Found problems: 916
Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]
Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?
Find all solutions of the equation $$p ^ 2 + q ^ 2 + 49r ^ 2 = 9k ^ 2-101$$ with $ p$, $q$ and $r$ positive prime numbers and $k$ a positive integer.
Prove that the system \begin{align*}
x^6+x^3+x^3y+y & = 147^{157} \\
x^3+x^3y+y^2+y+z^9 & = 157^{147}
\end{align*} has no solutions in integers $x$, $y$, and $z$.
The systems of equations
\[\left\{ \begin{array}{l}
2x_1 - x_2 = 1 \\
-x_1 + 2x_2 - x_3 = 1 \\
-x_2 + 2x_3 - x_4 = 1 \\
-x_3 + 3x_4 - x_5 =1 \\
\cdots\cdots\cdots\cdots\\
-x_{n-2} + 2x_{n-1} - x_n = 1 \\
-x_{n-1} + 2x_n = 1 \\
\end{array} \right.
\]
has a solution in positive integers $x_i$. Show that $n$ must be even.
Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$
There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$.
Given $a$ and $b$ distinct positive integers, show that the system of equations
$x y +zw = a$
$xz + yw = b$
has only finitely many solutions in integers $x, y, z,w$.
Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation
$$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$
Prove that $ x + y $ is the square of a natural number.
Let $a$, $b$, $c$, $d$, $e$ be distinct positive integers such that $a^4+b^4=c^4+d^4=e^5$. Show that $ac+bd$ is a composite number.
Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers
Determine all integers $ a$ and $ b$, so that $ (a^3\plus{}b)(a\plus{}b^3)\equal{}(a\plus{}b)^4$
For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$:
$$p_1\, p_2\, \cdots\, p_8
\left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right)
~~=~~ N $$
[i]Proposed by Gerhard Woeginger, Austria[/i]
Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$ has a solution in integers $a$ and $b$.
Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.
Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$
\frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$
Solve in integer:$x^6+x^2y=y^3+2y^2$
Find all positive integers m, n such that $m^3 - n^3 = 999$.
Find all the integer solutions $(x,y,z)$ of the equation $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$,
Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.
Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.
Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.
Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ .
a) Show that $n < 3$.
b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.
Find all non negative integers $(a, b)$ such that
$$a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|}.$$
Find the total number of integer solutions of the equation $$x^5-y^2=4$$
[i](Erken)[/i]