# Difference between revisions of "2016 IMO Problems/Problem 3"

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− | Let <math>P = A_1A_2 \cdots A_k</math> be a convex polygon in the plane. The vertices <math>A_1,A_2,\dots, A_k</math> have integral coordinates and lie on a circle. Let <math>S</math> be the area of <math>P</math>. And odd positive integer <math>n</math> is given such that the squares of the side | + | Let <math>P = A_1A_2 \cdots A_k</math> be a convex polygon in the plane. The vertices <math>A_1,A_2,\dots, A_k</math> have integral coordinates and lie on a circle. Let <math>S</math> be the area of <math>P</math>. And odd positive integer <math>n</math> is given such that the squares of the side lengths of <math>P</math> are integers divisible by <math>n</math>. Prove that <math>2S</math> is an integer divisible by <math>n</math>. |

## Revision as of 23:25, 19 April 2017

## Problem

Let be a convex polygon in the plane. The vertices have integral coordinates and lie on a circle. Let be the area of . And odd positive integer is given such that the squares of the side lengths of are integers divisible by . Prove that is an integer divisible by .