Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\] [i]Proposed by Luke Robitaille[/i]

Let the incircle of $\triangle ABC$ be tangent to $AB,BC,AC$ at points $M,N,P$, respectively. If $\measuredangle BAC=30^\circ$, find $\tfrac{[BPC]}{[ABC]}\cdot\tfrac{[BMC]}{[ABC]}$, where $[ABC]$ denotes the area of $\triangle ABC$.

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

For a given positive integer $n$, let $A_n$ be the set of all points $(x,y)$ in the coordinate plane with $x,y \in \{0,1,...,n\}$. A point $(i, j)$ is called internal if $0 < i, j < n$. A real function $f$ , defined on $A_n$, is called [i]good [/i] if it has the following property: For every internal point $x$, the value of $f(x)$ is the arithmetic mean of its values on the four neighboring points (i.e. the points at the distance $1$ from $x$). Prove that if $f$ and $g$ are good functions that coincide at the non-internal points of $A_n$, then $f \equiv g$.

Given is a convex polygon with $n\geq 5$ sides. Prove that there exist at most $\displaystyle \frac{n(2n-5)}3$ triangles of area 1 with the vertices among the vertices of the polygon.

There is a convex quadrangle $ABCD$ such that no three of its sides can form a triangle. Prove that: [list=a] [*]one of its angles is not greater than $60^\circ{}$; [*]one of its angles is at least $120^\circ$. [/list] [i]Maxim Didin[/i]

Let $ABC$ be an acute triangle with $AC > AB$ and circumcircle $\Gamma$. The tangent from $A$ to $\Gamma$ intersects $BC$ at $T$. Let $M$ be the midpoint of $BC$ and let $R$ be the reflection of $A$ in $B$. Let $S$ be a point so that $SABT$ is a parallelogram and finally let $P$ be a point on line $SB$ such that $MP$ is parallel to $AB$. Given that $P$ lies on $\Gamma$, prove that the circumcircle of $\triangle STR$ is tangent to line $AC$. [i]Proposed by Sam Bealing, United Kingdom[/i]

Let $f$ be the function defined by $f(x) = -2 \sin(\pi x)$. How many values of $x$ such that $-2 \le x \le 2$ satisfy the equation $f(f(f(x))) = f(x)$?

Suppose that $n$ people $A_1$, $A_2$, $\ldots$, $A_n$, ($n \geq 3$) are seated in a circle and that $A_i$ has $a_i$ objects such that \[ a_1 + a_2 + \cdots + a_n = nN \] where $N$ is a positive integer. In order that each person has the same number of objects, each person $A_i$ is to give or to receive a certain number of objects to or from its two neighbours $A_{i-1}$ and $A_{i+1}$. (Here $A_{n+1}$ means $A_1$ and $A_n$ means $A_0$.) How should this redistribution be performed so that the total number of objects transferred is minimum?

Do either $(1)$ or $(2)$: $(1)$ Let $ai = \sum_{n=0}^{\infty} \dfrac{x^{3n+i}}{(3n+i)!}$ Prove that $a_0^3 + a_1^3 + a_2^3 - 3 a_0a_1a_2 = 1.$ $(2)$ Let $O$ be the origin, $\lambda$ a positive real number, $C$ be the conic $ax^2 + by^2 + cx + dy + e = 0,$ and $C\lambda$ the conic $ax^2 + by^2 + \lambda cx + \lambda dy + \lambda 2e = 0.$ Given a point $P$ and a non-zero real number $k,$ define the transformation $D(P,k)$ as follows. Take coordinates $(x',y')$ with $P$ as the origin. Then $D(P,k)$ takes $(x',y')$ to $(kx',ky').$ Show that $D(O,\lambda)$ and $D(A,-\lambda)$ both take $C$ into $C\lambda,$ where $A$ is the point $(\dfrac{-c \lambda} {(a(1 + \lambda))}, \dfrac{-d \lambda} {(b(1 + \lambda))}) $. Comment on the case $\lambda = 1.$

A base-$ 10$ three-digit number $ n$ is selected at random. Which of the following is closest to the probability that the base-$ 9$ representation and the base-$ 11$ representation of $ n$ are both three-digit numerals? $ \textbf{(A)}\ 0.3 \qquad \textbf{(B)}\ 0.4 \qquad \textbf{(C)}\ 0.5 \qquad \textbf{(D)}\ 0.6 \qquad \textbf{(E)}\ 0.7$

Suppose that $a$ is an even positive integer and $A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}$ is a perfect square.Prove that $8\mid a$.

