Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$, then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$.

Let $[AB]$ be a chord of the circle $\Gamma$ not passing through its center and let $M$ be the midpoint of $[AB].$ Let $C$ be a variable point on $\Gamma$ different from $A$ and $B$ and $P$ be the point of intersection of the tangent lines at $A$ of circumcircle of $CAM$ and at $B$ of circumcircle of $CBM.$ Show that all $CP$ lines pass through a fixed point.

Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?

A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively. Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic. Babis

Let $p>2$ be a prime number and $n$ a positive integer. Prove that $pn^2$ has at most one positive divisor $d$ for which $n^2+d$ is a square number.

On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $ R$ be the region formed by the union of the square and all the triangles, and $ S$ be the smallest convex polygon that contains $ R$. What is the area of the region that is inside $ S$ but outside $ R$? $ \textbf{(A)} \; \frac{1}{4} \qquad \textbf{(B)} \; \frac{\sqrt{2}}{4} \qquad \textbf{(C)} \; 1 \qquad \textbf{(D)} \; \sqrt{3} \qquad \textbf{(E)} \; 2 \sqrt{3}$

When simplified the product $\left(1-\frac13\right)\left(1-\frac14\right)\left(1-\frac15\right)\cdots\left(1-\frac1n\right)$ becomes: $ \textbf{(A)}\ \frac1n \qquad\textbf{(B)}\ \frac2n\qquad\textbf{(C)}\ \frac{2(n-1)}{n}\qquad\textbf{(D)}\ \frac{2}{n(n+1)}\qquad\textbf{(E)}\ \frac{3}{n(n+1)} $

Given an acute triangle $ABC$ with $AB <AC$. Points $P$ and $Q$ lie on the angle bisector of $\angle BAC$ so that $BP$ and $CQ$ are perpendicular on that angle bisector. Suppose that point $E, F$ lie respectively at sides $AB$ and $AC$ respectively, in such a way that $AEPF$ is a kite. Prove that the lines $BC, PF$, and $QE$ intersect at one point.

A pair of integers $ (m,n)$ is called [i]good[/i] if \[ m\mid n^2 \plus{} n \ \text{and} \ n\mid m^2 \plus{} m\] Given 2 positive integers $ a,b > 1$ which are relatively prime, prove that there exists a [i]good[/i] pair $ (m,n)$ with $ a\mid m$ and $ b\mid n$, but $ a\nmid n$ and $ b\nmid m$.

Let $a$ and $b$ be integers. Define $a$ to be a power of $b$ if there exists a positive integer $n$ such that $a = b^n$. Define $a$ to be a multiple of $b$ if there exists an integer $n$ such that $a = bn$. Let $x$, $y$ and $z$ be positive integer such that $z$ is a power of both $x$ and $y$. Decide for each of the following statements whether it is true or false. Prove your answers. (a) The number $x + y$ is even. (b) One of $x$ and $y$ is a multiple of the other one. (c) One of $x$ and $y$ is a power of the other one. (d) There exist an integer $v$ such that both $x$ and $y$ are powers of $v$ (e) For each power of $x$ and for each power of $y$, an integer $w$ can be found such that $w$ is a power of each of these powers. (f) There exists a positive integer $k$ such that $x^k > y$.

Let $N$ be the smallest positive integer $N$ such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$. $\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l} \textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\ \textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\ \textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\ \textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\ \textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\ \textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\ \textbf{(S) }18&\textbf{(T) }19&\textbf{(U) }20\\\\ \textbf{(V) }21&\textbf{(W) }22 & \textbf{(X) }23 \end{array}$

Find all positive integers $d$ for which there exist polynomials $P(x)$ and $Q(x)$ with real coefficients such that degree of $P$ equals $d$ and $$P(x)^2+1=(x^2+1)Q(x)^2.$$

Following the previous set up, find the speed $v$ of the small block after it leaves the slope. (A) $v=v_\text{0}$ (B) $v=\frac{m}{m+M}v_\text{0}$ (C) $v=\frac{M}{m+M}v_\text{0}$ (D) $v=\frac{M-m}{m}v_\text{0}$ (E) $v=\frac{M-m}{m+M}v_\text{0}$

$AB$ and $AC$ are tangents to a circle $\omega$ with center $O$ at $B,C$ respectively. Point $P$ is a variable point on minor arc $BC$. The tangent at $P$ to $\omega$ meets $AB,AC$ at $D,E$ respectively. $AO$ meets $BP,CP$ at $U,V$ respectively. The line through $P$ perpendicular to $AB$ intersects $DV$ at $M$, and the line through $P$ perpendicular to $AC$ intersects $EU$ at $N$. Prove that as $P$ varies, $MN$ passes through a fixed point.

Mongolia TST 2011 Test 1 #2 Let $p$ be a prime number. Prove that: $\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)$ (proposed by B. Batbayasgalan, inspired by Putnam olympiad problem) Note: I believe they meant to say $p>2$ as well.

For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exacly $2004$ of other lines? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 1 $