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