] if $r ( a x_1) < r ( a x_2) <...< r ( a x_k)$. Show that there are at most $\frac{2 p}{k + 1}-{ 1}$ [i]good [/i] numbers.

2025 Euler Olympiad, Round 1, 8

Tags: ratio , algebra , Euler Olympiad

Let $S$ be the set of non-negative integer powers of $3$ and $5$, $S = \{1, 3, 5, 3^2, 5^2, \ldots \}$. For every $a$ and $b$ in $S$ satisfying $$ \left| \pi - \frac{a}{b} \right| < 0.1 $$ Find the minimum value of $ab$. [i]Proposed by Irakli Shalibashvili, Georgia [/i]

2024 Sharygin Geometry Olympiad, 8.5

Tags: geometry , rectangle , Sharygin Geometry Olympiad , Sharygin 2024

The vertices $M$, $N$, $K$ of rectangle $KLMN$ lie on the sides $AB$, $BC$, $CA$ respectively of a regular triangle $ABC$ in such a way that $AM = 2$, $KC = 1$. The vertex $L$ lies outside the triangle. Find the value of $\angle KMN$.

2023 Paraguay Mathematical Olympiad, 3

Tags: geometry

In the figure, points $A$, $B$, $C$ and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$ and segment $CD$ measures $4$. The circumferences woth centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$? [img]https://cdn.artofproblemsolving.com/attachments/d/4/378243b9f4203e103af266e551eadccfc96adf.png[/img]

2016 South East Mathematical Olympiad, 4

Tags: combinatorics

For any four points on a plane, if the areas of four triangles formed are different positive integer and six distances between those four points are also six different positive integers, then the convex closure of $4$ points is called a "lotus design." (1) Construct an example of "lotus design". Also what are areas and distances in your example? (2) Prove that there are infinitely many "lotus design" which are not similar.

2010 LMT, 15

Tags:

Determine the number of ordered pairs $(x,y)$ with $x$ and $y$ integers between $-5$ and $5,$ inclusive, such that $(x+y)(x+3y)=(x+2y)^2.$

2013 NIMO Problems, 10

Tags: algebra , polynomial

Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$. [i]Proposed by Evan Chen[/i]

2004 Iran MO (3rd Round), 3

Tags: vector , combinatorics proposed , combinatorics

Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$.We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$ Suppose $ n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places. Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$

2025 District Olympiad, P4

Tags: Sequence

Let $(x_n)_{n\geq 1}$ be an increasing and unbounded sequence of positive integers such that $x_1=1$ and $x_{n+1}\leq 2x_n$ for all $n\geq 1$. Prove that every positive integer can be written as a finite sum of distinct terms of the sequence. [i]Note:[/i] Two terms $x_i$ and $x_j$ of the sequence are considered distinct if $i\neq j$.

2013 Kosovo National Mathematical Olympiad, 4

Tags: induction

Let be $n$ positive integer than calculate: $1\cdot 1!+2\cdot2!+...+n\cdot n!$

2011 Junior Balkan MO, 1

Tags: inequalities , inequalities unsolved

Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that: $\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$

KoMaL A Problems 2021/2022, A. 824

Tags: komal , algebra

An infinite set $S$ of positive numbers is called thick, if in every interval of the form $\left [1/(n+1),1/n\right]$ (where $n$ is an arbitrary positive integer) there is a number which is the difference of two elements from $S$. Does there exist a thick set such that the sum of its elements is finite? Proposed by [i]Gábor Szűcs[/i], Szikszó

KoMaL A Problems 2021/2022, A. 816

Tags: komal , combinatorics

Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide whether there is the same number of both types of cars. He can try to fit together two cars in one try. What is the least number of tries needed? [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

2017-IMOC, G6

Tags: geometry , geometric inequality

A point $P$ lies inside $\vartriangle ABC$ such that the values of areas of $\vartriangle PAB, \vartriangle PBC, \vartriangle PCA$ can form a triangle. Let $BC = a,CA = b,AB = c, PA = x,PB = y, PC = z$, prove that $$\frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{x + y + z} \le a + b + c$$

2006 All-Russian Olympiad, 7

Tags: algebra , polynomial , algebra proposed

Assume that the polynomial $\left(x+1\right)^n-1$ is divisible by some polynomial $P\left(x\right)=x^k+c_{k-1}x^{k-1}+c_{k-2}x^{k-2}+...+c_1x+c_0$, whose degree $k$ is even and whose coefficients $c_{k-1}$, $c_{k-2}$, ..., $c_1$, $c_0$ all are odd integers. Show that $k+1\mid n$.

2019 IMC, 5

Tags: linear algebra , college contests

Determine whether there exist an odd positive integer $n$ and $n\times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions: [list=1] [*]$\det (B)=1$;[/*] [*]$AB=BA$;[/*] [*]$A^4+4A^2B^2+16B^4=2019I$.[/*] [/list] (Here $I$ denotes the $n\times n$ identity matrix.) [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]

2018 Azerbaijan Junior NMO, 1

Tags: combinatorics

First $20$ positive integers are written on a board. It is known that, after you erase a number from the board, there exists a number that is equal to the arithmetic mean of the rest of the numbers left on the board. Find all the numbers that could've been erased.