A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? $\textbf{(A) }\frac{5}{54}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{5}{27}\qquad\textbf{(D) }\frac{2}{9}\qquad \textbf{(E) }\frac{1}{3}$

Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

Function $f(x)$. which is defined on the set of non-negative real numbers, acquires real values. It is known that $f(0)\le 0$ and the function $f(x)/x$ is increasing for $x>0$. Prove that for arbitrary $x\ge 0$ and $y\ge 0$, holds the inequality $f(x+y)\ge f(x)+ f(y)$ .

Mykolka the numismatist possesses $241$ coins, each worth an integer number of turgiks. The total value of the coins is $360$ turgiks. Is it necessarily true that the coins can be divided into three groups of equal total value?

Each square of a $100\times 100$ board is painted with one of $100$ different colours, so that each colour is used exactly $100$ times. Show that there exists a row or column of the chessboard in which at least $10$ colours are used.

Is it possible to place a positive integer in every cell of a $10 \times 10$ array in such a way that both the following conditions are satisfied? $\bullet$ Each number (not in the top row) is a proper divisor of the number immediately above. $\bullet$ Each row consists of 1$0$ consecutive positive integers (but not necessarily in order).

Let $N \in \{10, 11, \ldots, 99\}$ be a two-digit positive integer. Compute the number of values of $N$ for which the last two digits in the decimal expansion of $N^{21}$ are the digits of $N$ in the same order.

Let $ G$ be a group such that if $ a,b\in \mathbb{G}$ and $ a^2b\equal{}ba^2$, then $ ab\equal{}ba$. i)If $ G$ has $ 2^n$ elements, prove that $ G$ is abelian. ii) Give an example of a non-abelian group with $ G$'s property from the enounce.

Let the triangle $ABC$ have area $1$. The interior bisectors of the angles $\angle BAC,\angle ABC, \angle BCA$ intersect the sides $(BC), (AC), (AB) $ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $E$. The lines $EF, FG,GE$ intersect the bisectors $(AL), (CQ) ,(BN)$ respectively at points $P, M, R$. Determine the area of the hexagon $LMNPR$.

A regular polygon with $1000$ sides has the vertices colored in red, yellow or blue. A move consists in choosing to adjiacent vertices colored differently and coloring them in the third color. Prove that there is a sequence of moves after which all the vertices of the polygon will have the same color. Marius Ghergu

Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex M from angle $CMD$. Prove that $ABCD$ is a cyclic quadrilateral. (M. Kungozhin)

Find the least positive integer $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$.

Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows: (1). $ \dfrac{1}{2} \in \mathbb{H}$, (2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1\plus{}x} \in \mathbb{H}$ and also $ \dfrac{x}{1\plus{}x} \in \mathbb{H}$. Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.

Prove that there are infinitely many integers which cannot be expressed as $2^a+3^b-5^c$ for non-negative integers $a,b,c$.

The fraction $\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}$ is (with suitable restrictions of the values of $a$, $b$, and $c$): $ \textbf{(A) }\text{irreducible}\qquad\textbf{(B) }\text{reducible to negative 1}\qquad$ $\textbf{(C) }\text{reducible to a polynomial of three terms} \qquad\textbf{(D) }\text{reducible to} \frac{a-b+c}{a+b-c} \qquad\textbf{(E) }\text{reducible to} \frac{a+b-c}{a-b+c} $

Let $H$ be the orthocenter of a nonisosceles triangle $ABC$. The bisector of angle $BHC$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. The perpendiculars to $AB$ and $AC$ from $P$ and $Q$ meet at $K$. Prove that $KH$ bisects the segment $BC$.

In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

Circles $o_1$ and $o_2$ with centers in $O_1$ and $O_2$, respectively, intersect in two different points $A$ and $B$, wherein angle $O_1AO_2$ is obtuse. Line $O_1B$ intersects circle $o_2$ in point $C \neq B$. Line $O_2B$ intersects circle $o_1$ in point $D \neq B$. Show that point $B$ is incenter of triangle $ACD$.

Evaluate $ \int_0^{2008} \left(3x^2 \minus{} 8028x \plus{} 2007^2 \plus{} \frac {1}{2008}\right)\ dx$.

Conside a positive odd integer $k$ and let $n_1<n_2<\ldots<n_k$ be $k$ positive odd integers. Prove that: \[n_1^2-n_2^2+n_3^2-n_4^2+\ldots+n_k^2\ge 2k^2-1\] [i]Titu Andreescu[/i]

Let $n$ be a fixed positive odd integer. Take $m+2$ [b]distinct[/b] points $P_0,P_1,\ldots ,P_{m+1}$ (where $m$ is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied: 1) $P_0=(0,1),P_{m+1}=(n+1,n)$, and for each integer $i,1\le i\le m$, both $x$- and $y$- coordinates of $P_i$ are integers lying in between $1$ and $n$ ($1$ and $n$ inclusive). 2) For each integer $i,0\le i\le m$, $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and is parallel to the $y$-axis if $i$ is odd. 3) For each pair $i,j$ with $0\le i<j\le m$, line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most $1$ point. Determine the maximum possible value that $m$ can take.

An ant traverses between vertices on a unit cube such that at each vertex, it uniformly at random chooses an adjacent vertex to travel to. What is the expected distance travelled by the ant until it returns to its starting vertex?

Two square sheets have areas equal to $ 2003$. Each of the sheets is arbitrarily divided into $ 2003$ nonoverlapping polygons, besides, each of the polygons has an unitary area. Afterward, one overlays two sheets, and it is asked to prove that the obtained double layer can be punctured $ 2003$ times, so that each of the $ 4006$ polygons gets punctured precisely once.

For a set $ P$ of five points in the plane, no three of them being collinear, let $ s(P)$ be the numbers of acute triangles formed by vertices in $ P$. Find the maximum value of $ s(P)$ over all such sets $ P$.

Prove for irrational number $\alpha$ and positive integer $n$ that \[ \left( \alpha + \sqrt{\alpha^2 - 1} \right)^{1/n} + \left(\alpha - \sqrt{\alpha^2 - 1} \right)^{1/n} \] is irrational.