This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 11

2022 Bolivia Cono Sur TST, P4
Tags: pythagorean triple , inradius , number theory , geometry
Find all right triangles with integer sides and inradius 6.

2019 IMAR Test, 3
Tags: abstract algebra , pythagorean triple , modular arithmetic , number theory
Consider a natural number $ n\equiv 9\pmod {25}. $ Prove that there exist three nonnegative integers $ a,b,c $ having the property that: $$ n=\frac{a(a+1)}{2} +\frac{b(b+1)}{2} +\frac{c(c+1)}{2} $$

2011 China Northern MO, 5
Tags: pythagorean triple , number theory
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 $30$.

2018 Polish Junior MO First Round, 1
Tags: algebra , pythagorean triple , system of equations
Numbers $a, b, c$ are such that $3a + 4b = 3c$ and $4a - 3b = 4c$. Show that $a^2 + b^2 = c^2$.

2000 Bosnia and Herzegovina Team Selection Test, 3
Tags: pythagorean triple , number theory
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]

2003 Dutch Mathematical Olympiad, 1
Tags: area , pythagorean triple , integer , perimeter , geometry
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.

1995 Tuymaada Olympiad, 6
Tags: number theory , right triangle , pythagorean triple , circles , geometry
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.

2018 Polish Junior MO Second Round, 1
Tags: system of equations , pythagorean triple , algebra
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?

2012 Czech-Polish-Slovak Junior Match, 5
Tags: number theory , pythagorean triple , perfect square
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.

2022 Korea Junior Math Olympiad, 8
Tags: number theory , rational number , diophantine equation , pythagorean triple , discriminant
Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$

2025 Philippine MO, P2
Tags: pythagorean triple , number theory
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.