Determine the number of three-digit numbers with the following property: The number formed by the first two digits is prime and the number formed by the last two digits is prime.

Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.

Let $\vartriangle ABC$ be an equilateral triangle with side $1$. Four points are marked $P_1$, $P_2$, $P_3$, $P_4$ on side $AC$ and four points $Q_1$, $Q_2$, $Q_3$, $Q_4$ on side $AB$ (see figure) of such a way to generate $9$ triangles of equal area. Find the length of segment $AP_4$. PS. Easier version of 2023 Chile NMO L2 P3

Let $\vartriangle ABC$ be an equilateral triangle with side $1$. The points $P$, $Q$, $R$ are chosen on the sides of the segments $AB$, $BC$, $AC$ respectively in such a way that $$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RA}=\frac25.$$Find the area of triangle $PQR$.

$1600$ bananas are distributed among $100$ monkeys (it is possible that some monkeys do not receive bananas). Prvove that at least four monkeys receive the same amount of bananas.

What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?