1991 USAMO Problems

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Problems from the 1991 USAMO. There were five questions administered in one three-and-a-half-hour session.

Problem 1

In triangle $\, ABC, \,$ angle $\,A\,$ is twice angle $\,B,\,$ angle $\,C\,$ is obtuse, and the three side lengths $\,a,b,c\,$ are integers. Determine, with proof, the minimum possible perimeter.

Solution

Problem 2

For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$ . Prove that $\sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1),$ where " $\Sigma$ " denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$ .

Solution

Problem 3

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence $2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots \pmod{n}$ is eventually constant.

[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$ . Also $a_i \pmod{n}$ means the remainder which results from dividing $\,a_i\,$ by $\,n$ .]

Solution

Problem 4

Let $\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \,$ where $\,m\,$ and $\,n\,$ are positive integers. Prove that $\,a^m + a^n \geq m^m + n^n$ .

[You may wish to analyze the ratio $\,(a^N - N^N)/(a-N),$ for real $\, a \geq 0 \,$ and integer $\, N \geq 1$ .]

Solution

Problem 5

Let $\, D \,$ be an arbitrary point on side $\, AB \,$ of a given triangle $\, ABC, \,$ and let $\, E \,$ be the interior point where $\, CD \,$ intersects the external common tangent to the incircles of triangles $\, ACD \,$ and $\, BCD$ . As $\, D \,$ assumes all positions between $\, A \,$ and $\, B \,$ , prove that the point $\, E \,$ traces the arc of a circle.