Difference between revisions of "1987 USAMO Problems"
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M is the midpoint of XY. The points P and Q lie on a line through Y on opposite sides of Y, such that <math>|XQ| = 2|MP|</math> and <math>\frac{|XY|}2 < |MP| < \frac{3|XY|}2</math>. For what value of <math>\frac{|PY|}{|QY|}</math> is <math>|PQ|</math> a minimum? | M is the midpoint of XY. The points P and Q lie on a line through Y on opposite sides of Y, such that <math>|XQ| = 2|MP|</math> and <math>\frac{|XY|}2 < |MP| < \frac{3|XY|}2</math>. For what value of <math>\frac{|PY|}{|QY|}</math> is <math>|PQ|</math> a minimum? | ||
− | [[1987USAMO Problems/Problem | + | [[1987USAMO Problems/Problem 4|Solution]] |
==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>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>\displaystyle T=\sum_{i=1}^n f(i)\cdot\frac{f(i)-1}2$. If n is odd, what is the smallest value of T? | <math>a_1, a_2, \cdots, a_n</math> is a sequence of 0's and 1's. T is the number of triples <math>(a_i, a_j, a_k) with </math>i<j<k<math> which are not equal to (0, 1, 0) or (1, 0, 1). For </math>1\le i\le n<math>, </math>f(i)<math> is the number of </math>j<i<math> with </math>a_j = a_i<math> plus the number of </math>j>i<math> with </math>a_j\neq a_i<math>. Show that </math>\displaystyle T=\sum_{i=1}^n f(i)\cdot\frac{f(i)-1}2$. If n is odd, what is the smallest value of T? | ||
− | [[1987USAMO Problems/Problem | + | [[1987USAMO Problems/Problem 5|Solution]] |
Revision as of 14:03, 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 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 i<j<k1\le i\le nf(i)j<ia_j = a_ij>ia_j\neq a_i\displaystyle T=\sum_{i=1}^n f(i)\cdot\frac{f(i)-1}2$. If n is odd, what is the smallest value of T?