Problems from AoPS.

Q1. In a sports league, each team uses a set of at most signature colors. A set of teams is* color-identifiable* if one can assign each team in one of their signature colors, such that no team in is assigned any signature color of a different team in .

For all positive integers and , determine the maximum integer such that: In

any sports league with exactly distinct colors present over all teams, one can always

find a color-identifiable set of size at least .

Q2. Let be an acute scalene triangle with circumcenter , and let be on line such that . The circle with diameter intersects the circumcircle of at two points and , where . Points , , , are defined analogously.

- Prove that , , are concurrent.
- Prove that , , are concurrent on the Euler line of . (E. Chen)

Q3. Let be relatively prime nonconstant polynomials. Show that there can be at most three real numbers such that is the square of a polynomial. (A. Miller)