Olympiad problems

2007 Tournament Of Towns, 3

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Two players in turns color the squares of a $4 \times 4$ grid, one square at the time. Player loses if after his move a square of $2\times2$ is colored completely. Which of the players has the winning strategy, First or Second? [i](3 points)[/i]

2005 Junior Balkan Team Selection Tests - Romania, 6

Let $ABC$ be an equilateral triangle and $M$ be a point inside the triangle. We denote by $A'$, $B'$, $C'$ the projections of the point $M$ on the sides $BC$, $CA$ and $AB$ respectively. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent if and only if $M$ belongs to an altitude of the triangle.

2004 Thailand Mathematical Olympiad, 10

Tags: number theory , combinatorics , Sum , divisible

Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.

2025 Bangladesh Mathematical Olympiad, P10

Tags: functional equation , algebra

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(x+f(y^2)) + f(xy) = f(x) + yf(x+y)$$ for all $x, y \in \mathbb{R}$. [i]Proposed by Md. Fuad Al Alam[/i]

2024 District Olympiad, P4

Tags: algebra , functional equation

Let $n\in\mathbb{N}\setminus\left\{0\right\}$ be a positive integer. Find all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying that : $$f(x+y^{2n})=f(f(x))+y^{2n-1}f(y),(\forall)x,y\in\mathbb{R},$$ and $f(x)=0$ has an unique solution.

2022 Assam Mathematical Olympiad, 10

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Let the vertices of the square $ABCD$ are on a circle of radius $r$ and with center $O$. Let $P, Q, R$ and $S$ are the mid points of $AB, BC, CD$ and $DA$ respectively. Then; (a) Show that the quadrilateral $P QRS$ is a square. (b) Find the distance from the mid point of $P Q$ to $O$.

2017 Turkey MO (2nd round), 1

Tags: combinatorics

A wedding is going to be held in a city with $25$ types of meals, to which some of the $2017$ citizens will be invited. All of the citizens like some meals and each meal is liked by at least one person. A "$suitable$ $list$" is a set of citizens, such that each meal is liked by at least one person in the set. A "$kamber$ $group$" is a set that contains at least one person from each "$suitable$ $list$". Given a "$kamber$ $group$", which has no subset (other than itself) that is also a "$kamber$ $group$", prove that there exists a meal, which is liked by everyone in the group.

1997 Israel National Olympiad, 5

Tags: coprime , number theory , primes

The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime. Show that at least one of these numbers is prime.

ICMC 4, 2

Tags: linear algebra , Matrices , Nilpotent , ICMC , college contests , matrix

Let $A$ be a square matrix with entries in the field $\mathbb Z / p \mathbb Z$ such that $A^n - I$ is invertible for every positive integer $n$. Prove that there exists a positive integer $m$ such that $A^m = 0$. [i](A matrix having entries in the field $\mathbb Z / p \mathbb Z$ means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of $p$.)[/i] [i]Proposed by Tony Wang[/i]

2020 JBMO Shortlist, 2

Tags: Junior , Balkan , shortlist , 2020 , algebra , Sequence

Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and $a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$ for $n \ge 1$. Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.

2009 China Team Selection Test, 1

Tags: inequalities , number theory proposed , number theory

Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$

2011 Tokio University Entry Examination, 2

Tags: algorithm , number theory , Euclidean algorithm

Define real number $y$ as the fractional part of real number $x$ such that $0\leq y<1$ and $x-y$ is integer. Denote this by $<x>$. For real number $a$, define an infinite sequence $\{a_n\}\ (n=1,\ 2,\ 3,\ \cdots)$ inductively as follows. (i) $a_1=<a>$ (ii) If $a\n\neq 0$, then $a_{n+1}=\left<\frac{1}{a_n}\right>$, if $a_n=0$, then $a_{n+1}=0$. (1) For $a=\sqrt{2}$, find $a_n$. (2) For any natural number $n$, find real number $a\geq \frac 13$ such that $a_n=a$. (3) Let $a$ be a rational number. When we express $a=\frac{p}{q}$ with integer $p$, natural number $q$, prove that $a_n=0$ for any natural number $n\geq q$. [i]2011 Tokyo University entrance exam/Science, Problem 2[/i]

I Soros Olympiad 1994-95 (Rus + Ukr), 9.8

Tags: algebra , inequalities

Let $f(x) =x^2-2x$. Find all $x$ for which $f(f(x))<3$.

