Found problems: 436
Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\
c + d = ab \end{cases}$ .
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.
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 $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$
Let $a$ and $b$ be positive integers such that $2a^2 + a = 3b^2 + b$. Prove that $a-b$ is a perfect square.
How many different integer solutions to the inequality $|x| + |y| < 100$ are there?
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.
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]
Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$
\frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$
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 $x,y$.
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]
How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$?
(A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.
Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that:
$$a^3 + b^3 + c^3 = 2016$$
Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1 \\
ax + 3y = 1\\
\end{array} \right.
\] has an integer solution.
Find all the integer solutions of the equation $ m^4 + 2n^2 = 9mn$.
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$
Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$
If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$.
Daniel Harrer
Find the triplets of natural numbers $(p,q,r)$ that satisfy the equality $$\frac{1}{p}+\frac{q}{q^r -1}=1.$$
Solve the equation in natural numbers $$ x+y+z+t=xyzt.
$$