Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$for all $x, y \in R$.

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \] Find all natural numbers $ n $ such that $ s(n) = 2010 $

$300$ parliament members are divided into $3$ chambers, each chamber consists of $100$ members. For every $2$ members, they either know each other or are strangers to each other.Show that no matter how they are divided into these $3$ chambers, it is always possible to choose $2$ members, each from different chamber such that there exist $17$ members from the third chamber so that all of them knows these two members, or all of them are strangers to these two members.

Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and $$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$$\forall n=1,2,...$ Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i + 1}) > a_{i - 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. Proposed by Morteza Saghafian, Iran

Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.

Given $3n$ cards, each of them will be written with a number from the following sequence: $$2, 3, ..., n, n + 1, n + 3, n + 4, ..., 2n + 1, 2n + 2, 2n + 4, ..., 3n + 3$$with each number used exactly once. Then every card is arranged from left to right in random order. Determine the probability such that for every $i$ with $1\le i \le 3n$, the number written on the $i$-th card, counted from the left, is greater than or equal to $i$.

Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $. Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]

Find maximal numbers of planes, such there are $6$ points and 1) $4$ or more points lies on every plane. 2) No one line passes through $4$ points.

Given acute triangle $ABC$ with circumcenter $O$ and the center of nine-point circle $N$. Point $N_1$ are given such that $\angle NAB = \angle N_1AC$ and $\angle NBC = \angle N_1BA$. Perpendicular bisector of segment $OA$ intersects the line $BC$ at $A_1$. Analogously define $B_1$ and $C_1$. Show that all three points $A_1,B_1,C_1$ are collinear at a line that is perpendicular to $ON_1$.

How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

Let $T$ be a tree with$ n$ vertices. Choose a positive integer $k$ where $1 \le k \le n$ such that $S_k$ is a subset with $k$ elements from the vertices in $T$. For all $S \in S_k$, define $c(S)$ to be the number of component of graph from $S$ if we erase all vertices and edges in $T$, except all vertices and edges in $S$. Determine $\sum_{S\in S_k} c(S)$, expressed in terms of $n$ and $k$.

Given a non-isosceles triangle $ABC$ with incircle $k$ with center $S$. $k$ touches the side $BC,CA,AB$ at $P,Q,R$ respectively. The line $QR$ and line $BC$ intersect at $M$. A circle which passes through $B$ and $C$ touches $k$ at $N$. The circumcircle of triangle $MNP$ intersects $AP$ at $L$. Prove that $S,L,M$ are collinear.

Given a positive integer $n$ and $I = \{1, 2,..., k\}$ with $k$ is a positive integer. Given positive integers $a_1, a_2, ..., a_k$ such that for all $i \in I$: $1 \le a_i \le n$ and $$\sum_{i=1}^k a_i \ge 2(n!).$$Show that there exists $J \subseteq I$ such that $$n! + 1 \ge \sum_{j \in J}a_j >\sqrt {n! + (n - 1)n}$$

Find all triplets of real numbers $(x, y, z)$ that satisfies the system of equations $x^5 = 2y^3 + y - 2$ $y^5 = 2z^3 + z - 2$ $z^5 = 2x^3 + x - 2$

A government’s land with dimensions $n \times n$ are going to be sold in phases. The land is divided into $n^2$ squares with dimension $1 \times 1$. In the first phase, $n$ farmers bought a square, and for each rows and columns there is only one square that is bought by a farmer. After one season, each farmer could buy one more square, with the conditions that the newly-bought square has a common side with the farmer’s land and it hasn’t been bought by other farmers. Determine all values of n such that the government’s land could not be entirely sold within $n$ seasons.

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.

Let $ a$, $ b$, and $ c$ be non-negative real numbers and let $ x$, $ y$, and $ z$ be positive real numbers such that $ a+b+c=x+y+z$. Prove that \[ \dfrac{a^3}{x^2}+\dfrac{b^3}{y^2}+\dfrac{c^3}{z^2} \ge a+b+c.\] Hery Susanto, Malang

Let $ A=\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \}$ and let $ S=\{n: n \in A,\gcd \left(n,2009^{2009}\right)=1\}$. Let $ P$ be the product of all elements of $ S$. Prove that \[ P \equiv 1 \pmod{2009^{2009}}.\] Nanang Susyanto, Jogjakarta