2014 Saudi Arabia GMO TST, 1

Tags: number theory , Diophantine equation , diophantine

Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$

2007 Today's Calculation Of Integral, 248

Tags: calculus , integration , trigonometry , function , trig identities , calculus computations

Evaluate $ \int_{\frac {\pi}{4}}^{\frac {3}{4}\pi } \cos \frac {1}{\sin \left(\frac {1}{\sin x}\right)}\cdot \cos \left(\frac {1}{\sin x}\right)\cdot \frac {\cos x}{\sin ^ 2 x\cdot \sin ^ 2 \left(\frac {1}{\sin x }\right)}\ dx$ Last Edited, Sorry kunny

2020 USOJMO, 5

Tags: USOMO , usojmo , USO(J)MO , 2020 USAMO , 2020 USAJMO

Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs. [i]Proposed by Ankan Bhattacharya[/i]

2025 Harvard-MIT Mathematics Tournament, 20

Tags: guts

Compute the $100$th smallest positive multiple of $7$ whose digits in base $10$ are all strictly less than $3.$

2014 India Regional Mathematical Olympiad, 1

Tags: geometry , circumcircle , Asymptote , perpendicular bisector

Let $ABC$ be a triangle with $\angle ABC $ as the largest angle. Let $R$ be its circumcenter. Let the circumcircle of triangle $ARB$ cut $AC$ again at $X$. Prove that $RX$ is perpendicular to $BC$.

2012 JBMO ShortLists, 3

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Let $a$ , $b$ , $c$ be positive real numbers such that $a+b+c=a^2+b^2+c^2$ . Prove that : \[\frac{a^2}{a^2+ab}+\frac{b^2}{b^2+bc}+\frac{c^2}{c^2+ca} \geq \frac{a+b+c}{2}\]

1986 AMC 12/AHSME, 23

Tags: algebra , polynomial , AMC

Let \[N = 69^{5} + 5\cdot 69^{4} + 10\cdot 69^{3} + 10\cdot 69^{2} + 5\cdot 69 + 1.\] How many positive integers are factors of $N$? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 216 $

1984 Polish MO Finals, 6

Tags: combinatorics

Cities $P_1,...,P_{1025}$ are connected to each other by airlines $A_1,...,A_{10}$ so that for any two distinct cities $P_k$ and $P_m$ there is an airline offering a direct flight between them. Prove that one of the airlines can offer a round trip with an odd number of flights.

2017 JBMO Shortlist, NT1

Tags: number theory

Determine all the sets of six consecutive positive integers such that the product of some two of them . added to the product of some other two of them is equal to the product of the remaining two numbers.

2008 Poland - Second Round, 2

Tags: geometry , geometric transformation , reflection , geometry proposed

We are given a triangle $ ABC$ such that $ AC \equal{} BC$. There is a point $ D$ lying on the segment $ AB$, and $ AD < DB$. The point $ E$ is symmetrical to $ A$ with respect to $ CD$. Prove that: \[\frac {AC}{CD} \equal{} \frac {BE}{BD \minus{} AD}\]

1991 Chile National Olympiad, 5

Tags: Fibonacci , number theory , algebra , Sum

The sequence $(a_k)$, $k> 0$ is Fibonacci, with $a_0 = a_1 = 1$. Calculate the value of $$\sum_{j = 0}^{\infty} \frac{a_j}{2^j}$$

2007 Princeton University Math Competition, 9

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Find $p+r$ if $p$ and $q$ are primes and $r$ is an integer such that \[ \left( r^2 + pr + 1 \right) \cdot \left( r^2 + \left( p^2 - q \right) r - p \right) = pq. \]