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1994 USAMO Problems

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Problems of the 1994 USAMO.

Contents

Problem 1

Let \, k_1 < k_2 < k_3 < \cdots \, be positive integers, no two consecutive, and let \, s_m = k_1 + k_2 + \cdots + k_m \, for \, m = 1,2,3, \ldots \; \;. Prove that, for each positive integer \, n, \, the interval \, [s_n, s_{n + 1}) \, contains at least one perfect square.

Solution

Problem 2

The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides are red, blue, red, blue, red, blue, red, yellow, blue?

Solution

Problem 3

A convex hexagon ABCDEF is inscribed in a circle such that AB = CD = EF and diagonals AD, BE, and CF are concurrent. Let P be the intersection of AD and CE. Prove that CP/PE = (AC/CE)^2.

Solution

Problem 4

Let \, a_1, a_2, a_3, \ldots \, be a sequence of positive real numbers satisfying \, \sum_{j = 1}^n a_j \geq \sqrt {n} \, for all \, n \geq 1. Prove that, for all \, n \geq 1, \,

\sum_{j = 1}^n a_j^2 > \frac {1}{4} \left( 1 + \frac {1}{2} + \cdots + \frac {1}{n} \right).

Solution

Problem 5

Let \, |U|, \, \sigma(U) \, and \, \pi(U) \, denote the number of elements, the sum, and the product, respectively, of a finite set \, U \, of positive integers. (If \, U \, is the empty set, \, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1.) Let \, S \, be a finite set of positive integers. As usual, let \, \binom{n}{k} \, denote \, n! \over k! \, (n - k)!. Prove that

\sum_{U \subseteq S} ( - 1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S)

for all integers \, m \geq \sigma(S).

Solution

Resources

1994 USAMO (Problems)
Preceded by
1993 USAMO 		1 • 2 • 3 • 4 • 5 Followed by
1995 USAMO
All USAMO Problems and Solutions
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Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.
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