Olympiad problems

2014 IMAC Arhimede, 5

Tags: number theory , numerical systems , system , binomial , product , modulo , prime number

Let $p$ be a prime number. The natural numbers $m$ and $n$ are written in the system with the base $p$ as $n = a_0 + a_1p +...+ a_kp^k$ and $m = b_0 + b_1p +..+ b_kp^k$. Prove that $${n \choose m} \equiv \prod_{i=0}^{k}{a_i \choose b_i} (mod p)$$

2018 Abels Math Contest (Norwegian MO) Final, 1

Tags: factorial , modulo , residue , number theory

For an odd number n, we write $n!! = n\cdot (n-2)...3 \cdot 1$. How many different residues modulo $1000$ do you get from $n!!$ for $n= 1, 3, 5, …$?

2024 Korea National Olympiad, 4

Tags: number theory , modulo , polynomial with integer coeffi

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 $.

2009 Ukraine Team Selection Test, 6

Tags: number theory , set , modulo , remainder

Find all odd prime numbers $p$ for which there exists a natural number $g$ for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal.