Olympiad problems

2024 Korea National Olympiad, 4

Find the smallest positive integer $ k \geq 2 $ for which there exists a polynomial $ f(x) $ of degree $ k $ with integer coefficients and a leading coefficient of $ 1 $ that satisfies the following condition: (Condition) For any two integers $ m $ and $ n $, if $ f(m) - f(n) $ is a multiple of $ 31 $, then $ m - n $ is a multiple of $ 31 $.

2010 IFYM, Sozopol, 8

Tags: number theory , modulo

Let $m, n,$ and $k$ be natural numbers, where $n$ is odd. Prove that $\frac{1}{m}+\frac{1}{m+n}+...+\frac{1}{m+kn}$ is not a natural number.

2015 Hanoi Open Mathematics Competitions, 5

Tags: number theory , modulo

Let $a,b,c$ and $m$ ($0 \le m \le 26$) be integers such that $a + b + c = (a - b)(b- c)(c - a) = m$ (mod $27$) then $m$ is (A): $0$, (B): $1$, (C): $25$, (D): $26$ (E): None of the above.

2007 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , modulo , prime

Let $p > 3$ be a prime number and $ x$ an integer, denote by $r ( x )\in \{ 0 , 1 , ... , p - 1 \}$ to the rest of $x$ modulo $p$ . Let $x_1, x_2, ... , x_k$ ( $2 < k < p$) different integers modulo $p$ and not divisible by $p$. We say that a number $a \in \{ 1 , 2 ,..., p -1 \}$ is [i]good [/i] if $r ( a x_1) < r ( a x_2) <...< r ( a x_k)$. Show that there are at most $\frac{2 p}{k + 1}-{ 1}$ [i]good [/i] numbers.