Categories
Intermediate

The Arcus arcade

A semicircle and an inscribed circle. Prove that the circle centre is on the parabola.

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Advanced

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.

Categories
Intermediate

The residue

A parabola is tangent to two sides of a square and to the hypotenuse of an isosceles triangle. The three tangency points are shown. What fraction of the square is green?

Categories
Beginner

Arc de Triomphe

A parabola passing through a square’s bottom vertices and tangent to its top side. What fraction is blue?

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Advanced

The screwball ice cream

A circle of radius r touching both sides of a parabola having focal length a. A line segment tangent to the circle is split in two equal segments of length d. What is d in terms of r and a?

Categories
Advanced

The rolling stones

If you roll a circle down a parabola, small circles can roll all the way down and up, whereas large circles get stuck between the sides at some point. What is the transition radius? In other words, what’s the maximum circle that keeps on rolling?