2011 USA Mathematical Olympiad - Day II
4:11 AM
Muhammad Yusuf
, Posted in
USAMO 4. Consider the assertion
that for each positive integer n ≥ 2, the remainder upon dividing 22^n by 2n−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 Q1 and Q2 are located within ABCD such that ∠Q1BC = ∠ABP, ∠Q1CB = ∠DCP, ∠Q2AD = ∠BAP, ∠Q2DA = ∠CDP. Prove that Q1Q2 ∥ AB if and only if Q1Q2 ∥ CD.
USAMO 6. Let A be a set with |A| = 225, meaning that A has 225 elements. Suppose further
that there are eleven subsets A1, ... ,A11 of A such that |Ai| = 45 for 1 ≤ i ≤ 11 and |Ai ∩ Aj | = 9 for 1 ≤ i <
j ≤
11. Prove that |A1 ∪ A2 ∪ ・ ・ ・ ∪ A11| ≥
165, and give an example
for which equality holds.
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