In the kindergarten there is a big box with balls of three colors: red, blue and green, 100 balls in total. Once Pasha took out of the box 30 red, 10 blue, and 20 green balls and played with them. Then he lost five balls and returned the others back into the box. The next day, Sasha took out of the box 8 red, 18 blue, and 48 green balls. Is it possible to determine the color of at least one lost ball?

Determine if there exist five consecutive positive integers such that their LCM is a perfect square.

Points $A'$ and $B'$ lie inside the parallelogram $ABCD$ and points $C'$ and $D'$ lie outside of it, so that all sides of 8-gon $AA'BB'CC'DD'$ are equal. Prove that $A'$, $B'$, $C'$, $D'$ are concyclic.

Vova has a square grid $72\times 72$. Unfortunately, $n$ cells are stained with coffee. Determine if Vova always can cut out a clean square $3\times 3$ without its central cell, if a) $n=699$; b) $n=750$.

Vasya has a numeric expression $$ \Box \cdot \Box +\Box \cdot \Box $$and 4 cards with numbers that can be put on 4 free places in the expression. Vasya tried to put cards in all possible ways and all the time obtained the same value as a result. Prove that equal numbers are written on three of his cards.

In a triangle $ABC$ with $\angle BAC = 90^{\circ}$ let $BL$ be the bisector, $L\in AC$. Let $D$ be a point symmetrical to $A$ with respect to $BL$. Let $M$ be the circumcenter of $ADC$. Prove that $CM$, $DL$, and $AB$ are concurrent.

15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible to have equal numbers of apricots in all the boxes after $k$ moves.

Determine if there exist positive integers $a_1,a_2,...,a_{10}$, $b_1,b_2,...,b_{10}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,10\}$ the sum $\sum\limits_{i\in S}a_i$ divides $\left( 12+\sum\limits_{i\in S}b_i \right)$.

Pasha placed numbers from 1 to 100 in the cells of the square $10\times 10$, each number exactly once. After that, Dima considered all sorts of squares, with the sides going along the grid lines, consisting of more than one cell, and painted in green the largest number in each such square (one number could be colored many times). Is it possible that all two-digit numbers are painted green?

In a triangle $ABC$ let $I$ be the incenter. Prove that the circle passing through $A$ and touching $BI$ at $I$, and the circle passing through $B$ and touching $AI$ at $I$, intersect at a point on the circumcircle of $ABC$.

Find all positive integers $n\geqslant 2$ such that there exists a permutation $a_1$, $a_2$, $a_3$, \ldots, $a_{2n}$ of the numbers $1, 2, 3, \ldots, 2n$ satisfying $$a_1\cdot a_2 + a_3\cdot a_4 + \ldots + a_{2n-3} \cdot a_{2n-2} = a_{2n-1} \cdot a_{2n}.$$

Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs.

Given a triangle $ABC$ with $BC=a$, $CA=b$, $AB=c$, $\angle BAC = \alpha$, $\angle CBA = \beta$, $\angle ACB = \gamma$. Prove that $$ a \sin(\beta-\gamma) + b \sin(\gamma-\alpha) +c\sin(\alpha-\beta) = 0.$$

15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible to have equal numbers of apricots in all the boxes after $k$ moves.

On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.

Determine if there exist pairwise distinct positive integers $a_1,a_2,\ldots,a_{101}$, $b_1$, $b_2$, \ldots, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,101\}$ the sum $\sum\limits_{i\in S}a_i$ divides $\left( 100!+\sum\limits_{i\in S}b_i \right)$.