In a party, each person knew exactly $ 22$ other persons. For each two persons $ X$ and $ Y$, if $ X$ and $ Y$ knew each other, there is no other person who knew both of them, and if $ X$ and $ Y$ did not know each other, there are exactly $ 6$ persons who knew both of them. Assume that $ X$ knew $ Y$ iff $ Y$ knew $ X$. How many people did attend the party? Yudi Satria, Jakarta

Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP=\angle PBQ=\angle QBC,\] (a) prove that $ ABC$ is a right-angled triangle, and (b) calculate $ \dfrac{BP}{CH}$. Soewono, Bandung

Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE = \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$, the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$. Utari Wijayanti, Bandung

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3+y^3)=xf(x^2)+yf(y^2)\] for all real numbers $ x$ and $ y$. Hery Susanto, Malang

Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}=\dfrac{2z}{2009}+\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take. Budi Surodjo, Jogjakarta

For each positive integer $ n$, define $ f(n)$ as the number of digits $ 0$ in its decimal representation. For example, $ f(2)=0$, $ f(2009)=2$, etc. Please, calculate \[ S=\sum_{k=1}^{n}2^{f(k)},\] for $ n=9,999,999,999$. Yudi Satria, Jakarta

Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)=41$ and $ f(b)=49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$. Nanang Susyanto, Jogjakarta

Given an equilateral triangle, all points on its sides are colored in one of two given colors. Prove that the is a right-angled triangle such that its three vertices are in the same color and on the sides of the equilateral triangle. Alhaji Akbar, Jakarta

Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1=1$, $ a_2=\dfrac{4}{3}$, and \[ a_{n+1}=\sqrt{1+a_na_{n-1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n-1}^2+\dfrac{1}{2}\] and \[ 1+\dfrac{1}{a_1}+\dfrac{1}{a_2}+\dots+\dfrac{1}{a_n}>2a_n.\] Fajar Yuliawan, Bandung

Let $ ABC$ be an acute-angled triangle such that there exist points $ D,E,F$ on side $ BC,CA,AB$, respectively such that the inradii of triangle $ AEF,BDF,CDE$ are all equal to $ r_0$. If the inradii of triangle $ DEF$ and $ ABC$ are $ r$ and $ R$, respectively, prove that \[ r+r_0=R.\] Soewono, Bandung

The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: choose two integers $ a$ and $ b$ such that $ a-b \ge 2$ and replace them with $ a-1$ and $ b+1$. Please, determine the maximum number of steps that can be done. Yudi Satria, Jakarta

Circles $ \Gamma_1$ and $ \Gamma_2$ are internally tangent to circle $ \Gamma$ at $ P$ and $ Q$, respectively. Let $ P_1$ and $ Q_1$ are on $ \Gamma_1$ and $ \Gamma_2$ respectively such that $ P_1Q_1$ is the common tangent of $ P_1$ and $ Q_1$. Assume that $ \Gamma_1$ and $ \Gamma_2$ intersect at $ R$ and $ R_1$. Define $ O_1,O_2,O_3$ as the intersection of $ PQ$ and $ P_1Q_1$, the intersection of $ PR$ and $ P_1R_1$, and the intersection $ QR$ and $ Q_1R_1$. Prove that the points $ O_1,O_2,O_3$ are collinear. Rudi Adha Prihandoko, Bandung

Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x+f(y)=af(y+f(x))\] for all real numbers $ x$ and $ y$. Hery Susanto, Malang

Prove that for all integers $ m$ and $ n$, the inequality \[ \dfrac{\phi(\gcd(2^m + 1,2^n + 1))}{\gcd(\phi(2^m + 1),\phi(2^n + 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\] holds. Nanang Susyanto, Jogjakarta

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? Nanang Susyanto, Jogjakarta

Consider a polynomial with coefficients of real numbers $ \phi(x)=ax^3+bx^2+cx+d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3+9a^2d-7abc \le 0.\] Hery Susanto, Malang

Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows: (1). $ \dfrac{1}{2} \in \mathbb{H}$, (2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1+x} \in \mathbb{H}$ and also $ \dfrac{x}{1+x} \in \mathbb{H}$. Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.

Prove that the number $ (\underbrace{9999 \dots 99}_{2005}) ^{2009}$ can be obtained by erasing some digits of $ (\underbrace{9999 \dots 99}_{2008}) ^{2009}$ (both in decimal representation). Yudi Satria, Jakarta