Olympiad problems

2010 Vietnam Team Selection Test, 2

We have $n$ countries. Each country have $m$ persons who live in that country ($n>m>1$). We divide $m \cdot n$ persons into $n$ groups each with $m$ members such that there don't exist two persons in any groups who come from one country. Prove that one can choose $n$ people into one class such that they come from different groups and different countries.

2016 Irish Math Olympiad, 1

Tags: number theory , Divisibility

If the three-digit number $ABC$ is divisible by $27$, prove that the three-digit numbers $BCA$ and $CAB$ are also divisible by $27$.

II Soros Olympiad 1995 - 96 (Russia), 10.5

Tags: geometry , 3D geometry , combinatorics , combinatorial geometry

Is there a six-link broken line in space that passes through all the vertices of a given cube?

2017 Greece National Olympiad, 4

Tags: polynomial , algebra

Let $u$ be the positive root of the equation $x^2+x-4=0$. The polynomial $$P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$ where $n$ is positive integer has non-negative integer coefficients and $P(u)=2017$. 1) Prove that $a_0+a_1+...+a_n\equiv 1\mod 2$. 2) Find the minimum possible value of $a_0+a_1+...+a_n$.

2013 China Team Selection Test, 3

Tags: inequalities , China TST , algebra

Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]

2020 Philippine MO, 4

Tags: PMO , geometry

Let $\triangle ABC$ be an acute triangle with circumcircle $\Gamma$ and $D$ the foot of the altitude from $A$. Suppose that $AD=BC$. Point $M$ is the midpoint of $DC$, and the bisector of $\angle ADC$ meets $AC$ at $N$. Point $P$ lies on $\Gamma$ such that lines $BP$ and $AC$ are parallel. Lines $DN$ and $AM$ meet at $F$, and line $PF$ meets $\Gamma$ again at $Q$. Line $AC$ meets the circumcircle of $\triangle PNQ$ again at $E$. Prove that $\angle DQE = 90^{\circ}$.

2004 Nicolae Coculescu, 1

Tags: Sequences , series , limits , harmonic series , calculus

Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $

2024 Romania National Olympiad, 4

Tags: function , real analysis , continuity

Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions with $g(x)=2f(x)+f(x^2),$ for all $x \in \mathbb{R}.$ a) Prove that, if $f$ is bounded in a neighbourhood of the origin and $g$ is continuous in the origin, then $f$ is continuous in the origin. b) Provide an example of function $f$, discontinuous in the origin, for which the function $g$ is continuous in the origin.

2022 Romania Team Selection Test, 1

Tags: combinatorics , geometry , AZE IMO TST

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

2003 Indonesia MO, 4

Tags: linear algebra , matrix , combinatorics unsolved , combinatorics

Given a $19 \times 19$ matrix where each component is either $1$ or $-1$. Let $b_i$ be the product of all components in the $i$-th row, and $k_i$ be the product of all components in the $i$-th column, for all $1 \le i \le 19$. Prove that for any such matrix, $b_1 + k_1 + b_2 + k_2 + \cdots + b_{19} + k_{19} \neq 0$.

2010 Sharygin Geometry Olympiad, 4

Tags: geometry , tangent , circles , equal angles , common tangents

Circles $\omega_1$ and $\omega_2$ inscribed into equal angles $X_1OY$ and $Y OX_2$ touch lines $OX_1$ and $OX_2$ at points $A_1$ and $A_2$ respectively. Also they touch $OY$ at points $B_1$ and $B_2$. Let $C_1$ be the second common point of $A_1B_2$ and $\omega_1, C_2$ be the second common point of $A_2B_1$ and $\omega_2$. Prove that $C_1C_2$ is the common tangent of two circles.

2011 Iran MO (3rd Round), 5

Tags: algebra , polynomial , trigonometry , algebra proposed

$f(x)$ is a monic polynomial of degree $2$ with integer coefficients such that $f(x)$ doesn't have any real roots and also $f(0)$ is a square-free integer (and is not $1$ or $-1$). Prove that for every integer $n$ the polynomial $f(x^n)$ is irreducible over $\mathbb Z[x]$. [i]proposed by Mohammadmahdi Yazdi[/i]

2023 MMATHS, 3

Tags: Yale , MMATHS

Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand $(m - a)(n - b),$ where $a, b, m, n$ are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for $m$ and $n,$ he realizes that the value of ChatSFFT’s result is the same as the real result in terms of $a$ and $b$. How many such pairs are there?

