Olympiad problems

Gheorghe Țițeica 2024, P4

A positive integer is called [i]joli[/i] if it can be written as the arithmetic mean of two or more (not necessarily distinct) powers of two, and [i]superjoli[/i] if it can be written as the arithmetic mean of two or more distinct powers of two. For instance $7$ and $92$ are superjoli because $7=\frac{2^4+2^2+1}{3}$ and $92=\frac{2^8+2^4+2^2}{3}$. a) Prove that every positive integer is joli. b) Prove that no power of two is superjoli. c) Find the smallest positive integer different from a power of two that is not superjoli. [i]France Olympiad[/i]

olympiad number theory number theory

Gheorghe Țițeica 2024, P3

We know there is some positive integer $k$ such that $\overline{3a\dots a20943}$ is prime (where $a$ appears $k$ times). Find the digit $a$. [i]Dorel Miheț[/i]

olympiad number theory number theory

2024 Turkey Team Selection Test, 4

Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.

factorial modular arithmetic olympiad number theory number theory perfect square perfect squares quadratic residues