Let $x$, $y$ and $z$ be positive real numbers such that $\sqrt{xy} + \sqrt{yz} + \sqrt{zx} = 3$. Prove that $\sqrt{x^3+x} + \sqrt{y^3+y} + \sqrt{z^3+z} \geq \sqrt{6(x+y+z)}$

Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.

Semicircle $\stackrel{\frown}{AB}$ has center $C$ and radius $1$. Point $D$ is on $\stackrel{\frown}{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\stackrel{\frown}{AE}$ and $\stackrel{\frown}{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\stackrel{\frown}{EF}$ has center $D$. The area of the shaded "smile", $AEFBDA$, is [asy] size(200); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair A=(-1,0), B=(1,0), D=(0,-1), C=(0,0), E=(1-sqrt(2),-sqrt(2)), F=(-1+sqrt(2),-sqrt(2)); fill(Arc((1,0),2,180,225)--Arc((0,-1),(2-sqrt(2)),225,315)--Arc((-1,0),2,315,360)--Arc((0,0),1,360,180)--cycle,mediumgray); draw(A--B^^C--D^^A--F^^B--E); draw(Arc((1,0),2,180,225)--Arc((0,-1),(2-sqrt(2)),225,315)--Arc((-1,0),2,315,360)--Arc((0,0),1,360,180)); label("$A$",A,N); label("$B$",B,N); label("$C$",C,N); label("$D$",(-0.1,-.7)); label("$E$",E,SW); label("$F$",F,SE); [/asy] $ \textbf{(A)}\ (2 - \sqrt{2})\pi\qquad\textbf{(B)}\ 2\pi - \pi\sqrt{2} - 1\qquad\textbf{(C)}\ \left(1 - \frac{\sqrt{2}}{2}\right)\pi\qquad\textbf{(D)}\ \frac{5\pi}{2} - \pi\sqrt{2} - 1\qquad\textbf{(E)}\ (3 - 2\sqrt{2})\pi $

The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome? $ \text{(A)}\ 0\qquad\text{(B)}\ 4\qquad\text{(C)}\ 9\qquad\text{(D)}\ 16\qquad\text{(E)}\ 25 $

For $S_n=\{1, 2, ..., n\}$, find the maximum value of $m$ that makes the following proposition true. [b]Proposition[/b] There exists $m$ different subsets of $S$, say $A_1, A_2, ..., A_m$, such that for every $i, j=1, 2, ..., m$, the set $A_i \cup A_j$ is not $S$.

a) 11 kayakers row on the Danube from Szentendre to Kopaszi-gát. They do not necessarily start at the same time, but we know that they all take the same route and that each kayaker rows with a constant speed. Whenever a kayaker passes another one, they do a high five. After they all arrive, everybody claims to have done precisely $10$ high fives in total. Show that it is possible for the kayakers to have rowed in such a way that this is true. b) At a different occasion $13$ kayakers rowed in the same manner; now after arrival everybody claims to have done precisely$ 6$ high fives. Prove that at least one kayaker has miscounted.

Michael walks at the rate of $ 5$ feet per second on a long straight path. Trash pails are located every $ 200$ feet along the path. A garbage truck travels at $ 10$ feet per second in the same direction as Michael and stops for $ 30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Player Zero and Player One play a game on a $n \times n$ board ($n \ge 1$). The columns of this $n \times n$ board are numbered $1,2,4,\dots,2^{n-1}$. Turn my turn, the players put their own number in one of the free cells (thus Player Zero puts a $0$ and Player One puts a $1$). Player Zero begins. When the board is filled, the game ends and each row yields a (reverse binary) number obtained by adding the values of the columns with a $1$ in that row. For instance, when $n=4$, a row with $0101$ yields the number $0 \cdot1+1 \cdot 2+0 \cdot 4+1 \cdot 8=10$. a) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $4$? b) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $3$?

Let $K$ be a point on the angle bisector, such that $\angle BKL=\angle KBL=30^\circ$. The lines $AB$ and $CK$ intersect in point $M$ and lines $AC$ and $BK$ intersect in point $N$. Determine $\angle AMN$.

A [i]cactus[/i] is a finite simple connected graph where no two cycles share an edge. Show that in a nonempty cactus, there must exist a vertex which is part of at most one cycle. [i]Kevin Yang[/i]

We are given a right-angled triangle $MNP$ with right angle in $P$. Let $k_M$ be the circle with center $M$ and radius $MP$, and let $k_N$ be the circle with center $N$ and radius $NP$. Let $A$ and $B$ be the common points of $k_M$ and the line $MN$, and let $C$ and $D$ be the common points of $k_N$ and the line $MN$ with with $C$ between $A$ and $B$. Prove that the line $PC$ bisects the angle $\angle APB$.

Several boys and girls are dancing a waltz at a ball. Is it possible that each girl can always get to dance the next dance with a boy who is either more handsome or more clever than for the previous dance, and that each time one of the girls gets to dance the next dance with a boy who is more handsome and more clever? (The numbers of boys and girls are equal and all are dancing.) (AY Belov)

A square and an equilateral triangle have the same perimeter. Let $ A$ be the area of the circle circumscribed about the square and $ B$ be the area of the circle circumscribed about the triangle. Find $ A/B$. $ \textbf{(A)}\ \frac{9}{16} \qquad \textbf{(B)}\ \frac{3}{4} \qquad \textbf{(C)}\ \frac{27}{32} \qquad \textbf{(D)}\ \frac{3\sqrt{6}}{8} \qquad \textbf{(E)}\ 1$

Find the integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$. [i]Proposed by Kevin Sun[/i]

Find all the functions $f:(0,+\infty) \to R $ that satisfy the equation $$f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)$$ for all pairs of $x,y$ real and positive numbers, where $\lambda$ is a positive real number such that $f(\lambda )=1$

There are $50$ coins in a row; each coin has a value. Two people are playing a game alternating moves. In one move a player can take either the leftmost or the rightmost coin. Who can always accumulate coins whose total value is at least the value of the coins of the opponent?

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?