Let $a_1,a_2,\ldots$ be an infinite sequence of integers such that $a_i$ divides $a_{i+1}$ for all $i\geq 1$, and let $b_i$ be the remainder when $a_i$ is divided by $210$. What is the maximal number of distinct terms in the sequence $b_1,b_2,\ldots$?

In a foreign island $5781$ sheep are sacrificed every year for the two deities of the island, Alice and Bob. Every deity wants as many sheep as he can to be sacrificed for him, and not for the other deity. Initially all $5781$ sheep are arranged around a circle with equal distances. At the first step, Alice puts one magic stone between several pairs of neighboring sheep, so that the total number of magic stones is odd. After that, Bob sacrifices one of the sheep for himself and replaces it by a food bucket. At the third step, Alice chooses a pair of neighboring sheep (not including the two which are separated by the bucket) and puts a border between them (the border may be at the same place as a magic stone). After that, all remaining sheep walk through an arc of the circle to the food bucket without crossing the border (so that there is only one possible route). Every sheep which walks on an odd number of magic stones is sacrificed for Alice, and every other sheep is for Bob. What is the maximal number of sheep which Alice can sacrifice for herself in a certain year, regardless of Bob's action?

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.

Triangle I is equilateral with side $A$, perimeter $P$, area $K$, and circumradius $R$ (radius of the circumscribed circle). Triangle II is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then: $ \textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad$ $\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad$ $\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes} $

$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$

Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy \[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]

Find the smallest positive $x$ for which holds the inequality $$\sin x \le \sin (x+1)\le \sin (x+2)\le sin (x+3)\le \sin (x+4) .$$

Let $n!_0$ denote the number obtained from $n!$ by deleting all the zeroes in the end of it decimal representation. Find all positive integers $a, b, c$, such that $a!_0+b!_0=c!_0$.

Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.

In the equation $x^2+z_1x+z_2+m=0$, $z_1,z_2,m$ are complex numbers, and $z_1^2-4z_2=16+20\text{i}$. Two roots of the equations are $\alpha,\beta$. If $|\alpha-\beta|=2\sqrt7$, find the maximum and minumum value of $|m|$.

Let $k$ and $s$ be odd positive integers such that \[\sqrt{3k-2}-1 \le s \le \sqrt{4k}.\] Show that there are nonnegative integers $t$, $u$, $v$, and $w$ such that \[k=t^{2}+u^{2}+v^{2}+w^{2}, \;\; \text{and}\;\; s=t+u+v+w.\]

Let $ABCD$ be a cyclic quadrilateral whose center of the circumscribed circle is inside this quadrilateral, and its diagonals intersect in point $S{}$. Let $P{}$ and $Q{}$ be the centers of the curcimuscribed circles of triangles $ABS$ and $BCS$. The lines through the points $P{}$ and $Q{}$, which are parallel to the sides $AD$ and $CD$, respectively, intersect at the point $R$. Prove that the point $R$ lies on the line $BD$.

On an acute triangle $ABC$ we draw the internal bisector of $<ABC$, $BE$, and the altitude $AD$, ($D$ on $BC$), show that $<CDE$ it's bigger than 45 degrees.

Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$. Suppose that a point $E$ on the side $AD$ satisfies the equality \[AE\cdot ED + BE^2=CD\cdot AE.\] Show that $\angle EBA=\angle DCB$.

Find all prime numbers $p$ such that $p^3-4p+9$ is a perfect square.