1992 USAMO Problems

Revision as of 10:51, 22 April 2010 by Moplam (talk | contribs) (Problem 2)

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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