1992 USAMO Problems

Problem 1

Find, as a function of $\, n, \,$ the sum of the digits of $9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right),$ where each factor has twice as many digits as the previous one.

Solution

Problem 2

Prove $\frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \cdots + \frac{1}{\cos 88^\circ \cos 89^\circ} = \frac{\cos 1^\circ}{\sin^2 1^\circ}.$

Solution

Problem 3

For a nonempty set $S$ of integers, let $\sigma(S)$ be the sum of the elements of $S$ . Suppose that $A = \{a_1, a_2, \ldots, a_{11}\}$ is a set of positive integers with $a_1 < a_2 < \cdots < a_{11}$ and that, for each positive integer $n \le 1500$ , there is a subset $S$ of $A$ for which $\sigma(S) = n$ . What is the smallest possible value of $a_{10}$ ?

Solution

Problem 4

Chords $AA'$ , $BB'$ , and $CC'$ of a sphere meet at an interior point $P$ but are not contained in the same plane. The sphere through $A$ , $B$ , $C$ , and $P$ is tangent to the sphere through $A'$ , $B'$ , $C'$ , and $P$ . Prove that $AA'=BB'=CC'$ .

Solution

Problem 5

Let $P(z)$ be a polynomial with complex coefficients which is of degree $1992$ and has distinct zeros.Prove that there exists complex numbers $a_1,a_2,\cdots,a_{1992}$ such that $P(z)$ divides the polynomial

$\left(\cdots\left(\left(z-a_1\right)^2-a_2\right)^2\cdots-a_{1991}\right)^2-a_{1992}$

Solution

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1992 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5