Found problems: 988
Solve the equation in natural numbers $$
xy+yz+zx = xyz + 2.
$$
FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$
Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$. Of these, $(100, 1 )$ is the pair with the smallest value for $n$. Find the pair with the smallest value for $m$.
Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$
For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$
Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that:
\[a^p+b^p=p^c.\]
Find all triples of non-negative integer numbers $(E, C, U)$ such that $EC \ge 1$ and:
$$2^{3^E}+3^{2^C}=593 \cdot 5^U$$
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$.
Daniel Harrer
Find all triples $(x,y,z)$ of natural numbers that verify the equation
$$2x^2y^2+2y^2z^2+2z^2x^2-x^4-y^4-z^4=576.$$
(Leo Moser) Show that the Diophantine equation \[\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}+\frac{1}{x_{1}x_{2}\cdots x_{n}}= 1\] has at least one solution for every positive integers $n$.
Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.
Prove that there are infinite triples of positive integers $(x,y,z)$ such that
$$x^2+y^2+z^2+xy+yz+zx=6xyz.$$
Find all integer solutions of the equation $a^3+3ab^2+7b^3=2011$.
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$
Solve the equation $28^x =19^y +87^z$, where $x, y, z$ are integers.
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy
$a \le b \le c$ and $abc = 2(a + b + c)$.
Determine the prime numbers $p, q, r$ with the property $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$
Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$
Find all pairs $(a,b)$ of integers $a$ and $b$ satisfying
\[(b^2+11(a-b))^2=a^3 b\]
Let $p, q, r$ be primes and let $n$ be a positive integer such that $p^n + q^n = r^2$. Prove that $n = 1$.
Laurentiu Panaitopol
Find all positive integers $m, n$ such that $n + (n+1) + (n+2) + ...+ (n+m) = 1000$.
Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.
Find all positive integers $a, b,c,d$ such that $a + b + c + d - 3 = ab + cd$.
Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square.
[i]Proposed by P. Boyvalenkov[/i]