Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?

Let $AB$ be a diameter of the circunference $\omega$, let $C$ and $D$ be point in this circunference, such that $CD$ is perpedicular to $AB$. Let $E$ be the point of intersection of the segment $CD$ and the segment $AB$, and a point $P$ that is in the segment $CD, P$ is different of $E$. The lines $AP$ and $BP$ intersects $\omega$, in $F$ and $G$ respectively. If $O$ is the circumcenter of triangle $EFG$, show that the area of triangle $OCD$ is invariant, independent of the position of the point $P$.

In a far-away country, travel between cities is only possible by bus or by train. One can travel by train or by bus between only certain cities, and there are not necessarily rides in both directions. We know that for any two cities $A$ and $B$, one can reach $B$ from $A$, [i]or[/i] $A$ from $B$ using only bus, or only train rides. Prove that there exists a city such that any other city can be reached using only one type of vehicle (but different cities may be reached with different vehicles).

Several black and white checkers (tokens?) are standing along the circumference. Two men remove checkers in turn. The first removes all the black ones that had at least one white neighbour, and the second -- all the white ones that had at least one black neighbour. They stop when all the checkers are of the same colour. a) Let there be $40$ checkers initially. Is it possible that after two moves of each man there will remain only one (checker)? b) Let there be $1000$ checkers initially. What is the minimal possible number of moves to reach the position when there will remain only one (checker)?

We say that a function \( f: \mathbb{R} \to \mathbb{R} \) is morally odd if its graph is symmetric with respect to a point, that is, there exists \((x_0, y_0) \in \mathbb{R}^2\) such that if \((u, v) \in \{(x, f(x)) : x \in \mathbb{R}\}\), then \((2x_0 - u, 2y_0 - v) \in \{(x, f(x)) : x \in \mathbb{R}\}\). On the other hand, \( f \) is said to be morally even if its graph \(\{(x, f(x)) : x \in \mathbb{R}\}\) is symmetric with respect to some line (not necessarily vertical or horizontal). If \( f \) is morally even and morally odd, we say that \( f \) is parimpar. (a) Let \( S \subset \mathbb{R} \) be a bounded set and \( f: S \to \mathbb{R} \) be an arbitrary function. Prove that there exists \( g: \mathbb{R} \to \mathbb{R} \) that is parimpar such that \( g(x) = f(x) \) for all \( x \in S \). (b) Find all polynomials \( P \) with real coefficients such that the corresponding polynomial function \( P: \mathbb{R} \to \mathbb{R} \) is parimpar.

There are $100$ people in a room. Some are [i]wise[/i] and some are [i]optimists[/i]. $\quad \bullet~$ A [i]wise[/i] person can look at someone and know if they are wise or if they are an optimist. $\quad \bullet~$ An [i]optimist[/i] thinks everyone is wise (including themselves). Everyone in the room writes down what they think is the number of wise people in the room. What is the smallest possible value for the average? $$ \mathrm a. ~ 10\qquad \mathrm b.~25\qquad \mathrm c. ~50 \qquad \mathrm d. ~75 \qquad \mathrm e. ~100 $$

Let $T=\text{TNFTPP}$. Triangle $ABC$ has $AB=6T-3$ and $AC=7T+1$. Point $D$ is on $BC$ so that $AD$ bisects angle $BAC$. The circle through $A$, $B$, and $D$ has center $O_1$ and intersects line $AC$ again at $B'$, and likewise the circle through $A$, $C$, and $D$ has center $O_2$ and intersects line $AB$ again at $C'$. If the four points $B'$, $C'$, $O_1$, and $O_2$ lie on a circle, find the length of $BC$.

Petya chooses $100$ pairwise distinct positive numbers less than $1$ and arranges them in a circle. In one operation, he may take three consecutive numbers \( a, b, c \) (in this order) and replace \( b \) with \( a - b + c \). What is the greatest value of \( k \) such that Petya could initially choose the numbers and perform several operations so that \( k \) of the resulting numbers are integers? \\

Prove that if the numbers $ x $, $ y $, $ z $ satisfy the equationw $$x + y + z = a,$$ $$ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{a},$$ then at least one of them is equal to $ a $.

The average age of the $6$ people in Room A is $40$. The average age of the $4$ people in Room B is $25$. If the two groups are combined, what is the average age of all the people? $\textbf{(A)}\ 32.5 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 33.5 \qquad \textbf{(D)}\ 34\qquad \textbf{(E)}\ 35$

We say that a triangle of sides $a,b,c$ is [i] virtual[/i] if such measures satisfy $$\begin{cases} a^{2024}+b^{2024}> c^{2024},\\ b^{2024}+c^{2024}> a^{2024},\\ c^{2024}+a^{2024}> b^{2024} \end{cases}$$ Find the number of ordered triples $(a,b,c)$ such that $a,b,c$ are integers between $1$ and $2024$ (inclusive) and $a,b,c$ are the sides of a [i]virtual [/i] triangle.

Let $\Omega$ be a set of $n$ points, where $n>2$. Let $\Sigma$ be a nonempty subcollection of the $2^n$ subsets of $\Omega$ that is closed with respect to the unions, intersections and complements. If $k$ is the number of elements of $\Sigma,$ what are the possible values of $k?$

If $a$ and $b$ are positive integers, prove that $11a+2b$ is a multiple of $19$ if and only if so is $18a+5b$ .

Five distinct points and five distinct lines are given in the plane. Prove that one can select two of the points and two of the lines so that none of the selected lines contains any of the selected points.

When $1999{}^2{}^0{}^0{}^0$ is divided by $5$ , the remainder is $ \text{(A)}\ 4\qquad\text{(B)}\ 1\qquad\text{(C)}\ 2\qquad\text{(D)}\ 3\qquad\text{(E)}\ 0 $

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.