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2 regular polygons, 3 sides more, 10^o more for an angle

Source: 2022 Paraguayan Mathematical Olympiad L3 p2 - OMAPA

October 24, 2022
geometryregular polygon

Problem Statement

Santiago, Daniel and Fátima practice for the Math Olympics. Santiago thinks of a regular polygon and Daniel of another, without telling Fatima what the polygons are. They just tell you that one of the polygons has 33 more sides than the other and that an angle of one of the polygons measures 1010 degrees more than one angle of the other. From this, and knowing that each interior angle of a regular polygon of nn sides measures 180(n2)n\frac{180(n-2)}{n} degrees, Fatima identifies what the polygons are. How many sides do the polygons that James and Daniel chose, have?