Found problems: 988
Determine all pairs of integer numbers $(x, y)$ such that:
$$\frac{(x - y)^2}{x + y} = x - y + 6$$
Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$. Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$, where $d(n)$ denotes the number of positive divisors of positive integer $n$.
For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)\equal{}4$ and $ p(14)\equal{}24$. Let $ k$ be a positive integer greater than $ 1$.
(a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)\equal{}k^2\minus{}k\plus{}1$.
(b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)\equal{}k^2\minus{}k\plus{}1$.
Solve the following equation in real numbers:
$\frac{(x^2-1)(|x|+1)}{x+sgnx}=[x+1].$
Find all pairs $(m,n)$ of positive integers such that $m+n$ and $mn+1$ are both powers of $2$.
Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$
Prove there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^3+y \mid x+y^3$.
([i]Dan Schwarz[/i])
If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$.
Daniel Harrer
Find all integer solutions of the equation $xy+5x-3y=27$.
The equation
\[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\]
is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$
Show that there are infinitely many pairs $(x, y)$ of rational numbers such that $x^3 +y^3 =9$.
Find the triplets of natural numbers $(p,q,r)$ that satisfy the equality $$\frac{1}{p}+\frac{q}{q^r -1}=1.$$
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
Solve the equation in natural numbers $$ x+y+z+t=xyzt.
$$
For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.
For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.
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 $3^m-7^n=2.$