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f(A)=f(B), sum of lengths of polygons - All-Russian MO 2001 Regional (R4) 10.1

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September 26, 2024
algebrageometrycombinatorial geometry

Problem Statement

The lengths of the sides of the polygon are a1a_1, a2a_2,. .... ,ana_n. The square trinomial f(x)f(x) is such that f(a1)=f(a2+...+an)f(a_1) = f(a_2 +...+ a_n). Prove that if AA is the sum of the lengths of several sides of a polygon, BB is the sum of the lengths of its remaining sides, then f(A)=f(B)f(A) = f(B).
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