Olympiad problems

2021 Winter Stars of Mathematics, 3

Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$ [i]Vlad Matei[/i]

2016 Spain Mathematical Olympiad, 2

Given a positive prime number $p$. Prove that there exist a positive integer $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist a positive integer $\beta$ such that $p|\beta(\beta-1)+25$.

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2020 Stars of Mathematics, 3

Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$ [i]Vlad Matei[/i]

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2008 Romania Team Selection Test, 3

Let $ m,\ n \geq 3$ be positive odd integers. Prove that $ 2^{m}\minus{}1$ doesn't divide $ 3^{n}\minus{}1$.

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2014 Contests, 3

Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$. [i]Warut Suksompong, Thailand[/i]

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PEN C Problems, 5

Let $p$ be an odd prime and let $Z_{p}$ denote (the field of) integers modulo $p$. How many elements are in the set \[\{x^{2}: x \in Z_{p}\}\cap \{y^{2}+1: y \in Z_{p}\}?\]

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2024 Durer Math Competition Finals, 5

Let $p{}$ be a fixed prime number. Determine the number of ordered $k$-tuples $(a_1,\ldots,a_k)$ of non-negative integers smaller than $p{}$ for which $p\mid a_1^2+\cdots+a_k^2$ where a) $k=3$ and b) $k$ is an arbitrary odd positive integer.

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PEN C Problems, 1

Find all positive integers $n$ that are quadratic residues modulo all primes greater than $n$.

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