Inequality on the number of local extrema and saddles
Source: Korean Summer Program Practice Test 2016 4
August 17, 2016
combinatorics
Problem Statement
Two integers and distinct real numbers are given. Define the sets as the following, where all indices are modulo .
\begin{align*}
A &= \{ 1 \le i \le n : a_i > a_{i-k}, a_{i-1}, a_{i+1}, a_{i+k} \text{ or } a_i < a_{i-k}, a_{i-1}, a_{i+1}, a_{i+k} \} \\
B &= \{ 1 \le i \le n : a_i > a_{i-k}, a_{i+k} \text{ and } a_i < a_{i-1}, a_{i+1} \} \\
C &= \{ 1 \le i \le n ; a_i > a_{i-1}, a_{i+1} \text{ and } a_i < a_{i-k}, a_{i+k} \}
\end{align*}
Prove that .