MMO2025 Round 4

Question

Let , , be the angles opposite the sides a, b, c of a triangle, respectively. Prove that if the lengths of a, b, c form an arithmetic progression in this order, then the values , 1 - , also form an arithmetic progression in that order. (Otgonbayar Uuye)

Accepted Answer

By the Law of Sines, the values , , form an arithmetic progression. Therefore 2 2 - 2 = 2 + 2 - 2 = + = 2 = 4 2 2. Since 2 > 0, it follows - 2 = 2 2. Then + = 2 + 2 - 2 = 4 ^2 2 = 2(1 - ). Thus, , 1 - , are in arithmetic progression. Alternatively, by the Law of Cosines, we have aligned + - 2(1 - ) &= b^2 + c^2 - a^22bc + a^2 + b^2 - c^22ab - 2(1 - a^2 + c^2 - b^22ac) &= (a+b-c)(b+c-a)2abc(a+c-2b) = 0. aligned