Olympiad problems

2017 Saudi Arabia BMO TST, 1

Tags: product , prime , right triangle , integer sidelengths , number theory , geometry

Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called [i]nice [/i] if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)

2008 Postal Coaching, 5

Tags: integer sidelengths , area , inradius , geometry

Prove that there are infinitely many positive integers $n$ such that $\Delta = nr^2$, where $\Delta$ and $r$ are respectively the area and the inradius of a triangle with integer sides.

1988 Austrian-Polish Competition, 9

Tags: combinatorial geometry , rectangle , integer sidelengths , covering , combinatorics

For a rectangle $R$ with integral side lengths, denote by $D(a, b)$ the number of ways of covering $R$ by congruent rectangles with integral side lengths formed by a family of cuts parallel to one side of $R$. Determine the perimeter $P$ of the rectangle $R$ for which $\frac{D(a,b)}{a+b}$ is maximal.

2016 Singapore Senior Math Olympiad, 4

Tags: geometry , integer , integer area , integer sidelengths , perpendicular

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90^o$ or $270^o$. If the lengths of its sides are odd integers, prove that its area is an even integer.

1987 Austrian-Polish Competition, 8

Tags: circles , curve , integer sidelengths , arc , combinatorial geometry , geometry

A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$. A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.