Consider a convex qudrilateral $ABCD$ and $M\in (AB),\ N\in (CD)$ such that $\frac{AM}{BM}=\frac{DN}{CN}=k$. Prove that $BC\parallel AD$ if and only if \[MN=\frac{1}{k+1} AD+\frac{k}{k+1} BC\] [i]***[/i]

Find all pairs $(x,y)$ of positive integers such that $y^{x^2}=x^{y+2}$.

Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$

In the $ ABC$ triangle, the bisector of $A $ intersects the $ [BC] $ at the point $ A_ {1} $ , and the circle circumscribed to the triangle $ ABC $ at the point $ A_ {2} $. Similarly are defined $ B_ {1} $ and $ B_ {2} $ , as well as $ C_ {1} $ and $ C_ {2} $. Prove that $$ \frac {A_{1}A_{2}}{BA_{2} + A_{2}C} + \frac {B_{1}B_{2}}{CB_{2} + B_{2}A} + \frac {C_{1}C_{2}}{AC_{2} + C_{2}B} \geq \frac {3}{4}$$

A circular disc with diameter $D$ is placed on an $8\times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is $\textbf{(A) }48\qquad\textbf{(B) }44\qquad\textbf{(C) }40\qquad\textbf{(D) }36\qquad \textbf{(E) }32$

A square of side $n \ge 2$ is divided into $n^2$ unit squares ($n \in N$). One draws $n-1$ lines so that the interior of each of the unit squares is cut by at least one of these lines. (a) Give an example of such a configuration for some $n$. (b) Show that some two of the lines must meet inside the square.

An equilateral polygon has unit side length and alternating interior angle measures of $15^o$ and $300^o$. Compute the area of this polygon.

Let P be the common point of 3 internal bisectors of a given ABC: The line passing through P and perpendicular to CP intersects AC and BC at M and N, respectively. If AP = 3cm, BP = 4cm, compute the value of $\frac{AM}{BN}$ ?

$ABC$ is a triangle and $\omega$ its incircle. Let $P,Q,R$ be the intersections with $\omega$ and the sides $BC,CA,AB$ respectively. $AP$ cuts $\omega$ in $P$ and $X$. $BX,CX$ cut $\omega$ in $M,N$ respectively. Show that $MR,NQ,AP$ are parallel or concurrent.

The teacher wrote a non-zero natural number on the board. The teacher explained students that they can delete the number written on the board and can write a number instead naturally new, whenever they want, applying one of the following each time rules: 1) Instead of the current number $n$ write $3n + 13$ 2) Instead of the current number $n$ write the number $\sqrt{n}$, if $n$ is a perfect square . a) If the number $256$ was originally written on the board, is it possible that after a finite number of steps to get the number $55$ on the board? b) If the number $55$ was originally written on the board, is it possible that after a number finished the steps to get the number $256$ on the board?

$ABC$ be a triangle. Its incircle touches the sides $CB, AC, AB$ respectively at $N_{A},N_{B},N_{C}$. The orthic triangle of $ABC$ is $H_{A}H_{B}H_{C}$ with $H_{A}, H_{B}, H_{C}$ are respectively on $BC, AC, AB$. The incenter of $AH_{C}H_{B}$ is $I_{A}$; $I_{B}$ and $I_{C}$ were defined similarly. Prove that the hexagon $I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}$ has all sides equal.

Consider a shape obtained from two equal squares with the same center. Prove that the ratio of the area of this shape to the perimeter does not change when the squares are rotated around their center. [img]http://4.bp.blogspot.com/-1AI4FxsNSr4/XovZWkvAwiI/AAAAAAAALvY/-kIzOgXB5rk3iIqGbpoKRCW9rwJPcZ3uQCK4BGAYYCw/s400/estonia%2B2000%2Bo.j.1.3.png[/img]