Prove that there are infinitely many positive odd integer $n$ such that $n!+1$ is composite number.

Find all functions $f : R \to R$ such that $x^2f(yf(x))= y^2f(x)f(f(x))$ for all real numbers $x$ and $y$.

Let $n$ be the square of an integer whose each prime divisor has an even number of decimal digits. Consider $P(x) = x^n - 1987x$. Show that if $x,y$ are rational numbers with $P(x) = P(y)$, then $x = y$.

There are $ k$ rooks on a $ 10 \times 10$ chessboard. We mark all the squares that at least one rook can capture (we consider the square where the rook stands as captured by the rook). What is the maximum value of $ k$ so that the following holds for some arrangement of $ k$ rooks: after removing any rook from the chessboard, there is at least one marked square not captured by any of the remaining rooks.

Let $f : N \to N$ be a function which satisfies $f(x)+ f(x+2) \le 2 f(x+1)$ for any $x \in N$. Prove that there exists a line in the coordinate plane containing infinitely many points of the form $(n, f(n)), n \in N$.

Given triangle $ ABC$ with $ AB>AC$. $ l$ is tangent line of the circumcircle of triangle $ ABC$ at $ A$. A circle with center $ A$ and radius $ AC$, intersect $ AB$ at $ D$ and $ l$ at $ E$ and $ F$. Prove that the lines $ DE$ and $ DF$ pass through the incenter and excenter of triangle $ ABC$.

Find the greatest common divisor of the numbers \[ 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.\]

The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded? [asy] size(5cm); defaultpen(linewidth(1pt)); draw(circle((3,3),3)); filldraw(circle((5.5,3),0.5),mediumgray*0.5 + lightgray*0.5); filldraw(circle((2,3),2),mediumgray*0.5 + lightgray*0.5); filldraw(circle((1,3),1),white); filldraw(circle((3,3),1),white); add(grid(6,6,mediumgray*0.5+gray*0.5+linetype("4 4"))); filldraw(circle((4.5,4.5),0.5),mediumgray*0.5 + lightgray*0.5); filldraw(circle((4.5,1.5),0.5),mediumgray*0.5 + lightgray*0.5); [/asy]$\textbf{(A) } \dfrac14\qquad\textbf{(B) } \dfrac{11}{36}\qquad\textbf{(C) } \dfrac13\qquad\textbf{(D) } \dfrac{19}{36}\qquad\textbf{(E) } \dfrac59$

Let $ABCD$ be a square with side length $8$. Let $M$ be the midpoint of $BC$ and let $\omega$ be the circle passing through $M, A$, and $D$. Let $O$ be the center of $\omega, X$ be the intersection point (besides A) of $\omega$ with $AB$, and $Y$ be the intersection point of $OX$ and $AM$. If the length of $OY$ can be written in simplest form as $\frac{m}{n}$ , compute $m + n$.

Consider the group $ \{f:\mathbb{C}\setminus\mathbb{Q}\longrightarrow\mathbb{C}\setminus\mathbb{Q} | f\text{ is bijective}\} $ under the composition of functions. Find the order of the smallest subgroup of it that: $ \text{(1)} $ contains the function $ z\mapsto \frac{z-1}{z+1} . $ $ \text{(2)} $ contains the function $ z\mapsto \frac{z-3}{z+1} . $ $ \text{(3)} $ contain both of the above functions.

For any positive integer $n$, define the subset $S_n$ of natural numbers as follow $$ S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}.$$ Find all positive integers $n$ such that there exists an element of $S_n$ which [u]doesn't belong[/u] to any of the sets $S_1, S_2,\dots,S_{n-1}$. [i]Proposed by Yahya Motevassel[/i]

Let $O$ and $R$ be the circumcenter and circumradius of a triangle $ABC$, and let $P$ be any point in the plane of the triangle. The perpendiculars $PA_1,PB_1,PC_1$ are drawn from $P$ on $BC,CA,AB$. Express $S_{A_1B_1C_1}/S_{ABC}$ in terms of $R$ and $d = OP$, where $S_{XYZ}$ is the area of $\triangle XYZ$.

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ and $Q$ be points on the segments $AB$ and $AC$, respectively, such that $BP : PQ : QC = AC : CB : BA$. Prove that the points $A$, $P$, $Q$ and $O$ lie on one circle. [i]Alternative formulation.[/i] Let $O$ be the center of the circumcircle of a triangle $ABC$. If $P$ and $Q$ are points on the sides $AB$ and $AC$, respectively, satisfying $\frac{BP}{PQ}=\frac{CA}{BC}$ and $\frac{CQ}{PQ}=\frac{AB}{BC}$, then show that the points $A$, $P$, $Q$ and $O$ lie on one circle.

$32$ competitors participate in a tournament. No two of them are equal and in a one against one match the better always wins. Show that the gold, silver, and bronze medal winners can be found in $39$ matches.

The equation $ \sqrt {x \plus{} 10} \minus{} \frac {6}{\sqrt {x \plus{} 10}} \equal{} 5$ has: $ \textbf{(A)}\ \text{an extraneous root between } \minus{} 5\text{ and } \minus{} 1 \\ \textbf{(B)}\ \text{an extraneous root between } \minus{} 10\text{ and } \minus{} 6 \\ \textbf{(C)}\ \text{a true root between }20\text{ and }25 \qquad\textbf{(D)}\ \text{two true roots} \\ \textbf{(E)}\ \text{two extraneous roots}$