Found problems: 700
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 prime numbers $p$, $q$ that satisfy the equation
$$p(p^4+p^2+10q)=q(q^2+3).$$
Let $p$ be an odd prime. Show that for every integer $c$, there exists an integer $a$ such that
$$a^{\frac{p+1}{2}} + (a+c)^{\frac{p+1}{2}} \equiv c\pmod p.$$
Let $a,b,k$ be positive integers such that
$gcd(a,b)^2+lcm(a,b)^2+a^2b^2=2020^k$
Prove that $k$ is an even number.
Let $n$ be a positive integer.
(a) Prove that there exists a set $S$ of $6n$ pairwise different positive integers, such that the least common multiple of any two elements of $S$ is no larger than $32n^2$.
(b) Prove that every set $T$ of $6n$ pairwise different positive integers contains two elements the least common multiple of which is larger than $9n^2$.
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.
A Pythagorean triple is a solution of the equation $x^2 + y^2 = z^2$ in positive integers such that $x < y$. Given any non-negative integer $n$ , show that some positive integer appears in precisely $n$ distinct Pythagorean triples.
For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let
\[\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad p\nmid 4a^3+27b^2\}\]
For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$. Find the maximal number of inequivalent elements in $\mathbb{E}_p$.
Let $p$ be a prime number of the form $4k+1$. Suppose that $2p+1$ is prime. Show that there is no $k \in \mathbb{N}$ with $k<2p$ and $2^k \equiv 1 \; \pmod{2p+1}$.
Prove that for each positive integer $ n$, there are pairwise relatively prime integers $ k_0,k_1,\ldots,k_n$, all strictly greater than $ 1$, such that $ k_0k_1\ldots k_n\minus{}1$ is the product of two consecutive integers.
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.
[i]Proposed by Stephan Wagner, South Africa[/i]
Prove that no perfect cube is of the form $y^2+108$ where $y \in \mathbb{Z}$.
Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.
Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 \minus{} 5x \equal{} 1728^{y}\cdot 1733^z \minus{} 17$.
[i](Paolo Leonetti)[/i]
A Pythagorean triple is a solution of the equation $x^2 + y^2 = z^2$ in positive integers such that $x < y$. Given any non-negative integer $n$ , show that some positive integer appears in precisely $n$ distinct Pythagorean triples.
Are there integers $m$ and $n$ such that
\[5m^2 - 6mn + 7n^2 = 1985 \ ?\]
Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.
For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$
[i]Proposed by Carl Schildkraut[/i]
Prove that, for an arbitrary positive integer $n \in Z_{>0}$, the number $n^2- n + 1$ does not have any prime factors of the form $6k + 5$, for $k \in Z_{>0}$.
Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.
Let a regular polygon with $p$ vertices be given, where $p$ is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes $p$ peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex ... after giving the $k$-th peanut, he skips the $2 \cdot k$ next vertices and gives $k+1$-th for the monkey at the next vertex. He does so until all $p$ peanuts are delivered.
[b]I.[/b] How many monkeys are there which does not receive peanuts?
[b]II.[/b] How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)?
Nine points form a grid of size $3\times 3$. How many triangles are there with $3$ vertices at these points?
Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and
\[x^3(y^3+z^3)=2012(xyz+2).\]