Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point [asy] size(80); defaultpen(linewidth(0.7)+fontsize(10)); draw(unitcircle); for(int i = 0; i < 5; ++i) { pair P = dir(90+i*72); dot(P); label("$"+string(i+1)+"$",P,1.4*P); }[/asy] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

$f(x)$ is continuous on the interval $[0, \pi]$ and satisfies \[ \int\limits_0^\pi f(x)dx=0, \qquad \int\limits_0^\pi f(x)\cos x dx=0 \] Show that $f(x)$ has at least two zeros in the interval $(0, \pi)$.

Determine all functions $f: R\to R$, such that $f (2x + f (y)) = x + y + f (x)$ for all $x, y \in R$. (Gerhard Kirchner)

On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?

Let $ABCD$ be a convex quadrilateral with $AB = AD.$ Let $T$ be a point on the diagonal $AC$ such that $\angle ABT + \angle ADT = \angle BCD.$ Prove that $AT + AC \geq AB + AD.$

Points $A_1$, $B_1$ and $C_1$ are taken on the respective edges $SA$, $SB$, $SC$ of a regular triangular pyramid $SABC$ so that the planes $A_1B_1C_1$ and $ABC$ are parallel. Let $O$ be the center of the sphere passing through $A$, $B$, $C_1$ and $S$. Prove that the line $SO$ is perpendicular to the plane $A_1B_1C$.

Let $n$ be a positive integer. Show that the equation \[\sqrt{x}+\sqrt{y}=\sqrt{n}\] have solution of pairs of positive integers $(x,y)$ if and only if $n$ is divisible by some perfect square greater than $1$. [i]Proposer: Nanang Susyanto[/i]

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. At point $A$ to $\omega_1$ and $\omega_2$ the tangents $\ell_1$ and $\ell_2$ are drawn respectively. The points $T_1$ and $T_2$ are chosen respectively on the circles $\omega_1$ and $\omega_2$ so that the angular measures of the arcs $T_1A$ and $AT_2$ are equal (the measure of the circular arc is calculated clockwise). The tangent $t_1$ at the point $ T_1$ to the circle $\omega_1$ intersects $\ell_2$ at the point $M_1$. Similarly, the tangent $t_2$ at the point $T_2$ to the circle $\omega_2$ intersects $\ell_1$ at point $M_2$. Prove that the midpoints of the segments $M_1M_2$ are on the same a straight line that does not depend on the position of points $T_1$, $T_2$.

Let $ n $ be a positive integer. Consider words of length $n$ composed of letters from the set $ \{ M, E, O \} $. Let $ a $ be the number of such words containing an even number (possibly 0) of blocks $ ME $ and an even number (possibly 0) blocks of $ MO $ . Similarly let $ b $ the number of such words containing an odd number of blocks $ ME $ and an odd number of blocks $ MO $. Prove that $ a>b $.

Find all natural numbers $n>3$, such that $2^{2000}$ is divisible by $1+C^1_n+C^2_n+C^3_n$.

Let $a_n$ be a sequence such that $a_1 = 1$ and $a_{n+1} = \lfloor a_n +\sqrt{a_n} +\frac12 \rfloor $, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. What are the last four digits of $a_{2012}$?

Let $a_1=5$ and $a_{n+1}= a^2_{n}-2$ for any $n=1,2,...$. a) Find $\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_1a_2 ...a_{n}}$ b) Find $\lim_{\nu \rightarrow \infty}\left(\frac{1}{a_1}+\frac{1}{a_1a_2}+...+\frac{1}{a_1a_2 ...a_{\nu}}\right)$

Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $D$ and $E$ are chosen on $AC$ and $AB$, respectively, such that quadrilateral $BCDE$ is cyclic and when the triangle is folded along segment $DE$, point $A$ lies on side $BC$. If the length of $DE$ can be expressed as $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $100m+n$. [i]Proposd by Joseph Heerens[/i]

Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$

Find all functions $f$ from integers to integers such that \[ f(m+n) + f(m-n) - 2f(m) = 6mn^2\] for all integers $m$ and $n$.

There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?

In there $2018\times 2018$ square cells colored in white or black. It is known, that exists $10 \times 10$ square with only white cells and $10\times 10$ square with only black cells. For what minimal $d$ always exists square $10\times 10$ such that the number of black and white cells differs by no more than $d$?

Chords $AB$ and $CD$, which do not intersect, are drawn in a circle. On the chord $AB$ or on its extension is taken the point $E$. Using a compass and construct the point $F$ on the arc $AB$ , such that $\frac{PE}{EQ} = \frac{m}{n}$, where $m,n$ are given natural numbers, $P$ is the point of intersection of the chord $AB$ with the chord $FC$, $Q$ is the point of intersection of the chord $AB$ with the chord $FD$. Consider cases where $E\in PQ$ and $E \notin PQ$.

Two fixed circles are given on the plane, one of them lies inside the other one. From a point $C$ moving arbitrarily on the external circle, draw two chords $CA, CB$ of the larger circle such that they tangent to the smalaler one. Find the locus of the incenter of triangle $ABC$.

Given $n \ge 3$ positive integers not exceeding $100$, let $d$ be their greatest common divisor. Show that there exist three of these numbers whose greatest common divisor is also equal to $d$.

Prove that the numbers $${{2^n-1} \choose {i}}, i = 0, 1, . . ., 2^{n-1} - 1,$$ have pairwise different residues modulo $2^n$

Find all polynomials $P(x)$ with real coefficients such that \[P(x)P(x + 1) = P(x^2), \quad \forall x \in \mathbb R.\]

Let $ABC$ be a triangle. Point $P$ lies in the interior of $\triangle ABC$ such that $\angle ABP = 20^\circ$ and $\angle ACP = 15^\circ$. Compute $\angle BPC - \angle BAC$.

Two players play the following game on a circular board with 2009 houses. The two plays put, alternatively, on an empty house, one of three pieces, called [i]explorer (E)[/i], [i]trap (T)[/i] or [i]stone (S)[/i]. A treasure is a sequence of three consecutive filled houses such that the first one (on any direction) has an explorer and the middle one doesn't have a trap. For example, [i]STE[/i] is not a treasure, while [i]TEE[/i] is a treasure. The first player forming a treasure wins. Can any of the players guarantee the victory? And, in affirmative case, who?

Let $P(x),Q(x)$ be monic polynomials with integer coeeficients. Let $a_n=n!+n$ for all natural numbers $n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for all positive integer $n$ then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n\neq0$. \\ [i]Hint (given in question): Try applying division algorithm for polynomials [/i]