Found problems: 916
Find all triplets of nonnegative integers $(x,y,z)$ and $x\leq y$ such that
$x^2+y^2=3 \cdot 2016^z+77$
Find all non-negative integers $n$ for which the equation
\[
{\left( x^2 + y^2 \right)}^n = {(xy)}^{2018}
\]
admits positive integral solutions.
Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$
Find all integers $x,y$ that satisfy the equation $x+y=x^2-xy+y^2$
Prove that for every integer $n > 1$ there exist integers $x$ and $y$ such that $$\frac{1}{n}=\frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+...+\frac{1}{y(y+1)}.$$
Are there any (not necessarily positive) integers $m$ and $n$ such that
a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ?
b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$
Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$
Show that $(w, x, y, z)=(0,0,0,0)$ is the only integer solution to the equation
$$w^{2}+11 x^{2}-8 y^{2}-12 y z-10 z^{2}=0$$
Solve $ x^3-y^3=2xy+7 $ in integers.
Let $k, l, m, n$ be positive integers. Given that $k+l+m+n=km=ln$, find all possible values of $k+l+m+n$.
For $r> 0$ denote by $B_r$ the set of points at distance at most $r$ length units from the origin.
If $P_r$ is the set of the points in $B_r$ whit integer coordinates, show that the equation $$xy^3z + 2x^3z^3-3x^5y = 0$$
has an odd number of solutions $(x, y, z)$ in $P_r$.
Find all possible non-negative integer solution ($x,$ $y$) of the following equation-
$$x!+2^y=z!$$
Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.
Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$
Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.
Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that
\[\frac{a^p - a}{p}=b^2.\]
Suppose that $m, n, k$ are positive integers satisfying $$3mk=(m+3)^n+1.$$
Prove that $k$ is odd.
Determine all pairs of integers $a, b$ for which they apply $4^a + 4a^2 + 4 = b^2$ .
Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.
Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$.
[i]Proposed by Viktor Simjanoski[/i]
Find all positive integers $n$ such that the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ has exactly $2011$ positive integer solutions $(x,y)$ where $x \leq y$.
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$
Find all rational solutions of
\[a^2 + c^2 + 17(b^2 + d^2) = 21,\]\[ab + cd = 2.\]
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$
a) Factorize $A= x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2$
b) Prove that there are no integers $x,y,z$ such that $x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 $