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convex quadrilateral with angles and sides in 2 arithmetic sequences

Source: 2021 D&uuml;rer Math Competition Finals Day2 E+16 https://artofproblemsolving.com/community/c2749870_

January 9, 2022

geometryanglesarithmetic sequence

Problem Statement

The angles of a convex quadrilateral form an arithmetic sequence in clockwise order, and its side lengths also form an arithmetic sequence (but not necessarily in clockwise order). If the quadrilateral is not a square, and its shortest side has length

1

, then its perimeter is

a + \sqrt{b}4

, where

a

and

b

are positive integers. What is the value of

a + b

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