Difference between revisions of "1987 USAMO Problems"
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==Problem 5== | ==Problem 5== | ||
| − | <math>a_1, a_2, \cdots, a_n</math> is a sequence of 0's and 1's. T is the number of triples <math>(a_i, a_j, a_k) with < | + | <math>a_1, a_2, \cdots, a_n</math> is a sequence of 0's and 1's. T is the number of triples <math>(a_i, a_j, a_k)</math> with <math>i<j<k</math> which are not equal to (0, 1, 0) or (1, 0, 1). For <math>1\le i\le n</math>, <math>f(i)</math> is the number of <math>j<i</math> with <math>a_j = a_i</math> plus the number of <math>j>i</math> with <math>a_j\neq a_i</math>. Show that <math>T=\sum_{i=1}^n f(i)\cdot\frac{f(i)-1}2</math>. If n is odd, what is the smallest value of T? |
[[1987USAMO Problems/Problem 5|Solution]] | [[1987USAMO Problems/Problem 5|Solution]] | ||
Revision as of 14:04, 24 July 2011
Problem 1
Find all solutions to 
, where m and n are non-zero integers.
Problem 2
The feet of the angle bisectors of the triangle ABC form a right-angled triangle. If the right-angle is at X, where AX is the bisector of angle A, find all possible values for angle A.
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?
