Olympiad problems

2018 JBMO Shortlist, G2

Let $ABC$ be a right angled triangle with $\angle A = 90^o$ and $AD$ its altitude. We draw parallel lines from $D$ to the vertical sides of the triangle and we call $E, Z$ their points of intersection with $AB$ and $AC$ respectively. The parallel line from $C$ to $EZ$ intersects the line $AB$ at the point $N$. Let $A' $ be the symmetric of $A$ with respect to the line $EZ$ and $I, K$ the projections of $A'$ onto $AB$ and $AC$ respectively. If $T$ is the point of intersection of the lines $IK$ and $DE$, prove that $\angle NA'T = \angle ADT$.

2024 Junior Balkan Team Selection Tests - Romania, P3

Tags: number theory

[b]Version 1.[/b] Find all primes $p$ satisfying the following conditions: (i) $\frac{p+1}{2}$ is a prime number. (ii) There are at least three distinct positive integers $n$ for which $\frac{p^2+n}{p+n^2}$ is an integer. [b]Version 2.[/b] Let $p \neq 5$ be a prime number such that $\frac{p+1}{2}$ is also a prime. Suppose there exist positive integers $a <b$ such that $\frac{p^2+a}{p+a^2}$ and $\frac{p^2+b}{p+b^2}$ are integers. Show that $b=(a-1)^2+1$.

2009 Romania National Olympiad, 4

Tags: function , calculus , derivative , real analysis

Let $f,g,h:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is differentiable, $g$ and $h$ are monotonic, and $f'=f+g+h$. Prove that the set of the points of discontinuity of $g$ coincides with the respective set of $h$.

2012 Saint Petersburg Mathematical Olympiad, 5

Tags: combinatorics , number theory

In the $100 \times 100$ table in every cell there is natural number. All numbers in same row or column are different. Can be that for every square sum of numbers, that are in angle cells, is square number ?

2025 Korea Winter Program Practice Test, P4

Tags: geometry

Let $\omega$ be the circumcircle of triangle $ABC$ with center $O$, and the $A$ inmixtilinear circle is tangent to $AB, AC, \omega$ at $D,E,T$ respectively. $P$ is the intersection of $TO$ and $DE$ and $X$ is the intersection of $AP$ and $\omega$. Prove that the isogonal conjugate of $P$ lies on the line passing through the midpoint of $BC$ and $X$.

1998 All-Russian Olympiad Regional Round, 11.2

Tags: perpendicular , geometry

Circle $S$ with center $O$ and circle $S'$ intersect at points $A$ and $B$. Point $C$ is taken on the arc of a circle $S$ lying inside $S'$. Denote the intersection points of $AC$ and $BC$ with $S'$, other than $A$ and $B$, as $E$ and $D$, respectively. Prove that lines $DE$ and $OC$ are perpendicular.

2002 Junior Balkan Team Selection Tests - Moldova, 7

Tags: geometry , locus , perimeter , angle

The side of the square $ABCD$ has a length equal to $1$. On the sides $(BC)$ ¸and $(CD)$ take respectively the arbitrary points $M$ and $N$ so that the perimeter of the triangle $MCN$ is equal to $2$. a) Determine the measure of the angle $\angle MAN$. b) If the point $P$ is the foot of the perpendicular taken from point $A$ to the line $MN$, determine the locus of the points $P$.

2008 Romania National Olympiad, 3

Tags: inequalities , algebra

Let $ a,b \in [0,1]$. Prove that \[ \frac 1{1\plus{}a\plus{}b} \leq 1 \minus{} \frac {a\plus{}b}2 \plus{} \frac {ab}3.\]

2011 Korea - Final Round, 3

Tags: combinatorics

There are $n$ boys $a_1, a_2, \ldots, a_n$ and $n$ girls $b_1, b_2, \ldots, b_n $. Some pairs of them are connected. Any two boys or two girls are not connected, and $a_i$ and $b_i$ are not connected for all $i \in \{ 1,2,\ldots,n\}$. Now all boys and girls are divided into several groups satisfying two conditions: (i) Every groups contains an equal number of boys and girls. (ii) There is no connected pair in the same group. Assume that the number of connected pairs is $m$. Show that we can make the number of groups not larger than $\max\left \{2, \dfrac{2m}{n} +1\right \}$.

