Difference between revisions of "1992 USAMO Problems"
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<cmath> \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}. </cmath> | <cmath> \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}. </cmath> | ||
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[[1992 USAMO Problems/Problem 2 | Solution]] | [[1992 USAMO Problems/Problem 2 | Solution]] | ||
Revision as of 09:51, 22 April 2010
Problem 1
Find, as a function of 
the sum of the digits of
![\[9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right),\]](http://latex.artofproblemsolving.com/3/9/5/395f3a29a3769af826b23ad1d2ca73fe825dd3d4.png)
where each factor has twice as many digits as the previous one.
Problem 2
Prove
![\[\frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \cdots + \frac{1}{\cos 88^\circ \cos 89^\circ} = \frac{\cos 1^\circ}{\sin^2 1^\circ}.\]](http://latex.artofproblemsolving.com/c/7/a/c7a719b049c276e55a58cfb22c3e914b3cce1a7f.png)
Problem 3
For a nonempty set 
of integers, let 
be the sum of the elements of . Suppose that 
is a set of positive integers with 
and that, for each positive integer 
, there is a subset of 
for which 
. What is the smallest possible value of 
?
Problem 4
Chords 
, 
, and 
of a sphere meet at an interior point 
but are not contained in the same plane. The sphere through , 
, 
, and is tangent to the sphere through 
, 
, 
, and . Prove that 
.
Problem 5
Let 
be a polynomial with complex coefficients which is of degree 
and has distinct zeros.Prove that there exists complex numbers 
such that divides the polynomial
