Found problems: 988
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.
[i]Proposed by Lewis Chen[/i]
Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.
Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$, where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$.
If $R$ and $S$ are two rectangles with integer sides such that the perimeter of $R$ equals the area of $S$ and the perimeter of $S$ equals the area of $R$, then we call $R$ and $S$ a friendly pair of rectangles. Find all friendly pairs of rectangles.
Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.
Prove that equation $y^2=x^3+7$ doesn't have any solution on integers.
Determine the integers $a, b, c$ for which
$$\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}$$
Let $ p$ be a prime number such that $ p\minus{}1$ is a perfect square. Prove that the equation
$ a^{2}\plus{}(p\minus{}1)b^{2}\equal{}pc^{2}$
has infinite many integer solutions $ a$, $ b$ and $ c$ with $ (a,b,c)\equal{}1$
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 positive integers $k$ such that there exists positive integers $a, b$ such that
\[a^2+4=(k^2-4)b^2.\]
Find all the values of $n \in N$ such that $n^2 = 2^n$.
Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $
[i]Dragoș Crișan[/i]
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.
Let $a, b$ be coprime positive integers. A positive integer $n$ is said to be [i]weak[/i] if there do not exist any nonnegative integers $x, y$ such that $ax+by=n$. Prove that if $n$ is a [i]weak[/i] integer and $n < \frac{ab}{6}$, then there exists an integer $k \geq 2$ such that $kn$ is [i]weak[/i].
For which integers $n = 1, \ldots , 6$ does the equation $$a^n + b^n = c^n + n$$ have a solution in integers?
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}. $
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.
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 $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $