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## The hemispheric battle

C fires at city B and D fires at city A, but their rockets collide in mid-air and fall into the ground at F. Then C fires a laser bouncing off the ground at F at the same angle as incidence. Will the laser beam hit combatant D?

Author: Marshall W. Buck

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## The skew bridge

Two points are on opposite sides of a channel. A skew bridge, which has a fixed direction (for instance North East), can be placed anywhere along the channel. Where to place it such that the path from A to B over the bridge is the shortest?

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## The prisoners dilemma

Two prisoners are chained to the walls of a triangular cell. The base is covered with a mirror and a perpendicular wall is separating them. They can each see the opposite corner through a hole. Proof they can see each other through another hole at the base of the perpendicular.

Proposed by Marshall W. Buck. Note that this problem is known as the Blanchet Theorem.

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A ladder seen from the side is sliding down a wall. Prove that a fixed point on the ladder traces part of an ellipse.

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## The flat earth

An observer is standing on a plane at h metres below the centre of a reflecting sphere of radius r. Another object on the plane is perceived at an angle α from the vertical. What is α of the horizon in terms of h and r?

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## Camera obscura

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?

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## Pulling the strings

Two rods of length x and y are connected by two strings with a knot exactly halfway. The rods are placed parallel both with uncrossed and crossed strings. The distance between the knots is measured as a and b respectively. Express x and y in terms of a and b.

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## The arena

In a circular arena two contestants are leaving from points A and B. Their respective speeds are such that they would collide at point C, but just before they do they exchange directions. Then they hit the edge of the arena and return to their starting points. During the whole race, A and B move at constant but not necessarily equal speeds. Which contestant returns to his starting point first?

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## 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?

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## The magic mirror

You are standing in front of a mirror. Your shoulder width is 50 cm and your eyes are 6 cm apart. What is the minimum mirror width for a stereo view of yourself?