Malaysia IMONST R2 2020 p2

geometry problems from Malaysia IMO National Selection Test , Round 2, with aops links in the names

aops post collections

2020

2020 Malaysia IMONST 2 Juniors 1

Prove that if $a$ and $b$ are legs, $c$ is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula $r = \frac12 (a + b - c)$.

2020 Malaysia IMONST 2 Juniors 2

Prove that for any integer $n\ge 6$ we can divide an equilateral triangle completely into $n$ smaller equilateral triangles.

2020 Malaysia IMONST 2 Juniors 3, Seniors 1

Given a trapezium with two parallel sides of lengths $m$ and $n$, where $m$, $n$ are integers, prove that it is possible to divide the trapezium into several congruent triangles.

2020 Malaysia IMONST 2 Seniors p4

Given are four circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$. Circles $\Gamma_1$ and $\Gamma_2$ are externally tangent at point $A$. Circles $\Gamma_2$ and $\Gamma_3$ are externally tangent at point $B$. Circles $\Gamma_3$ and $\Gamma_4$ are externally tangent at point $C$. Circles $\Gamma_4$ and $\Gamma_1$ are externally tangent at point $D$. Prove that $ABCD$ is cyclic.