A quarter circular room is covered with mirrors. A ray of light is emitted from one corner and reflected somewhere on the opposite wall, then on the arc, before reaching the adjacent wall at a point a distance a from the light source and b from the circle centre. What is the maximum of a/b?

# Category: Advanced

## The freight elevator

Four points A, B, C and D form a quadrilateral. Prove that it is a rectangle if and only if for an arbitrary point P we have PA2+PC2=PB2+PD2.

## The Eightfold Path

A semicircle, a square and an isosceles triangle. What’s the angle α?

## Fun with flags

A triangle with two cevians. What is d/c-b/a?

## The Olympic bowl

A parabola is tangent to the equal sides of an isosceles triangle in its equal vertices. Another tangent is drawn as shown. Prove that the red line segments are congruent.

## As above, so below

A semicircle, a square, a triangle and two circles. A centre and a tangency point are highlighted. Prove that the circles are congruent.

## Mind the gap

A regular hexagon with an equilateral triangle sharing a vertex and having a vertex somewhere along the opposite side. What is the maximum length of line segment g?

## The crowded house

Three circles are inscribed in a rectangle. What is the area X in terms of A and B?

## The letter box

A rectangle with a diameter and two inscribed squares. What is its area in terms of the square areas A and B?

## A close encounter

Two semicircles are tangent to each other’s diameter. Are the line segments connecting their corners parallel?