1992 USAMO

Part of USAMO

Subcontests

(5)

1

Polynomial division

\, P(z) \,

be a polynomial with complex coefficients which is of degree

\, 1992 \,

and has distinct zeros. Prove that there exist complex numbers

\, a_1, a_2, \ldots, a_{1992} \,

\, P(z) \,

divides the polynomial

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

1

Value of integer sums

For a nonempty set

\, S \,

of integers, let

\, \sigma(S) \,

be the sum of the elements of

\, S

\, 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\leq 1500, \,

there is a subset

\, S \,

\, A \,

\, \sigma(S) = n

. What is the smallest possible value of

\, a_{10}

1

Trig equality

\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}.

1

Sum of 9's

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.

1

Tangent spheres [AA' = BB' = CC']

AA^{\prime}

BB^{\prime}

CC^{\prime}

of a sphere meet at an interior point

P

but are not contained in a plane. The sphere through

A

B

C

P

is tangent to the sphere through

A^{\prime}

B^{\prime}

C^{\prime}

P

\, AA' = BB' = CC'