Sequence $(a_n)_{n\geq 0}$ is defined as $a_{0}=0, a_1=1, a_2=2, a_3=6$, and $ a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0$. Prove that $n^2$ divides $a_n$ for infinite $n$. (Romania)

A bag is filled with quarters and nickels. The average value when pulling out a coin is $10$ cents. What is the least number of nickels in the bag possible?

Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \ne A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ be the circumcircle of the triangle $\triangle AEZ$. Let $D$ be the second point of intersection of $(c)$ with $ZC$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $FE$ and $CZ$. If the tangent to $(c)$ at $Z$ meets $PA$ at $T$, prove that the points $T$, $E$, $B$, $Z$ are concyclic. Proposed by [i]Theoklitos Parayiou, Cyprus[/i]

Consider the sequence $s_0 = (1, 2008)$. Define new sequences $s_i$ inductively by inserting the sum of each pair of adjacent terms in $s_{i-1}$ between them — for instance, $s_1 = (1, 2009, 2008)$. For some $n, s_n$ has exactly one term that appears twice. Find this repeated term.

NASA is launching a spaceship at the south pole, but a sudden earthquake shock caused the spaceship to be launched at an angle of $\theta$ from vertical ($0 < \theta < 90^\circ$). The spaceship crashed back to Earth, and NASA found the debris floating in the ocean in the northern hemisphere. NASA engineers concluded that $\tan \theta > M$, where $M$ is maximal. Find $M$. Assume that the Earth is a sphere, and the trajectory of the spaceship (in the reference frame of Earth) is an ellipse with the center of the Earth one of the foci. [i]Proposed by Kevin You[/i]

Prove that if $t$ is a natural number then there exists a natural number $n>1$ such that $(n,t)=1$ and none of the numbers $n+t,n^2+t,n^3+t,....$ are perfect powers.

Let $ a_n$ be the last nonzero digit in the decimal representation of the number $ n!.$ Does the sequence $ a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?

For what values of $ n$ does the group $ \textsl{SO}(n)$ of all orthogonal transformations of determinant $ 1$ of the $ n$-dimensional Euclidean space possess a closed regular subgroup?($ \textsl{G}<\textsl{SO}(n)$ is called $ \textit{regular}$ if for any elements $ x,y$ of the unit sphere there exists a unique $ \varphi \in \textsl{G}$ such that $ \varphi(x)\equal{}y$.) [i]Z. Szabo[/i]

The numbers $1, 2\ldots11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.

Given a positive integer $ n$, find all functions $ F: \mathbb{N} \rightarrow \mathbb{R}$ such that $ F(x\plus{}y)\equal{}F(xy\minus{}n)$ whenever $ x,y \in \mathbb{N}$ satisfy $ xy>n$.

How many real solutions are there to the equation $ |x \minus{} |2x \plus{} 1\parallel{} \equal{} 3.$ (Here, $ |x|$ denotes the absolute value of $ x$: i.e., if $ x \geq 0,$ then $ |x| \equal{} |\minus{}x| \equal{} x.$) A. 0 B. 1 C. 2 D. 3 E. 4

Let $ x,y,z$ real numbers such that $ x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx$. Find the minimum value of \[ {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}\]

Find the number of 8-digit base-6 positive integers $(a_1a_2a_3a_4a_5a_6a_7a_8)_6$ (with leading zeros permitted) such that $(a_1a_2\ldots a_8)_6\mid(a_{i+1}a_{i+2}\ldots a_{i+8})_6$ for $i=1,2,\ldots,7$, where indices are taken modulo $8$ (so $a_9=a_1$, $a_{10}=a_2$, and so on). [i]Victor Wang[/i]

Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.

Christopher and Robin are playing a game in which they take turns tossing a circular token of diameter 1 inch onto an infinite checkerboard whose squares have sides of 2 inches. If the token lands entirely in a square, the player who tossed the token gets 1 point; otherwise, the other player gets 1 point. A player wins as soon as he gets two more points than the other player. If Christopher tosses first, what is the probability that he will win? Express your answer as a fraction.

Let $A$ be a subset of the edges of an $n\times n $ table. Let $V(A)$ be the set of vertices from the table which are connected to at least on edge from $A$ and $j(A)$ be the number of the connected components of graph $G$ which it's vertices are the set $V(A)$ and it's edges are the set $A$. Prove that for every natural number $l$: $$\frac{l}{2}\leq min_{|A|\geq l}(|V(A)|-j(A)) \leq \frac{l}{2}+\sqrt{\frac{l}{2}}+1$$

$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.

The point $P$ lies inside the triangle $ABC$. A line is drawn through $P$ parallel to each side of the triangle. The lines divide $AB$ into three parts length $c, c', c"$ (in that order), and $BC$ into three parts length $a, a', a"$ (in that order), and $CA$ into three parts length $b, b', b"$ (in that order). Show that $abc = a'b'c' = a"b"c"$.

A positive integer $n \ge 2$ is called [i]peculiar [/i] if the number $n \choose i$ + $n \choose j $ $-i-j$ is even for all integers $i$ and $j$ such that $0 \le i \le j \le n$. Determine all peculiar numbers.

Let $f : \mathbb R \to \mathbb R$ be a function such that $f(1)=1$ and \[f(x+y)=f(x)+f(y)\] And for all $x \in \mathbb R / \{0\}$ we have $f\left( \frac 1x \right) = \frac{1}{f(x)}.$ Find all such functions $f.$

Let $ L$ denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points $ A,B,C$ of $ L$ there is a fourth point $ D,$ different from $ A,B,C,$ such that the interiors of the segments $ AD,BD,CD$ contain no points of $ L.$ Is the statement true if one considers four points of $ L$ instead of three?

Find all polynomials $ P\in\mathbb{R}[x]$ such that for all $ r\in\mathbb{Q}$,there exist $ d\in\mathbb{Q}$ such that $ P(d)\equal{}r$

Let $ABC$ be an acute-angled triangle, $H$ the intersection point of its altitudes , and $AA'$ the diameter of the circumcircle of triangle $ABC$. Prove that the quadrilateral $HB A'C$ is a parallelogram.

Is it true that for all natural $n$, it is always possible to give each of the $n$ children a hat painted in one of $100$ colors so that if a girl is known to more than $2015$ boys, then not all of these boys have hats of the same color, and, if a boy is acquainted with more than $2015$ girls, don't all these girls have hats of the same color? [hide=original wording]Vai tiesa, ka visiem naturāliem n vienmēr iespējams katram no n bērniem iedot pa cepurei, kas nokrāsota vienā no 100 krāsām tā, ka, ja kāda meitene ir pazīstama ar vairāk nekā 2015 zēniem, tad ne visiem šiem zēniem cepures ir vienā krāsā, un, ja kāds zēns ir pazīštams ar vairāk nekā 2015 meitenēm, tad ne visām šīm meitenēm cepures ir vienā krāsā?[/hide]

Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$.