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a&radic;b-c=length of side of octagon, find a+b+c

Source: Bangladesh Mathematical Olympiad 2021 Problem 10

February 26, 2022

geometryconstruction

Problem Statement

A_1A_2A_3A_4A_5A_6A_7A_8

is a regular octagon. Let

P

be a point inside the octagon such that the distances from

P

A_1A_2, A_2A_3

and

A_3A_4

are

24, 26

and

27

respectively. The length of

A_1A_2

can be written as

a \sqrt{b} -c

, where

a,b

and

c

are positive integers and

b

is not divisible by any square number other than

1

. What is the value of

(a+b+c)