In a sequence of natural numbers, the first number is $a$, and each subsequent number is the smallest number coprime to all the previous ones and greater than all of them. Prove that in this sequence from some place all numbers will be primes.

On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $K, L$ and $M$ are selected, respectively, such that $AK = AM$ and $BK = BL$. If $\angle{MLB} = \angle{CAB}$, Prove that $ML = KI$, where $I$ is the incenter of triangle $CML$.

$200$ sticks are given whose lengths are $1, 2, 4, \ldots , 2^{199}$. What is the smallest number of sticks needed to be broken so that out of all the resulting sticks, several triangles could be created, if each stick could be broken only once, and each triangle can be created out of only three sticks?

Non-negative integers are placed on the vertices of a $100$-gon, the sum of the numbers is $99$. Every minute at one of the vertices that is equal to $0$ will be replaced by $2$ and both its neighboring numbers are subtracted by $1$. Prove that after a while a negative number will appear on the board.

Natural numbers $a$ and $b$ are given such that the number $$ P = \frac{[a, b]}{a + 1} + \frac{[a, b]}{b + 1} $$Is a prime. Prove that $4P + 5$ is the square of a natural number.

The Feuerbach point (the tangent point of the inscribed circle and the nine-point circle of triangle $ABC$) $F$ is marked in triangle $ABC$. $A_1$ is on the side $BC$ such that $AA_1$ is the altitude of triangle $ABC$. Prove that the line symmetric to $FA_1$ with respect to $BC$ is perpendicular to $IO$, where $O$ is the center of the circumcircle of the triangle $ABC$ and $I$ is the center of its incircle.

Alireza multiplied a billion consecutive natural numbers, and Matin multiplied two million consecutive natural numbers. Prove that these two got different results or one of them has made a mistake.

Kostya drives a car from a village to a city, driving along three roads. Moreover, on each of these roads, he drives at a constant speed. Is it possible that a third of the distance traveled was completed earlier than a third of the time, half of the distance traveled later than half of the time, and two-thirds of the distance was earlier than two-thirds of the time?

The company has $100$ people. For any $k$, we can find a group of $k$ people such that there are two (different from them) strangers, each of them knows all of these $k$ people. At what maximum $k$ is this possible?

The natural numbers $x, y > 1$, are such that $x^2 + xy -y$ is the square of a natural number. Prove that $x + y + 1$ is a composite number.

In the triangle $ABC$, the cevians $AA_1$, $BB_1$ and $CC_1$ intersect at the point $O$. It turned out that $AA_1$ is the bisector, and the point $O$ is closer to the straight line $AB$ than to the straight lines $A_1C_1$ and $B_1A_1$. Prove that $\angle{BAC} > 120^{\circ}$.

Each of the $11$ girls wants to mail each of the other a gift for Christmas. The packages contain no more than two gifts. If they have enough time, what is the smallest possible number of packages that they have to send?