1987 USAMO Problems/Problem 5

Problem

$a_1, a_2, \cdots, a_n$ is a sequence of 0's and 1's. T is the number of triples $(a_i, a_j, a_k)$ with $i<j<k$ which are not equal to (0, 1, 0) or (1, 0, 1). For $1\le i\le n$ , $f(i)$ is the number of $j<i$ with $a_j = a_i$ plus the number of $j>i$ with $a_j\neq a_i$ . Show that $T=\sum_{i=1}^n f(i)\cdot\left(\frac{f(i)-1}2\right)$ . If n is odd, what is the smallest value of T?

Solution

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1987 USAMO (Problems • Resources)
Preceded by Problem 4	Followed by Last Question
1 • 2 • 3 • 4 • 5
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1987 USAMO Problems/Problem 5

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