ĐỀ THI TOÁN APMO (CHÂU Á THÁI BÌNH DƯƠNG)_ĐỀ 24 pptx
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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
