Week 6 October 26, 2015

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.