Let $\Delta ABC$ be a triangle with an inscribed circle centered at $I$. The line perpendicular to $AI$ at $I$ intersects $\odot (ABC)$ at $P,Q$ such that, $P$ lies closer to $B$ than $C$. Let $\odot (BIP) \cap \odot (CIQ) =S$. Prove that, $SI$ is the angle bisector of $\angle PSQ$

Determine all integers $ n>1$ for which the inequality \[ x_1^2\plus{}x_2^2\plus{}\ldots\plus{}x_n^2\ge(x_1\plus{}x_2\plus{}\ldots\plus{}x_{n\minus{}1})x_n\] holds for all real $ x_1,x_2,\ldots,x_n$.

Let $ABCD$ be a convex quadrilateral, and $M$ and $N$ the midpoints of the sides $CD$ and $AD$, respectively. The lines perpendicular to $AB$ passing through $M$ and to $BC$ passing through $N$ intersect at point $P$. Prove that $P$ is on the diagonal $BD$ if and only if the diagonals $AC$ and $BD$ are perpendicular.

If $x^5 - x ^3 + x = a,$ prove that $x^6 \geq 2a - 1$.

Let $I$, $I_{a}$ be the incenter and the $A$-excenter of a triangle $ABC$; $E$, $F$ be the touching points of the incircle with $AC$, $AB$ respectively; $G$ be the common point of $BE$ and $CF$. The perpendicular to $BC$ from $G$ meets $AI$ at point $J$. Prove that $E$, $F$, $J$, $I_{a}$ are concyclic. Proposed by:Y.Shcherbatov

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that: $$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$ For what kind of triangles does the equality hold?

Prove that $$ (n+2)^n=\prod_{k=1}^{n+1} \sum_{l=1}^{n+1} le^{\frac{2i\pi k (n-l+1)}{n+2}} , $$ for any natural number $ n. $ [i]Mihai Piticari[/i]

Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul chooses one card to turn face-up, looks at it, and then chooses another to turn face-up and looks at it. If the two face-up cards match, the game ends. If not, Rahul flips both cards face-down and keeps repeating this process. Initially, Rahul doesn't know which cards are which. Assuming that he has perfect memory, find the smallest number of moves after which he can guarantee that the game has ended.

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

Prove that there does not exist a function $f : \mathbb R^+\to\mathbb R^+$ such that \[f(f(x)+y)=f(x)+3x+yf(y)\] for all positive reals $x,y$.

Friends Alice, Betty, and Cathy are playing a game. Betty and Cathy are each given a square number, such that Betty knows Cathy's number and Cathy knows Betty's, but neither of them know their own. Alice then says: "The sum of the numbers is less than 100." Betty says: "If Cathy knew the number of possibilities for my number, she would know her own." Cathy then says: "Now I know my number." What is Cathy's number? $\textbf{(A) }16\qquad\textbf{(B) }25\qquad\textbf{(C) }36\qquad\textbf{(D) }49\qquad\textbf{(E) }64$

For all real numbers $x$, we denote by $\lfloor x \rfloor$ the largest integer that does not exceed $x$. Find all functions $f$ that are defined on the set of all real numbers, take real values, and satisfy the equality \[f(x + y) = (-1)^{\lfloor y \rfloor} f(x) + (-1)^{\lfloor x \rfloor} f(y)\] for all real numbers $x$ and $y$. [i]Navneel Singhal, India[/i]

Let $D$ be a point in the interior of $\bigtriangleup ABC$ such that $AB = ab$, $AC = ac$, $AD = ad$, $BC = bc$, $BD = bd$ and $CD = cd$. Prove that $\angle ABD + \angle ACD = \frac{\pi}{3}$.

$\textbf{Bake Sale}$ Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy] $\circ$ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy] $\circ$ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy] $\circ$ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Who gets the fewest cookies from one batch of cookie dough? $ \textbf{(A)}\ \text{Art}\qquad\textbf{(B)}\ \text{Roger}\qquad\textbf{(C)}\ \text{Paul}\qquad\textbf{(D)}\ \text{Trisha}\qquad\textbf{(E)}\ \text{There is a tie for fewest.}$

Place the integers $1,2 , \ldots, n^{3}$ in the cells of a $n\times n \times n$ cube such that every number appears once. For any possible enumeration, write down the maximal difference between any two adjacent cells (adjacent means having a common vertex). What is the minimal number noted down?

Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$