We colored the $n^2$ unit squares of an $n\times n$ square lattice such that in each $2\times 2$ square, at least two of the four unit squares have the same color. What is the largest number of colors we could have used? [i]Based on a problem of the Dürer Competition[/i]

There is a convex polyhedron with $k$ faces. Show that if more than $k/2$ of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere.

Penelope plays a game where she adds $25$ points to her score each time she wins a game and deducts $13$ points from her score each time she loses a game. Starting with a score of zero, Penelope plays $m$ games and has a total score of $2007$ points. What is the smallest possible value for $m$?

Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$, $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers.

Let $n$ be a positive integer. A number $A$ consists of $2n$ digits, each of which is 4; and a number $B$ consists of $n$ digits, each of which is 8. Prove that $A+2B+4$ is a perfect square.

Let $AE$ be a diameter of the circumcircle of triangle $ABC$. Join $E$ to the orthocentre, $H$, of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$. Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$. Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.

Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have \[a_n = \begin{cases} a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\ 2n & \text{if } (a_{n-1},n) > 1 \end{cases} \] Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.

Let $a,b,c$ be positive real numbers, such that: $a+b+c \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$ Prove that: \[a+b+c \geq \frac{3}{a+b+c}+\frac{2}{abc}. \]

At a position $O$ of an airport in a plateau there is a gun which can rotate arbitrarily. Two tanks moving along two given segments $AB$ and $CD$ attack the airport. Determine, by a ruler and a compass, the reach of the gun, knowing that the total length of the parts of the trajectories of the two tanks reachable by the gun is equal to a given length $\ell$.

Let $a,b$ be positive real numbers such that $a^3+b^3=2(a^2+b^2)$. Prove the following inequality: $$ \sqrt{a^3+1} + \sqrt{b^3+1} \leq a+b+2. $$ When is equality achieved?

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

If $1+2x+3x^2 +...=9$, find $x$.

A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$. [i]AMM Magazine[/i]

Show that $\tan \frac{\pi}{3n}$ is irrational for all positive integers $n$.

Given are two circles $\omega_1,\omega_2$ which intersect at points $X,Y$. Let $P$ be an arbitrary point on $\omega_1$. Suppose that the lines $PX,PY$ meet $\omega_2$ again at points $A,B$ respectively. Prove that the circumcircles of all triangles $PAB$ have the same radius.

Let $ABC$ and $XYZ$ be two triangles. Define \[A_1=BC\cap ZX, A_2=BC\cap XY,\]\[B_1=CA\cap XY, B_2=CA\cap YZ,\]\[C_1=AB\cap YZ, C_2=AB\cap ZX.\] Hereby, the abbreviation $g\cap h$ means the point of intersection of two lines $g$ and $h$. Prove that $\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}$ holds if and only if $\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}$.

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

Given is a cyclic pentagon $ABCDE$, inscribed in a circle $k$. The line $CD$ intersects $AB$ and $AE$ in $X$ and $Y$ respectively. Segments $EX$ and $BY$ intersect again at $P$, and they intersect $k$ in $Q$ and $R$, respectively. Point $A'$ is reflection of $A$ across $CD$. The circles $(PQR)$ and $(A'XY)$ intersect at $M$ and $N$. Prove that $CM$ and $DN$ intersect on $(PQR)$.

In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$. Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively. Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that \[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}. \]

Are there numbers $a, b$ such that $| a -b |\le 1979$ and the equation $ax^2 + (a + b) x + b = x$ has no roots?

[b]a)[/b] Let be three natural numbers, $ a>b\ge 3\le 3n, $ such that $ b^n|a^n-1. $ Prove that $ a^b>2^n. $ [b]b)[/b] Does there exist positive real numbers $ m $ which have the property that $ \log_8 (1+3\sqrt x) =\log_{27} (mx) $ if and only if $ 2^{x} +2^{1/x}\le 4? $

Let $a, b$ be positive real numbers satisfying $a + b = 1$. Show that if $x_1,x_2,...,x_5$ are positive real numbers such that $x_1x_2...x_5 = 1$, then $(ax_1+b)(ax_2+b)...(ax_5+b)>1$

Let $a$, $b$, $c$ be real numbers. Prove that \[ab + bc + ca + \max\{|a - b|, |b - c|, |c - a|\} \le 1 + \frac{1}{3} (a + b + c)^2.\]

Let $f(x) = x-\tfrac1{x}$, and define $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.

The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled? $ \textbf{(A)}\ 80\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108\qquad\textbf{(E)}\ 192 $