1996 USAMO Problems

Problems of the 1996 USAMO.

Problem 1

Prove that the average of the numbers $n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)$ is $\cot 1^{\circ}$ .

Problem 2

For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$ . Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$ . Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.

Solution

Problem 3

Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$ ) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$ .

Solution

Problem 4

An $n$ -term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$ . Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$ .

Solution

Problem 5

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$ , $\angle MBA=20^\circ$ , $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$ . Prove that the triangle is isosceles.

Solution

Problem 6

Determine (with proof) whether there is a subset $X$ of the integers with the following property: for any integer $n$ there is exactly one solution of $a + 2b = n$ with $a,b \in X$ .

Solution

1996 USAMO (Problems • Resources)
Preceded by 1995 USAMO	Followed by 1997 USAMO
1 • 2 • 3 • 4 • 5 • 6
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1996 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

See Also