40 th United States of America Mathematical Olympiad Day II 12:30 PM – 5 PM EDT April 28, 2011 USAMO 4. Consider the assertion that for each positive integer n ≥ 2, the remainder upon dividing 2 2 n by 2 n −1 is a power of 4. Either prove the assertion or find (with proof) a counterexample. USAMO 5. Let P be a given point inside quadrilateral ABCD. Points Q 1 and Q 2 are located within ABCD such that ∠Q 1 BC = ∠ABP, ∠Q 1 CB = ∠DCP, ∠Q 2 AD = ∠BAP, ∠Q 2 DA = ∠CDP. Prove that Q 1 Q 2 ∥ AB if and only if Q 1 Q 2 ∥ CD. USAMO 6. Let A be a set with |A| = 225, meaning that A has 225 elements. Suppose further that there are eleven subsets A 1 , . . . , A 11 of A such that |A i | = 45 for 1 ≤ i ≤ 11 and |A i ∩ A j | = 9 for 1 ≤ i < j ≤ 11. Prove that |A 1 ∪ A 2 ∪ · · · ∪ A 11 | ≥ 165, and give an example for which equality holds. Copyright c ⃝ Committee on the American Mathematics Competitions, Mathematical Association of America . counterexample. USAMO 5. Let P be a given point inside quadrilateral ABCD. Points Q 1 and Q 2 are located within ABCD such that ∠Q 1 BC = ∠ABP, ∠Q 1 CB = ∠DCP, ∠Q 2 AD = ∠BAP, ∠Q 2 DA = ∠CDP. Prove that Q 1 Q 2 ∥