Found problems: 436
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.
Find the positive integer solutions of the equation $$ x+ \frac{1}{y+ \frac{1}{z}}= \frac{10}{7}$$
(G. Galperin)
Solve in natural numbers the equation $x^2-y^2=105$.
Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$
Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that
$a + b + c = 3k + 1$,
$ab + bc + ca = 3k^2 + 2k$.
Slovakia
Let $\omega = e^{2\pi i /5}$ be a primitive fifth root of unity. Prove that there do not exist integers $a, b, c, d, k$ with $k > 1$ such that \[(a + b \omega + c \omega^2 + d \omega^3)^{k}=1+\omega.\]
[i]Carl Lian[/i]
Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.
Starting with a positive integer $M$ written on the board , Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x+2$.Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he replaces it with $2x+27$.Given that Alice and Bob reach the same number after playing $4$ moves each, find the smallest value of $M+N$
Solve in integers the equation $xy + 3x - 5y = - 3$.
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$
Prove that no integer $x$ and $y$ satisfy: $$3x^2 = 9 + y^3.$$
Let $x$ and $y$ be integers with $x + y \ne 0$. Find all pairs $(x, y)$ such that $$\frac{x^2 + y^2}{x + y}= 10.$$
(Walther Janous)
Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$
Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$.
Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to
$(m - n)^2 = kmn + m + n$.
Determine all positive integers$ n$ that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers.
In the formula: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.
Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$
Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$,
which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square .
Remark: $(7, 4) \ne (4, 7)$
Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.
Solve the following system of equations in integer numbers:
$$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$
Determine all solutions of the diophantine equation $a^2 = b \cdot (b + 7)$ in integers $a\ge 0$ and $b \ge 0$.
(W. Janous, Innsbruck)
Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$
Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.