Olympiad problems

2009 Postal Coaching, 6

Find all functions $f : N \to N$ such that $$\frac{f(x+y)+f(x)}{2x+f(y)}= \frac{2y+f(x)}{f(x+y)+f(y)}$$ , for all $x, y$ in $N$.

1978 IMO Shortlist, 11

Tags: function , algebra , concave , mean , IMO Shortlist

A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.

2020 Estonia Team Selection Test, 2

Tags: geometry , areas , geometric inequality

The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$. Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.

2013 Saudi Arabia GMO TST, 3

Tags: geometry , incenter , Circumcenter , orthocenter , equal segments , circumcircle

$ABC$ is a triangle, $H$ its orthocenter, $I$ its incenter, $O$ its circumcenter and $\omega$ its circumcircle. Line $CI$ intersects circle $\omega$ at point $D$ different from $C$. Assume that $AB = ID$ and $AH = OH$. Find the angles of triangle $ABC$.

2006 Polish MO Finals, 2

Tags: quadratics , geometry , 3D geometry , number theory proposed , number theory

Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$.

2019 Azerbaijan Senior NMO, 3

Tags: Diophantine equation , number theory

Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$

2014 Saudi Arabia IMO TST, 3

Tags: combinatorics unsolved , combinatorics

There are $2015$ coins on a table. For $i = 1, 2, \dots , 2015$ in succession, one must turn over exactly $i$ coins. Prove that it is always possible either to make all of the coins face up or to make all of the coins face down, but not both.

2023 Iberoamerican, 6

Tags: algebra , polynomial

Let $P$ be a polynomial of degree greater than or equal to $4$ with integer coefficients. An integer $x$ is called $P$-[i]representable[/i] if there exists integer numbers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \geq 0$, more than half of the integers of the set $\{0,1,\dots,N\}$ are $P$-[i]representable[/i], then all the even integers are $P$-[i]representable[/i] or all the odd integers are $P$-[i]representable[/i].

2012 HMNT, 9

Tags: geometry

Triangle $ABC$ satisfies $\angle B > \angle C$. Let $M$ be the midpoint of $BC$, and let the perpendicular bisector of $BC$ meet the circumcircle of $\vartriangle ABC$ at a point $D$ such that points $A$, $D$, $C$, and $B$ appear on the circle in that order. Given that $\angle ADM = 68^o$ and $\angle DAC = 64^o$ , find $\angle B$.

1996 Denmark MO - Mohr Contest, 1

Tags: geometry , right triangle , areas

In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.

1960 AMC 12/AHSME, 5

Tags: AMC

The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is: $ \textbf{(A) }\text{infinitely many} \qquad\textbf{(B) } \text{four}\qquad\textbf{(C) }\text{two}\qquad\textbf{(D) }\text{one}\qquad\textbf{(E) }\text{none} $

2014 PUMaC Combinatorics A, 8

Tags:

There are $60$ friends who want to visit each others home during summer vacation. Everyday, they decide to either stay home or visit the home of everyone who stayed home that day. Find the minimum number of days required for everyone to have visited their friends’ homes.

2023 Iran MO (3rd Round), 2

Tags: geometry

In triangle $\triangle ABC$ , $M$ is the midpoint of arc $(BAC)$ and $N$ is the antipode of $A$ in $(ABC)$. The line through $B$ perpendicular to $AM$ , intersects $AM , (ABC)$ at $D,P$ respectively and a line through $D$ perpendicular to $AC$ , intersects $BC,AC$ at $F,E$ respectively. Prove that $PE,MF,ND$ are concurrent.

2010 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics , number theory

$600$ integer numbers from $[1,1000]$ colored in red. Natural segment $[n,k]$ is called yummy if for every natural $t$ from $[1,k-n]$ there are two red numbers $a,b$ from $[n,k]$ and $b-a=t$ . Prove that there is yummy segment with $[a,b]$ with $b-a \geq 199$

2020 Sharygin Geometry Olympiad, 3

Tags: geometry , circumcircle

Let $ABC$ be a triangle with $\angle C=90^\circ$, and $D$ be a point outside $ABC$, such that $\angle ADC=\angle BAC$. The segments $CD$ and $AB$ meet at point $E$. It is known that the distance from $E$ to $AC$ is equal to the circumradius of triangle $ADE$. Find the angles of triangle $ABC$.

2019 BAMO, B

Tags: geometry , parallelogram , areas

In the figure below, parallelograms $ABCD$ and $BFEC$ have areas $1234$ cm$^2$ and $2804$ cm$^2$, respectively. Points $M$ and $N$ are chosen on sides $AD$ and $FE$, respectively, so that segment $MN$ passes through $B$. Find the area of $\vartriangle MNC$. [img]https://cdn.artofproblemsolving.com/attachments/b/6/8b57b632191bdb3a27ab7c59e2376dab23950b.png[/img]

2011 Turkey Junior National Olympiad, 2

Tags: geometry , circumcircle , analytic geometry , graphing lines , slope , power of a point , radical axis

Let $ABC$ be a triangle with $|AB|=|AC|$. $D$ is the midpoint of $[BC]$. $E$ is the foot of the altitude from $D$ to $AC$. $BE$ cuts the circumcircle of triangle $ABD$ at $B$ and $F$. $DE$ and $AF$ meet at $G$. Prove that $|DG|=|GE|$

2023 ISL, N2

Tags: number theory , IMO Shortlist , AZE IMO TST , TST

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

2016 Mathematical Talent Reward Programme, MCQ: P 2

Tags: function , limit , Summation , infinite sum

Let $f$ be a function satisfying $f(x+y+z)=f(x)+f(y)+f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1)=1$, $f(2)=2$. Then $\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals [list=1] [*] 4 [*] 6 [*] 12 [*] 24 [/list]

1978 AMC 12/AHSME, 14

Tags: AMC

If an integer $n > 8$ is a solution of the equation $x^2 - ax+b=0$ and the representation of $a$ in the base-$n$ number system is $18$, then the base-$n$ representation of $b$ is $\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 80 \qquad \textbf{(D)}\ 81 \qquad \textbf{(E)}\ 280$

2010 LMT, 5

Tags:

Once J and his cheetah collide, J dies a very slow and painful death. The cheetahs come back for his funeral, which is held in a circular stadium with $10$ rows. The first row has $10$ seats in a circle, and each subsequent row has $3$ more seats. However, no two adjacent seats may be occupied due to the size of the cheetahs. What is the maximum number of cheetahs that can fit in the stadium?

2020 LMT Fall, A7 B15

Tags:

Let $S$ denote the sum of all rational numbers of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive divisors of $1300$. If $S$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then find $m+n$. [i]Proposed by Ephram Chun[/i]

2020 Princeton University Math Competition, A4/B5

Tags: combinatorics

Let $P$ be the power set of $\{1, 2, 3, 4\}$ (meaning the elements of P are the subsets of $\{1, 2, 3, 4\}$). How many subsets $S$ of $P$ are there such that no two distinct integers $a, b \in \{1, 2, 3, 4\}$ appear together in exactly one element of $S$?

2021 Malaysia IMONST 1, 12

Tags: number theory , Diophantine equation , diophantine

Determine the number of positive integer solutions $(x,y, z)$ to the equation $xyz = 2(x + y + z)$.

2014 Saudi Arabia IMO TST, 4

Tags: probability , combinatorics unsolved , combinatorics

Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win?