LES SUJETS 2018-2019
Subject 1: Light trap
Consider a plane figure f made of two sides AB and BC of a triangle.
In the picture you see a ray of light that slides into f and then emerges after 3 reflections.
Problem 1: Determine a concrete example of points A, B, C and a ray of light , so that we obtain 4, 5, 6, … n reflections.
Problem 2: Is it possible, that the ray never comes out, that is to say, the number of reflections is infinite? ─ That would mean that our simple figure f is really a light trap...
For both the above tasks, we additionally assume that:
before and after each reflection, the light ray moves along a straight line;
during each reflection, the principle is that the angle of incidence of the ray equals the angle of its reflection;
the ray has an intersection with the interval AC sliding into f and it does not hit the point C.
Extra problem(s): If you manage to solve these problems, we may think similar problems for slightly more complex traps ...
Subject 2: „leaky choice” game
Two players alternately pull stones out of a hat which contains 7 stones (our players know that there are 7 of them). With each move, each player has only two possibilities: he draws either 1 or 3 stones (he cannot pull out 2 stones – that is why the choice is “leaky”). The one who pulls out the last stone – wins.
Problem 1 (not difficult): Does any of these players have a winning strategy? If yes – please find it.
Problem 2: Let’s generalize the problem – instead of 7 stones, let’s introduce n of them. Therefore we analyze an analogous game, but with n stones in the hat.
Please find the solution for every n natural.
Extra problem(s): If you manage to solve these problems, we may think about similar games for slightly different “leaky choices”.