Balls of $3$ colours — red, blue and white — are placed in two boxes. If you take out $3$ balls from the first box, there would definitely be a blue one among them. If you take out $4$ balls from the second box, there would definitely be a red one among them. If you take out any $5$ balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.

The rhombuses $ABDK$ and $CBEL$ are arranged so that $B$ lies on the segment $AC$ and $E$ lies on the segment $BD$. Point $M$ is the midpoint of $KL$. Prove that $\angle DME=90^{\circ}$.

Given $10$ positive integers with a sum equal to $1000$. The product of their factorials is a $10$-th power of an integer. Prove that all these numbers are equal.

Given a set $P$ of $n>100$ points on the plane such that no three of them are collinear, and a set $S$ of $20n$ distinct segments, each joining a pair of points from $P$. Prove that there exists a line not passing through a point from $P$ and intersecting at least $200$ segments from $S$.

Alex calculated the value of function $f(n) = n^2 + n + 1$ for each integer from $1$ to $100$. Marina calculated the value of function $g(n) = n^2-n+1$ for the same numbers. Who of them has greater product of values and what is their ratio?

The integers from $1$ to $320000$ are placed in the cells of a $8 \times 40000$ board. Prove that it is possible to permute the rows of the table so that the numbers in each column will not be sorted from the top to the bottom in increasing order.

The positive numbers $a_1, a_2, \ldots , a_{2024}$ are placed on a circle clockwise in this order. Let $A_i$ be the arithmetic mean of the number $a_i$ and one or several following it clockwise. Prove that the largest of the numbers $A_1, A_2, \ldots , A_{2024}$ is not less than the arithmetic mean of all numbers $a_1, a_2, \ldots , a_{2024}$.

There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$. (a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles. (b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.

Let $a, b, c, d$ be positive real numbers. It is given that at least one of the following two conditions holds: $$ab >\min(\frac{c}{d}, \frac{d}{c}), cd >\min(\frac{a}{b}, \frac{b}{a}).$$Show that at least one of the following two conditions holds: $$bd>\min(\frac{c}{a}, \frac{a}{c}), ca >\min(\frac{d}{b}, \frac{b}{d}).$$

In an acute-angled triangle $ABC$ let $BL$ be the bisector, and let $BK$ be the altitude. Let the lines $BL$ and $BK$ meet the circumcircle of $ABC$ again at $W$ and $T$, respectively. Given that $BC = BW$, prove that $TL \perp BC$.

Let $n$ be a $d$-digit (i.e., having $d$ digits in its decimal representation) positive integer not divisible by $10$. Writing all the digits of $n$ in reverse order, we obtain the number $n'$. Determine if it is possible that the decimal representation of the product $n\cdot n'$ consists of digits $8$ only, if (a) $d = 9998$; (b) $d = 9999?$

Yasha writes in the cells of the table $99 \times 99$ all positive integers from 1 to $99^2$ (each number once). Grisha looks at the table and selects several cells, among which there are no two cells sharing a common side, and then sums up the numbers in all selected cells. Find the largest sum Grisha can guarantee to achieve.

Let $a, b, c$ be reals and consider three lines $y=ax+b, y=bx+c, y=cx+a$. Two of these lines meet at a point with $x$-coordinate $1$. Show that the third one passes through a point with two integer coordinates.

Given is a permutation of $1, 2, \ldots, 2023, 2024$ and two positive integers $a, b$, such that for any two adjacent numbers, at least one of the following conditions hold: 1) their sum is $a$; 2) the absolute value of their difference is $b$. Find all possible values of $b$.

Find the largest positive integer $n$, such that there exists a finite set $A$ of $n$ reals, such that for any two distinct elements of $A$, there exists another element from $A$, so that the arithmetic mean of two of these three elements equals the third one.

Let $ABC$ be an acute triangle and let $X$ be a variable point on $AC$. The incircle of $\triangle ABX$ touches $AX, BX$ at $K, P$, respectively. The incircle of $\triangle BCX$ touches $CX, BX$ at $L, Q$, respectively. Find the locus of $KP \cap LQ$.