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 $p$ .

Let $p$ be an odd prime number. Show that the smallest positive quadratic nonresidue of $p$ is smaller than $\sqrt{p}+1$.

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 $p$.

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

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

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

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]

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]

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

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

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 $p\nmid x_1x_2x_3x_4$.

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 $1+\lfloor \sqrt{3n} \rfloor$.

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

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

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

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 \[\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).\]

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]

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

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]

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.

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 $p$ .

$p=3k+1$ is a prime number. For each $m \in \mathbb Z_p$, define function $L$ as follow: $L(m) = \sum_{x \in \mathbb{Z}_p}^{ } \left ( \frac{x(x^3 + m)}{p} \right )$ [i]a)[/i] For every $m \in \mathbb Z_p$ and $t \in {\mathbb Z_p}^{*}$ prove that $L(m) = L(mt^3)$. (5 points) [i]b)[/i] Prove that there is a partition of ${\mathbb Z_p}^{*} = A \cup B \cup C$ such that $|A| = |B| = |C| = \frac{p-1}{3}$ and $L$ on each set is constant. Equivalently there are $a,b,c$ for which $L(x) = \left\{\begin{matrix} a & & &x \in A \\ b& & &x \in B \\ c& & & x \in C \end{matrix}\right.$ . (7 points) [i]c)[/i] Prove that $a+b+c = -3$. (4 points) [i]d)[/i] Prove that $a^2 + b^2 + c^2 = 6p+3$. (12 points) [i]e)[/i] Let $X= \frac{2a+b+3}{3},Y= \frac{b-a}{3}$, show that $X,Y \in \mathbb Z$ and also show that :$p= X^2 + XY +Y^2$. (2 points) (${\mathbb Z_p}^{*} = \mathbb Z_p \setminus \{0\}$)

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?

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

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 $p\nmid x_1x_2x_3x_4$.