Difference between revisions of "1987 USAMO Problems"
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==Problem 3== | ==Problem 3== | ||
− | X is the smallest set of polynomials <math>p(x)</math> such that: | + | <math>X</math> is the smallest set of polynomials <math>p(x)</math> such that: |
− | 1. <math>p(x) = x</math> belongs to X | + | 1. <math>p(x) = x</math> belongs to <math>X</math>. |
− | 2. If <math>r(x)</math> belongs to X, then <math>x\cdot r(x)</math> and <math>(x + (1 - x) \cdot r(x) )</math> both belong to X. | + | 2. If <math>r(x)</math> belongs to <math>X</math>, then <math>x\cdot r(x)</math> and <math>(x + (1 - x) \cdot r(x) )</math> both belong to <math>X</math>. |
− | Show that if <math>r(x)</math> and <math>s(x)</math> are distinct elements of X, then <math>r(x) \neq s(x)</math> for any <math>0 < x < 1</math>. | + | Show that if <math>r(x)</math> and <math>s(x)</math> are distinct elements of <math>X</math>, then <math>r(x) \neq s(x)</math> for any <math>0 < x < 1</math>. |
[[1987USAMO Problems/Problem 3|Solution]] | [[1987USAMO Problems/Problem 3|Solution]] |
Revision as of 14:06, 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
is the smallest set of polynomials such that:
1. belongs to . 2. If belongs to , then and both belong to .
Show that if and are distinct elements of , 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?