1992 India National Olympiad, 7

Tags: combinatorics , counting , Chessboard , Arithmetic Progression

Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.

2006 IMO, 2

Tags: combinatorics , polygon , Extremal combinatorics , IMO , IMO 2006 , IMO Shortlist , Dusan Djukic

Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i]. Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

2017 Harvard-MIT Mathematics Tournament, 8

Tags: probability , number theory , relatively prime

Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is [I]cool[/I] if for each of the $16$ frogs, the number of friends they made during the meeting is a multiple of $4$. Say that the probability of the situation being cool can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime. Find $a$.

2012 Cono Sur Olympiad, 4

Tags: number theory proposed , number theory

4. Find the biggest positive integer $n$, lesser thar $2012$, that has the following property: If $p$ is a prime divisor of $n$, then $p^2 - 1$ is a divisor of $n$.

2021 Durer Math Competition Finals, 1

Tags: number theory , prime

Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder $1$ after division by $6$.

2022 Francophone Mathematical Olympiad, 3

Tags: geometry

Let $\triangle ABC$ a triangle, and $D$ the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$. the line parallel to $AC$ passing by the point $B$, intersect the line $AD$ at $X$. the line parallel to $CX$ passing by the point $B$, intersect $AC$ at $Y$. $E = (AYB) \cap BX$ . prove that $C$ , $D$ and $E$ collinear.

2013 Gheorghe Vranceanu, 1

Tags: function , integration , trigonometry , IVT , real analysis

Find the pairs of functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ with $ f $ continuous, $ g $ differentiable and satisfying: $$ -\sin g(x) + \int \cos f(x)dx =\cos g(x) +\int \sin f(x)dx $$

2006 Federal Math Competition of S&M, Problem 3

Tags: number theory

Determine the largest natural number whose all decimal digits are different and which is divisible by each of its digits.

KoMaL A Problems 2020/2021, A. 796

Tags: geometry , komal

Let $ABCD$ be a cyclic quadrilateral. Let lines $AB$ and $CD$ intersect in $P,$ and lines $BC$ and $DA$ intersect in $Q.$ The feet of the perpendiculars from $P$ to $BC$ and $DA$ are $K$ and $L,$ and the feet of the perpendiculars from $Q$ to $AB$ and $CD$ are $M$ and $N.$ The midpoint of diagonal $AC$ is $F.$ Prove that the circumcircles of triangles $FKN$ and $FLM,$ and the line $PQ$ are concurrent. [i]Based on a problem by Ádám Péter Balogh, Szeged[/i]

2021 VIASM Math Olympiad Test, Problem 3

Tags: combinatorics

Given the positive integer $n$. Let $X = \{1, 2,..., n\}$. For each nonempty subset $A$ of $X$, set $r(A) = max_A - min_A$, where $max_A, min_A$ are the greatest and smallest elements of $A$, respectively. Find the mean value of $r(A)$ when $A$ runs on subsets of $X$.

PEN O Problems, 12

Tags: induction , modular arithmetic

Let $m$ and $n$ be positive integers. If $x_1$, $x_2$, $\cdots$, $x_m$ are positive integers whose arithmetic mean is less than $n+1$ and if $y_1$, $y_2$, $\cdots$, $y_n$ are positive integers whose arithmetic mean is less than $m+1$, prove that some sum of one or more $x$'s equals some sum of one or more $y$'s.

1967 IMO Longlists, 53

Tags: geometry , construction , bisector , midpoint , Center , IMO Shortlist , IMO Longlist

In making Euclidean constructions in geometry it is permitted to use a ruler and a pair of compasses. In the constructions considered in this question no compasses are permitted, but the ruler is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the rule. Then the distance between the parallel lines is equal to the breadth of the ruler. Carry through the following constructions with such a ruler. Construct: [b]a)[/b] The bisector of a given angle. [b]b)[/b] The midpoint of a given rectilinear line segment. [b]c)[/b] The center of a circle through three given non-collinear points. [b]d)[/b] A line through a given point parallel to a given line.