1992 Flanders Math Olympiad, 2

Tags: probability

It has come to a policeman's ears that 5 gangsters (all of different height) are meeting, one of them is the clan leader, he's the tallest of the 5. He knows the members will leave the building one by one, with a 10-minute break between them, and too bad for him Belgium has not enough policemen to follow all gangsters, so he's on his own to spot the clanleader, and he can only follow one member. So he decides to let go the first 2 people, and then follow the first one that is taller than those two. What's the chance he actually catches the clan leader like this?

2001 China Western Mathematical Olympiad, 3

Tags: relatively prime , number theory

Let $ n, m$ be positive integers of different parity, and $ n > m$. Find all integers $ x$ such that $ \frac {x^{2^n} \minus{} 1}{x^{2^m} \minus{} 1}$ is a perfect square.

2000 Moldova National Olympiad, Problem 5

Tags: combinatorics , graph theory

An airline offer $2000$ two-way routes connecting $64$ towns in a country. Show that it is possible to reach any town from any other town using the offered routes.

2019 China Team Selection Test, 1

Tags: geometry , circumcircle

$ABCDE$ is a cyclic pentagon, with circumcentre $O$. $AB=AE=CD$. $I$ midpoint of $BC$. $J$ midpoint of $DE$. $F$ is the orthocentre of $\triangle ABE$, and $G$ the centroid of $\triangle AIJ$.$CE$ intersects $BD$ at $H$, $OG$ intersects $FH$ at $M$. Show that $AM\perp CD$.

2016 Saudi Arabia Pre-TST, 2.4

Tags: number theory , product , divide , divisible

Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b > n$ such that $$\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)$$.

2016 Balkan MO Shortlist, A1

Tags: algebra , inequality

Let $a, b,c$ be positive real numbers. Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$

2003 China Team Selection Test, 2

Tags: function , functional equation , algebra

Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

2021 OMMock - Mexico National Olympiad Mock Exam, 6

Tags: number theory

Let $a$ and $b$ be fixed positive integers. We say that a prime $p$ is [i]fun[/i] if there exists a positive integer $n$ satisfying the following conditions: [list] [*]$p$ divides $a^{n!} + b$. [*]$p$ divides $a^{(n + 1)!} + b$. [*]$p < 2n^2 + 1$. [/list] Show that there are finitely many fun primes.

1999 Mongolian Mathematical Olympiad, Problem 2

Tags: functional equation , fe

Find all functions $f:\mathbb R\to\mathbb R$ such that (i) $f(0)=1$; (ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$; (iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.

2021 Math Prize for Girls Problems, 7

Tags:

Compute the value of the infinite series \[ \sum_{k=0}^{\infty} \frac{\cos(k \pi / 4)}{2^k} \, . \]

2021 Belarusian National Olympiad, 11.7

Tags: number theory

Prove that for any positive integer $n$, there exist pairwise distinct positive integers $a,b,c$, not equal to $n$, such that $ab+n, ac+n, bc+n$ are all perfect squares.

2016 Romania National Olympiad, 3

Tags: geometry , geometric inequality , geometric inequalities

If $a, b$ and $c$ are the length of the sides of a triangle, show that $$\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.$$

2012 Romanian Masters In Mathematics, 4

Tags: modular arithmetic , function , number theory

Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not. [i](Russia) Valery Senderov[/i]

2014 AMC 12/AHSME, 3

Tags: symmetry

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? ${ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6$

2002 All-Russian Olympiad, 2

Tags: incenter , angle bisector , geometry

Point $A$ lies on one ray and points $B,C$ lie on the other ray of an angle with the vertex at $O$ such that $B$ lies between $O$ and $C$. Let $O_1$ be the incenter of $\triangle OAB$ and $O_2$ be the center of the excircle of $\triangle OAC$ touching side $AC$. Prove that if $O_1A = O_2A$, then the triangle $ABC$ is isosceles.

1972 IMO Longlists, 23

Tags: number theory

Does there exist a $2n$-digit number $\overline{a_{2n}a_{2n-1}\cdots a_1}$(for an arbitrary $n$) for which the following equality holds: \[\overline{a_{2n}\cdots a_1}= (\overline{a_n \cdots a_1})^2?\]