Found problems: 3
2023 Polish MO Finals, 5
Give a prime number $p>2023$. Let $r(x)$ be the remainder of $x$ modulo $p$. Let $p_1<p_2< \ldots <p_m$ be all prime numbers less that $\sqrt[4]{\frac{1}{2}p}$. Let $q_1, q_2, \ldots, q_n$ be the inverses modulo $p$ of $p_1, p_2, \ldots p_n$. Prove that for every integers $0 < a,b < p$, the sets
$$\{r(q_1), r(q_2), \ldots, r(q_m)\}, ~~ \{r(aq_1+b), r(aq_2+b), \ldots, r(aq_m+b)\}$$
have at most $3$ common elements.
2025 Turkey Team Selection Test, 9
Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?
2022 Poland - Second Round, 3
Positive integers $a,b,c$ satisfying the equation
$$a^3+4b+c = abc,$$
where $a \geq c$ and the number $p = a^2+2a+2$ is a prime. Prove that $p$ divides $a+2b+2$.