Problem 3

In an

n

-gon all interior angles are equal, and the lengths of consecutive sides satisfy

a_1\ge a_2\ge\cdots\ge a_n

. Prove that

a_1=a_2=\cdots=a_n

Step 4 of 5: Use monotonicity to equalize all but the first side

In plain words

The first and last entries of a decreasing chain are equal, so every entry in between is equal too.

a_2=a_n\quad\text{and}\quad a_2\ge a_3\ge\cdots\ge a_n\Longrightarrow a_2=\cdots=a_n=t

Detailed analysis

The assumed chain $a_2\ge a_3\ge\cdots\ge a_n$ together with $a_2=a_n$ forces $a_2=\cdots=a_n=t$ .