Olympiad problems

2002 Romania Team Selection Test, 3

Let $a,b$ be positive real numbers. For any positive integer $n$, denote by $x_n$ the sum of digits of the number $[an+b]$ in it's decimal representation. Show that the sequence $(x_n)_{n\ge 1}$ contains a constant subsequence. [i]Laurentiu Panaitopol[/i]

2017 AMC 12/AHSME, 9

Tags: geometry , circles

A circle has center $ (-10,-4) $ and radius $13$. Another circle has center $(3,9) $ and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$? $\textbf{(A)} \text{ 3} \qquad \textbf{(B)} \text{ } 3 \sqrt{3} \qquad \textbf{(C)} \text{ } 4\sqrt{2} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ }\frac{13}{2}$

2007 Vietnam National Olympiad, 2

Tags: function , algebra , limit

Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that: $f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$

Kyiv City MO Seniors 2003+ geometry, 2009.10.4

Tags: rhombus , circumcircle , geometry , incenter

In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that: a) quadrilateral $IKCF$ is rhombus; b) the side of this rhombus is $\sqrt {DF \cdot EK}$. (Rozhkova Maria)

2018 Irish Math Olympiad, 6

Tags: functional equation , algebra

Find all real-valued functions $f$ satisfying $f(2x + f(y)) + f(f(y)) = 4x + 8y$ for all real numbers $x$ and $y$.

1968 All Soviet Union Mathematical Olympiad, 095

Tags: algebra

What is greater, $31^{11}$ or $17^{14}$ ?

2021 MOAA, 17

Tags: team

Compute the remainder when $10^{2021}$ is divided by $10101$. [i]Proposed by Nathan Xiong[/i]

2023 Yasinsky Geometry Olympiad, 1

Tags: geometry

Let $O$ be the circumcenter of triangle $ABC$ and the line $AO$ intersects segment $BC$ at point $T$ . Assume that lines $m$ and $\ell$ passing through point $T$ are perpendicular to $AB$ and $AC$ respectively. If $E$ is the point of intersection of $m$ and $OB$ and $F$ is the point of intersection of $\ell$ and $OC$, prove that $BE = CF$. (Oleksii Karliuchenko)

2003 All-Russian Olympiad, 3

Tags: inequalities , induction , combinatorics

A tree with $n\geq 2$ vertices is given. (A tree is a connected graph without cycles.) The vertices of the tree have real numbers $x_1,x_2,\dots,x_n$ associated with them. Each edge is associated with the product of the two numbers corresponding to the vertices it connects. Let $S$ be a sum of number across all edges. Prove that \[\sqrt{n-1}\left(x_1^2+x_2^2+\dots+x_n^2\right)\geq 2S.\] (Author: V. Dolnikov)

Russian TST 2022, P1

Tags: algebra , trigonometry

Let $a{}$ and $b{}$ be positive integers. Prove that for any real number $x{}$ \[\sum_{j=0}^a\binom{a}{j}\big(2\cos((2j-a)x)\big)^b=\sum_{j=0}^b\binom{b}{j}\big(2\cos((2j-b)x)\big)^a.\]

2005 USAMO, 6

Tags: inequalities , ceiling function , floor function , pigeonhole principle , logarithm , induction , modular arithmetic

For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with \[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]

2000 JBMO ShortLists, 14

Tags: algebra

Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$.

2012 USAMO, 3

Tags: function , modular arithmetic , floor function

Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \ldots$ of nonzero integers such that the equality \[a_k+2a_{2k}+\ldots+na_{nk}=0\]holds for every positive integer $k$.

2013 VJIMC, Problem 3

Tags: polynomial , number theory , algebra

Prove that there is no polynomial $P$ with integer coefficients such that $P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$.

2014 District Olympiad, 3

Tags: similar triangles , incenter , angle bisector , geometry , trigonometry

Let $ABC$ be a triangle in which $\measuredangle{A}=135^{\circ}$. The perpendicular to the line $AB$ erected at $A$ intersects the side $BC$ at $D$, and the angle bisector of $\angle B$ intersects the side $AC$ at $E$. Find the measure of $\measuredangle{BED}$.

2024 Iranian Geometry Olympiad, 3

Tags: geometry

Inside a convex quadrilateral $ABCD$ with $BC>AD$, a point $T$ is chosen. $S$ lies on the segment $AT$ such that $DT = BC, \angle TSD = 90^\circ$. Prove that if $\angle DTA + \angle TAB + \angle ABC = 180^\circ$, then $AB + ST \geqslant CD + AS$. [i]Proposed by Alexander Tereshin - Russia[/i]

1968 Yugoslav Team Selection Test, Problem 1

Tags: inequalities , combinatorics , geometry

Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.

1989 Irish Math Olympiad, 1

Tags: geometry

Suppose $L$ is a fixed line, and $A$ is a fixed point not on $L$. Let $k$ be a fixed nonzero real number. For $P$ a point on $L$, let $Q$ be a point on the line $AP$ with $|AP|\cdot |AQ|=k^2$. Determine the locus of $Q$ as $P$ varies along the line $L$.

2021 Turkey Team Selection Test, 1

Tags: legendre symbol , prime number , divisibility , number theory

Let $n$ be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.

2005 Nordic, 2

Tags: inequalities , calculus , algebra

Let $a,b,c$ be positive real numbers. Prove that \[\frac{2a^2}{b+c} + \frac{2b^2}{c+a} + \frac{2c^2}{a+b} \geq a+b+c\](this is, of course, a joke!) [b]EDITED with exponent 2 over c[/b]

2023 Bosnia and Herzegovina Junior BMO TST, 4.

Tags: combinatorics

Let $n$ be a positive integer. A board with a format $n*n$ is divided in $n*n$ equal squares.Determine all integers $n$≥3 such that the board can be covered in $2*1$ (or $1*2$) pieces so that there is exactly one empty square in each row and each column.

1994 AMC 8, 10

Tags:

For how many positive integer values of $N$ is the expression $\dfrac{36}{N+2}$ an integer? $\text{(A)}\ 7 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 12$

2016 Harvard-MIT Mathematics Tournament, 7

Tags: geometry

Let ABC be a triangle with $AB = 13, BC = 14, CA = 15$. The altitude from $A$ intersects $BC$ at $D$. Let $\omega_1$ and $\omega_2$ be the incircles of $ABD$ and $ACD$, and let the common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersect $AD$ at $E$. Compute the length of $AE$.

V Soros Olympiad 1998 - 99 (Russia), 9.5

Tags: locus , geometry , isosceles

An angle with vertex $A$ is given on the plane. Points $K$ and $P$ are selected on its sides so that $AK + AP = a$, where $a$ is a given segment. Let $M$ be a point on the plane such that the triangle $KPM$ is isosceles with the base $KP$ and the angle at the vertex $M$ equal to the given angle. Find the locus of points $M$ (as $K$ and $P$ move).

1992 Poland - First Round, 4

Tags: functional equation , function , algebra

Determine all functions $f: R \longrightarrow R$ such that $f(x+y)-f(x-y)=f(x)*f(y)$ for $x,y \in R$