For each positive integer $k$, define $r(k)$ as the number of runs of $k$ in base-$2$, where a run is a collection of consecutive $0$s or consecutive $1$s without a larger one containing it. For example, $(11100100)_2$ has $4$ runs, namely $111-00-1-00$. Also, $r(0) = 0$. Given a positive integer $n$, find all functions $f : \mathbb{Z} \rightarrow\mathbb{Z}$ such that \[\sum_{k=0}^{2^n-1} 2^{r(k)}f(k+(-1)^{k} x)=(-1)^{x+n}\text{ for all integer $x$.}\] [i]Proposed by YaWNeeT[/i]

Prove that a plane that passes: a) through the centers of two opposite edges of the tetrahedron and b) through the center of one of the other edges of the tetrahedron divides the tetrahedron into two parts of equal volumes. Will the thesis remain true if we reject assumption (b) ?

In many computer languages, the division operation ignores remainders. Let’s denote this operation by $//$, so for instance $13//3 = 4$. If, for some $b$, $a//b = c$, then we say that $c$ is a [i]near factor[/i] of $a$. Thus, the near factors of $13$ are $1$, $2$, $3$, $4$, and $6$. Let $a$ be a positive integer. Prove that every positive integer less than or equal to $\sqrt{a}$ is a near factor of $a$.

Point $P$ and equilateral triangle $ABC$ satisfy $|AP|=2$, $|BP|=3$. Maximize $|CP|$.

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(x+y)\le f(x)+f(y)\le x+y$ for all $x,y\in\mathbb{R}$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^{\circ}$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $AC$ and let $\omega$ be the circumcircle of triangle $BPD$. Segments $BM$ and $DM$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $YF$. Prove that $PQ \perp AC$.

Two circles $C_1$ and $C_2$ intersect each other at points $A$ and $B$. Their external common tangent (closer to $B$) touches $C_1$ at $P$ and $C_2$ at $Q$. Let $C$ be the reflection of $B$ in line $PQ$. Prove that $\angle CAP=\angle BAQ$.

The inscribed circle of the triangle $ABC$ touches the sides $BC, CA, AB$ at points $A_1, B_1, C_1$ respectively. The points $P_b, Q_b, R_b$ are the points of the segments $BC_1, C_1A_1, A_1B$, respectively, such that $BP_bQ_bR_b$ is parallelogram. In the same way, the points $P_c, Q_c, R_c$ are the points of the sections $CB_1, B_1A_1, A_1C$, respectively such that $CP_cQ_cR_c$ is a parallelogram. The intersection of the lines $P_bR_b$ and $P_cR_c$ is $T$. Show that $TQ_b = TQ_c$.

There are written $n$ distinct positive integers. An operation is defined as follows: we chose two numers $a$ and $b$ written on the table; we erase them; we write at their places $a+1$ and $b-1$. Find the smallest value of the difference the biggest and the smallest written numbers after some operations.

Sabine rolls a fair $14$-sided die numbered $1$ to $14$ and gets a value of $x$. She then draws $x$ cards uniformly at random (without replacement) from a deck of $14$ cards, each of which labeled a different integer from $1$ to $14$. She finally sums up the value of her die roll and the value on each card she drew to get a score of $S$. Let $A$ be the set of all obtainable scores. Compute the probability that $S$ is greater than or equal to the median of $A$.

There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? $ \text{(A)}\ 30\qquad\text{(B)}\ 40\qquad\text{(C)}\ 41\qquad\text{(D)}\ 60\qquad\text{(E)}\ 119 $

The positive integer $N$ is said[i] amiable [/i]if the set $\{1,2,\ldots,N\}$ can be partitioned into pairs of elements, each pair having the sum of its elements a perfect square. Prove there exist infinitely many amiable numbers which are themselves perfect squares. ([i]Dan Schwarz[/i])

John takes seven positive integers $a_1,a_2,...,a_7$ and writes the numbers $a_i a_j$, $a_i+a_j$ and $|a_i -a_j |$ for all $i \ne j$ on the blackboard. Find the greatest possible number of distinct odd integers on the blackboard.

Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.

Prove that two natural numbers whose digits are all ones are relatively prime if and only if the numbers of their digits are relatively prime.

Show that there exists a constant $a > 1$ such that, for any positive integers $m$ and $n$, $\frac{m}{n} < \sqrt7$ implies that $$7-\frac{m^2}{n^2} \ge \frac{a}{n^2} .$$

There exist $ r$ unique nonnegative integers $ n_1 > n_2 > \cdots > n_r$ and $ r$ unique integers $ a_k$ ($ 1\le k\le r$) with each $ a_k$ either $ 1$ or $ \minus{} 1$ such that \[ a_13^{n_1} \plus{} a_23^{n_2} \plus{} \cdots \plus{} a_r3^{n_r} \equal{} 2008. \]Find $ n_1 \plus{} n_2 \plus{} \cdots \plus{} n_r$.

Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

If $a, b, x$ and $y$ are real numbers such that $ax + by = 3,$ $ax^2+by^2=7,$ $ax^3+bx^3=16$, and $ax^4+by^4=42,$ find $ax^5+by^5$.

In triangle $ ABC$, angle $ C$ is a right angle and $ CB > CA$. Point $ D$ is located on $ \overline{BC}$ so that angle $ CAD$ is twice angle $ DAB$. If $ AC/AD \equal{} 2/3$, then $ CD/BD \equal{} m/n$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 22\qquad \textbf{(E)}\ 26$

Determine all natural numbers $n$ for which there are $2n$ distinct positive integers $x_1,…,x_n,y_1,…,y_n$ such that the product $$(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n)$$ is a perfect square.

Color the positive integers by four colors $c_1,c_2,c_3,c_4$. (1)Prove that there exists a positive integer $n$ and $i,j\in\{1,2,3,4\}$,such that among all the positive divisors of $n$, the number of divisors with color $c_i$ is at least greater than the number of divisors with color $c_j$ by $3$. (2)Prove that for any positive integer $A$,there exists a positive integer $n$ and $i,j\in\{1,2,3,4\}$,such that among all the positive divisors of $n$, the number of divisors with color $c_i$ is at least greater than the number of divisors with color $c_j$ by $A$.

In a basketball tournament any two of the $n$ teams $S_1,S_2,...,S_n$ play one match (no draws). Denote by $v_i$ and $p_i$ the number of victories and defeats of team $S_i$ ($i = 1,2,...,n$), respectively. Prove that $v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n$

Square $ BDEC$ with center $ F$ is constructed to the out of triangle $ ABC$ such that $ \angle A \equal{} 90{}^\circ$, $ \left|AB\right| \equal{} \sqrt {12}$, $ \left|AC\right| \equal{} 2$. If $ \left[AF\right]\bigcap \left[BC\right] \equal{} \left\{G\right\}$ , then $ \left|BG\right|$ will be $\textbf{(A)}\ 6 \minus{} 2\sqrt {3} \qquad\textbf{(B)}\ 2\sqrt {3} \minus{} 1 \qquad\textbf{(C)}\ 2 \plus{} \sqrt {3} \\ \qquad\textbf{(D)}\ 4 \minus{} \sqrt {3} \qquad\textbf{(E)}\ 5 \minus{} 2\sqrt {2}$

To get to the Stromboli Volcano from the observatory, one has to take a road and a passway, each taking $4$ hours. There are two craters on the top. The first crater erupts for $1$ hour, stays silent for $17$ hours, then repeats the cycle. The second crater erupts for $1$ hour, stays silent for $9$ hours, erupts for $1$ hour, stays silent for $17$ hours, and then repeats the cycle. During the eruption of the first crater, it is dangerous to take both the passway and the road, but the second crater is smaller, so it is still safe to take the road. At noon, scout Vanya saw both craters erupting simultaneously. Will it ever be possible for him to mount the top of the volcano without risking his life?