Olympiad problems

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.

2014 Contests, 3

Tags: divisibility , hensel , surjective , quadratic residue

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]

PEN C Problems, 5

Tags: analytic geometry , graphing lines , slope , hyperbola , conic , quadratic residue

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

2016 Iran Team Selection Test, 3

Tags: number theory , prime number , modular arithmetic , quadratic residue

Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.

PEN C Problems, 4

Tags: floor function , quadratic , inequalities , number theory , prime number , quadratic residue

Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.

PEN C Problems, 2

Tags: gauss , modular arithmetic , algebra , system of equations , number theory , quadratic residue

The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

2007 Bulgaria Team Selection Test, 4

Tags: number theory , quadratic residue

Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$

2016 Spain Mathematical Olympiad, 2

Tags: number theory , prime number , quadratic residue

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 $p|\beta(\beta-1)+25$.

2014 Contests, 2

Tags: quadratic , induction , modular arithmetic , number theory , quadratic residue

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

2017 China Team Selection Test, 3

Tags: combinatorics , set theory , quadratic residue

Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions: ($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$; ($ii$)$2017|x_1+...+x_{100}$; ($iii$)$2017|x_1^2+...+x_{100}^2$.

2014 APMO, 3

Tags: divisibility , hensel , surjective , quadratic residue

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]

2018 Serbia National Math Olympiad, 2

Tags: sequence , quadratic residue , number theory

Let $n>1$ be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by $n$. Prove that the number of consecutive beautiful numbers is less or equal to $1+\lfloor \sqrt{3n} \rfloor$.

2008 Romania Team Selection Test, 3

Tags: modular arithmetic , number theory , quadratic reciprocity , quadratic residue

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

2024 Brazil Cono Sur TST, 3

Tags: hensel s lemma , chinese remainder theorem , divisibility , quadratic residue , number theory

Find all positive integers $m$ that have some multiple of the form $x^2+5y^2+2024$, with $x$ and $y$ integers.

2022 Korea -Final Round, P5

Tags: number theory , divisibility , chinese remainder theorem , quadratic residue , hensel s lemma

Find all positive integers $m$ such that there exists integers $x$ and $y$ that satisfies $$m \mid x^2+11y^2+2022.$$

KoMaL A Problems 2023/2024, A. 858

Tags: number theory , diophantine equation , quadratic residue

Prove that the only integer solution of the following system of equations is $u=v=x=y=z=0$: $$uv=x^2-5y^2, (u+v)(u+2v)=x^2-5z^2$$

2016 Iran Team Selection Test, 3

Tags: number theory , prime number , modular arithmetic , quadratic residue

Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.

2023 Silk Road, 3

Tags: number theory , prime number , quadratic , quadratic residue , directed graph

Let $p$ be a prime number. We construct a directed graph of $p$ vertices, labeled with integers from $0$ to $p-1$. There is an edge from vertex $x$ to vertex $y$ if and only if $x^2+1\equiv y \pmod{p}$. Let $f(p)$ denotes the length of the longest directed cycle in this graph. Prove that $f(p)$ can attain arbitrarily large values.

2017 Balkan MO Shortlist, N5

Tags: arithmetic mean , fractional part , odd , prime , quadratic residue , number theory

Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

2017 Romania Team Selection Test, P4

Tags: arithmetic mean , fractional part , odd , prime , quadratic residue , number theory

Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

2020 Stars of Mathematics, 3

Tags: quadratic residue , number theory

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]

PEN C Problems, 1

Tags: quadratic , quadratic residue

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

PEN C Problems, 6

Tags: algebra , polynomial , absolute value , quadratic residue

Let $a, b, c$ be integers and let $p$ be an odd prime with \[p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.\] Show that \[\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).\]

2015 Iran MO (3rd round), 3

Tags: number theory , prime number , quadratic residue

Let $p>5$ be a prime number and $A=\{b_1,b_2,\dots,b_{\frac{p-1}{2}}\}$ be the set of all quadratic residues modulo $p$, excluding zero. Prove that there doesn't exist any natural $a,c$ satisfying $(ac,p)=1$ such that set $B=\{ab_1+c,ab_2+c,\dots,ab_{\frac{p-1}{2}}+c\}$ and set $A$ are disjoint modulo $p$. [i]This problem was proposed by Amir Hossein Pooya.[/i]

2007 Bulgaria Team Selection Test, 4

Tags: number theory , quadratic residue

Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$