Found problems: 916
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 $abc = 2(a + b + c)$.
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]
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
Solve the equation $x^{2}+y^{2}=10xy$ for integers $x$ and $y$
Solve in integers the following equation:
\[y=2x^2+5xy+3y^2\]
Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$
Solve into $N$: $$a^2 = 2^b +c^4$$
Find all non-negative integers $a,b,c,d$ such that $7^a= 4^b + 5^c + 6^d$.
Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation
\[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\]
where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]
Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.
Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.