Olympiad problems

2023 Canadian Junior Mathematical Olympiad, 2

An acute triangle is a triangle that has all angles less that $90^{\circ}$ ($90^{\circ}$ is a Right Angle). Let $ABC$ be a right-angled triangle with $\angle ACB =90^{\circ}.$ Let $CD$ be the altitude from $C$ to $AB,$ and let $E$ be the intersection of the angle bisector of $\angle ACD$ with $AD.$ Let $EF$ be the altitude from $E$ to $BC.$ Prove that the circumcircle of $BEF$ passes through the midpoint of $CE.$

2022 Germany Team Selection Test, 1

Tags: combinatorics , graph theory , greatest common divisor

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

1993 China National Olympiad, 2

Tags: inequalities

Given a natural number $k$ and a real number $a (a>0)$, find the maximal value of $a^{k_1}+a^{k_2}+\cdots +a^{k_r}$, where $k_1+k_2+\cdots +k_r=k$ ($k_i\in \mathbb{N} ,1\le r \le k$).

2018 District Olympiad, 3

Tags: inequalities , logarithm

Let $a, b, c$ be strictly positive real numbers such that $1 < b \le c^2 \le a^{10}$, and \[\log_ab + 2\log_bc + 5\log_ca = 12.\] Prove that \[2\log_ac + 5\log_cb + 10\log_ba \ge 21.\]

2020 Mediterranean Mathematics Olympiad, 4

Tags: geometry , collinear

Let $P,Q,R$ be three points on a circle $k_1$ with $|PQ|=|PR|$ and $|PQ|>|QR|$. Let $k_2$ be the circle with center in $P$ that goes through $Q$ and $R$. The circle with center $Q$ through $R$ intersects $k_1$ in another point $X\ne R$ and intersects $k_2$ in another point $Y\ne R$. The two points $X$ and $R$ lie on different sides of the line through $PQ$. Show that the three points $P$, $X$, $Y$ lie on a common line.

2010 AMC 10, 5

Tags: geometry

The area of a circle whose circumference is $ 24\pi$ is $ k\pi$. What is the value of $ k$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 144$

2012 German National Olympiad, 3

Tags: geometry , circumcircle , tangent , triangle , circles

Let $ABC$ a triangle and $k$ a circle such that: (1) The circle $k$ passes through $A$ and $B$ and touches the line $AC.$ (2) The tangent to $k$ at $B$ intersects the line $AC$ in a point $X\ne C.$ (3) The circumcircle $\omega$ of $BXC$ intersects $k$ in a point $Q\ne B.$ (4) The tangent to $\omega$ at $X$ intersects the line $AB$ in a point $Y.$ Prove that the line $XY$ is tangent to the circumcircle of $BQY.$

1988 Bundeswettbewerb Mathematik, 4

Tags: number theory , diophantine equation , equation , algebra

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

1974 All Soviet Union Mathematical Olympiad, 200

Tags: algebra , combinatorics

a) Prove that you can rearrange the numbers $1, 2, ... , 32$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean. b) Can you rearrange the numbers $1, 2, ... , 100$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean?

1993 Moldova Team Selection Test, 9

Tags: number theory

Positive integer $q{}$ is $m-additive$ $(m\in\mathbb{N}, m\geq2)$ if there exist pairwise distinct positive integers $a_1,a_2,\ldots,a_m$ such that $q=a_1+a_2+\ldots+a_m$ and $a_i | a_{i+1}$ for $i=1,2,\ldots,m-1$. [b]a)[/b] Prove that $1993$ is $8$-additive, but $9$-additive. [b]b)[/b] Determine the greatest integer $m$ for which $2102$ is $m$-additive.

2016 Regional Olympiad of Mexico Center Zone, 2

Tags: game , combinatorics , winning strategy , game strategy

There are seven piles with $2014$ pebbles each and a pile with $2008$ pebbles. Ana and Beto play in turns and Ana always plays first. One move consists of removing pebbles from all the piles. From each pile is removed a different amount of pebbles, between $1$ and $8$ pebbles. The first player who cannot make a move loses. a) Who has a winning strategy? b) If there were seven piles with $2015$ pebbles each and a pile with $2008$ pebbles, who has a winning strategy?

