Found problems: 436
Find all pairs $(a, b)$ of positive integers that satisfy the equation $a^2 -3 \cdot 2^b = 1$.
If $p$ is a prime number and $x, y$ are positive integers, find in terms of $p$, all pairs $(x, y)$ that satisfy the equation: $$p(x -2) = x(y -1).$$
If $x+y = 21$, find all triples $(x, y, p)$ that satisfy this equation.
Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$
Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities.
Remember that the number $1$ is not prime.
Solve, for integers $x$ and $y$ : $$2x^2y = (x+2)^2(y + 1), $$ provided that $(x+2)^2(y + 1)> 1000$.
Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.
Solve the equation in nonnegative integers $a,b,c$:
$3^a+2^b+2015=3c!$
I.Gorodnin
Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\
a^2 = 2(a+b+c)\\
\end{array} \right.
\]
Solve, in the positive integers, the equation $5^m + n^2 = 3^p$ .
Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.
Find all pairs $(m, n)$ of positive integers such that $m^2 + n^2 =(m + 1)(n + 1).$
Find all pairs of integers $(x,y)$ such that $$(x+1)(y+1)(x+y)(x^2+y^2)=16x^2y^2$$
Find all pairs $(x, y)$ of integers numbers such that $y^3+5=x(y^2+2)$
Find all pairs of integers $(a, b)$ that satisfy the equation
$$(a + 1)(b- 1) = a^2b^2.$$
Find all the values of $n \in N$ such that $n^2 = 2^n$.
Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.
Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$
Find all integers m, n such that $m^3 = n^3 + n$.
Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $
Prove that the equation
$$\frac{1}{\sqrt{x} +\sqrt{1006}}+\frac{1}{\sqrt{2012 -x} +\sqrt{1006}}=\frac{2}{\sqrt{x} +\sqrt{2012 -x}}$$
has $2013$ integer solutions.
Are there integers $m$ and $n$ such that $m^2 + 1954 = n^2$?
Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.
Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation
$k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.
Determine all pairs of integer numbers $(a, b)$ that satisfy the relation:
$$(a + b)(a + b + 6) = 34 \cdot 3^{|2a-b|}- 7$$
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.