Difference between revisions of "1987 USAMO Problems"
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==Problem 2== | ==Problem 2== | ||
− | The feet of the angle bisectors of | + | The feet of the angle bisectors of <math>\Delta ABC</math> form a right-angled triangle. If the right-angle is at <math>X</math>, where <math>AX</math> is the bisector of <math>\angle A</math>, find all possible values for <math>\angle A</math>. |
[[1987USAMO Problems/Problem 2|Solution]] | [[1987USAMO Problems/Problem 2|Solution]] |
Revision as of 14:05, 24 July 2011
Contents
[hide]Problem 1
Find all solutions to , where m and n are non-zero integers.
Problem 2
The feet of the angle bisectors of form a right-angled triangle. If the right-angle is at , where is the bisector of , find all possible values for .
Problem 3
X is the smallest set of polynomials such that:
1. belongs to X 2. If belongs to X, then and both belong to X.
Show that if and are distinct elements of X, then for any .
Problem 4
M is the midpoint of XY. The points P and Q lie on a line through Y on opposite sides of Y, such that and . For what value of is a minimum?
Problem 5
is a sequence of 0's and 1's. T is the number of triples with which are not equal to (0, 1, 0) or (1, 0, 1). For , is the number of with plus the number of with . Show that . If n is odd, what is the smallest value of T?