2016 Romanian Master of Mathematics, 1

Tags: geometry

Let $ABC$ be a triangle and let $D$ be a point on the segment $BC, D\neq B$ and $D\neq C$. The circle $ABD$ meets the segment $AC$ again at an interior point $E$. The circle $ACD$ meets the segment $AB$ again at an interior point $F$. Let $A'$ be the reflection of $A$ in the line $BC$. The lines $A'C$ and $DE$ meet at $P$, and the lines $A'B$ and $DF$ meet at $Q$. Prove that the lines $AD, BP$ and $CQ$ are concurrent (or all parallel).

2008 Portugal MO, 5

Tags: geometry

Let $ABC$ be a right-angled triangle in $A$ such that $AB<AC$. Let $M$ be the midpoint of $BC$ and let $D$ be the intersection of $AC$ with the perpendicular line to $BC$ which passes through $M$. Let $E$ be the intersection point of the parallel line to $AC$ which passes through $M$ with the perpendicular line to $BD$ which passes through $B$. Prove that triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC=60^{\circ}$.

2014 Greece Team Selection Test, 4

Tags: binomial coefficient , combinatorics

Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.

1960 Miklós Schweitzer, 8

Tags:

[b]8.[/b] Let $f$ be a bounded real function defined on the unit cube $H$ of the $n$-dimensional space and, for a given $y$, let $A_y$ and $B_y$ denote the parts of the interior of $H$ on which $f>y$ and $f<y$, respectively. Show that $f$ is integrable in the Riemannian sense if and only if for every $y$ almost all points of $A_y$ and $B_y$ are inner points. [b](R. 9)[/b]

1964 Miklós Schweitzer, 9

Tags: function , topology , real analysis

Let $ E$ be the set of all real functions on $ I\equal{}[0,1]$. Prove that one cannot define a topology on $ E$ in which $ f_n\rightarrow f$ holds if and only if $ f_n$ converges to $ f$ almost everywhere.

2018 Peru Iberoamerican Team Selection Test, P7

Tags: combinatorics

There is a finite set of points in the plane, where each point is painted in any of $ n $ different colors $ (n \ge 4) $. It is known that there is at least one point of each color and that the distance between any pair of different colored points is less than or equal a 1. Prove that it is possible to choose 3 colors so that, by removing all points of those colors, the remaining set of points can be covered with a radius circle $ \frac {1} {\sqrt {3}} $.

1965 AMC 12/AHSME, 1

Tags:

The number of real values of $ x$ satisfying the equation $ 2^{2x^2 \minus{} 7x \plus{} 5} \equal{} 1$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{more than 4}$

2006 Mexico National Olympiad, 1

Tags: number theory

Let $ab$ be a two digit number. A positive integer $n$ is a [i]relative[/i] of $ab$ if: [list] [*] The units digit of $n$ is $b$. [*] The remaining digits of $n$ are nonzero and add up to $a$.[/list] Find all two digit numbers which divide all of their relatives.

2024 Thailand TST, 2

Tags: geometry

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$. Prove that $\angle BXP = \angle CXQ$. [i]Kian Moshiri, United Kingdom[/i]

2019 Korea USCM, 6

Tags: real analysis , inequalities , integral inequality , function

A function $f:[0,\infty)\to[0,\infty)$ is integrable and $$\int_0^\infty f(x)^2 dx<\infty,\quad \int_0^\infty xf(x) dx <\infty$$ Prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)$$

2005 MOP Homework, 7

Tags: polynomial , algebra

Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$ over the integers for every $i$.

2004 Iran Team Selection Test, 6

Tags: algebra , polynomial , limit , number theory

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

2017 ITAMO, 2

Tags: algebra , number theory

Let $n\geq 2$ be an integer. Consider the solutions of the system $$\begin{cases} n=a+b-c \\ n=a^2+b^2-c^2 \end{cases}$$ where $a,b,c$ are integers. Show that there is at least one solution and that the solutions are finitely many.

2025 Harvard-MIT Mathematics Tournament, 16

Tags: guts

The [i]Cantor set[/i] is defined as the set of real numbers $x$ such that $0 \le x < 1$ and the digit $1$ does not appear in the base-$3$ expansion of $x.$ Two numbers are uniformly and independently selected at random from the Cantor set. Compute the expected value of their difference. (Formally, one can pick a number $x$ uniformly at random from the Cantor set by first picking a real number $y$ uniformly at random from the interval $[0, 1)$, writing it out in binary, reading its digits as if they were in base-$3,$ and setting $x$ to $2$ times the result.)