MathDB

Let

A_k

be the number of pairs

(a_1, a_2, ..., a_{2k})

for

k\leq 50

, where

a_1, a_2, ..., a_{2k}

are all different positive integers that satisfy the following.

\cdot

a_1, a_2, ..., a_{2k} \leq 100

\cdot

For an odd number less or equal than

2k-1

, we have

a_i > a_{i+1}

\cdot

For an even number less or equal than

2k-2

, we have

a_i < a_{i+1}

Prove that

A_1 \leq A_2 \leq \cdots \leq A_{49}

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