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Week 6 October 26, 2015 1. Write 203 as the sum of numbers whose product is also 203 2. I cram infinitely many people into a room and randomly put red and blue hats on their heads (I have lots of both), and they can all see all the infinitely many other people. Then I get them to all simultaneously guess the colour of their hat. Assuming, once again, that my eardrums do not burst, can only finitely many of them be wrong? 3. Show that, given 4 points in the plane, and 4 point-like lights that can illuminate a 90 degree sector, that there is a way on placing the 4 lights on the 4 points so that the entire plane is illuminated. 4. A convex regular polyhedron can stand stably on any face, because its center of gravity is at the center. Its easy to construct an irregular polyhedron thats unstable on certain faces, so that it topples over. Is it possible to make a model of an irregular polyhedron thats unstable on every face? 5. Find 3x2 y 2 if x, y are integers such that y 2 + 3x2 y 2 = 30x2 + 517. 6. Three frogs are placed at 3 of the 4 vertices of a square. Every second, one frog leaps over another, so that the leaped frog is the midpoint of the line starting at the start position of the leaper and ending with the end position of the leaper. Will a frog ever occupy the 4th vertex of the square? (Assume no order on the frogs leaps) 7. If we have 2015 points in the plane such that any triangle with vertices in these point has area at most 1, show that all 2015 points can be fit in a triangle of area 4 8. If a, b, c, d, e ∈ R are such that a + 2b + 3c + d4 + 5e = 0 show that a + bx + cx2 + dx3 + ex4 has at least one real root. 9. f : Z → Z satisfies f (n) = n − 3 for n ≥ 1000 and f (n) = f (f (n + 5)) if n < 1000. Find f (84) 10. Find a convincing proof that π is rational and tear it to shreds. 1