Found problems: 11
Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$
Find all right triangles with integer sides and inradius 6.
A Pythagorean triangle is a right triangle whose three sides are integers.
The best known example is the triangle with rectangular sides $3$ and $4$ and hypotenuse $5$.
Determine all Pythagorean triangles whose area is twice the perimeter.
Do positive reals $a, b, c, x$ such that $a^2+ b^2 = c^2$ and $(a + x)^2+ (b +x)^2 = (c + x)^2$ exist?
A positive integer is written on a blackboard. Carmela can perform the following operation as many times as she wants: replace the current integer $x$ with another positive integer $y$, as long as $|x^2 - y^2|$ is a perfect square. For example, if the number on the blackboard is $17$, Carmela can replace it with $15$, because $|17^2 - 15^2| = 8^2$, then replace it with $9$, because $|15^2 - 9^2| = 12^2$. If the number on the blackboard is initially $3$, determine all integers that Carmela can write on the blackboard after finitely many operations.
Positive integers $a, b, c$ satisfying the equality $a^2 + b^2 = c^2$.
Show that the number $\frac12(c - a) (c - b)$ is square of an integer.
We call [i]Pythagorean triple[/i] a triple $(x,y,z)$ of positive integers such that $x<y<z$ and $x^2+y^2=z^2$. Prove that for all $n \in \mathbb{N}$ number $2^{n+1}$ is in exactly $n$ [i]Pythagorean triples[/i]
Consider a natural number $ n\equiv 9\pmod {25}. $ Prove that there exist three nonnegative integers $ a,b,c $ having the property that:
$$ n=\frac{a(a+1)}{2} +\frac{b(b+1)}{2} +\frac{c(c+1)}{2} $$
Numbers $a, b, c$ are such that $3a + 4b = 3c$ and $4a - 3b = 4c$. Show that $a^2 + b^2 = c^2$.
Given a circle of radius $r= 1995$. Show that around it you can describe exactly $16$ primitive Pythagorean triangles. The primitive Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.
If the positive integers $a, b, c$ satisfy $a^2+b^2=c^2$, then $(a, b, c)$ is called a Pythagorean triple. Find all Pythagorean triples containing $30$.