Found problems: 988
Find the integers $x, y, z$ for which $$\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z}}}=\dfrac{7}{17}$$
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$
holds.
[i](Walther Janous)[/i]
Determine all positive integer solutions $(x, y, z, t)$ of the equation \[(x+y)(y+z)(z+x)=xyzt\] for which $\gcd(x, y)=\gcd(y, z)=\gcd(z, x)=1$.
Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.
Find all solutions in integers of $x^{3}+2y^{3}=4z^{3}$.
What is the smallest perfect square that ends in $9009$?
Solve the equation in integer numbers $$y^3-x^3=91$$
Show that the equation
$$a^2b=2017(a+b)$$
has no solutions for positive integers $a$ and $b$.
[i]Proposed by Oriol Solé[/i]
Let $x, y$ be integers and $p$ be a prime for which
\[ x^2-3xy+p^2y^2=12p \]
Find all triples $(x,y,p)$.
Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.
If $p$ is a prime positive integer and $x,y$ are positive integers,
find , in terms of $p$, all pairs $(x,y)$ that are solutions of the equation: $p(x-2)=x(y-1)$. (1)
If it is also given that $x+y=21$, find all triplets $(x,y,p)$ that are solutions to equation (1).
Find the number of positive integer solutions $(a,b,c,d)$ to the equation \[(a^2+b^2)(c^2-d^2)=2020.\]
Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.
Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.
$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.
Determine all solutions in non-zero integers $a$ and $b$ of the equation \[(a^2+b)(a+b^2) = (a-b)^3.\]
Find all positive integers $k$ such that there exists positive integers $a, b$ such that
\[a^2+4=(k^2-4)b^2.\]
Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are
a) natural numbers,
b) integers
a) Prove that the equation $2^x + 21^x = y^3$ has no solution in the set of natural numbers.
b) Solve the equation $2^x + 21^y = z^2y$ in the set of non-negative integer numbers.
Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.
Find all pairs of nonnegative integers $(x, y)$ for which $(xy + 2)^2 = x^2 + y^2 $.
Find all natural numbers $n$ for which the equation $\frac1x+\frac1y=\frac1n$ has exactly five solutions $(x,y)$ in the set of natural numbers.
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$
Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.
Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called [i]quadratic [/i] iff at least one of the numbers $ a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.