1. Let ABC be a triangle with inradius r and circumradius R. Prove that: X sin3 A + sin3 B sin5 A + sin5 B ≤ R r 2 The sum is over all cyclic permutation of (A, B, C). Proposed by George Apostolopoulos Messolonghi Greece JP.122. Prove that in ∆ABC the following relationship holds: min a s − a , b s − b , c s − c ≤ 2 R r − 1 ≤ max a s − a , b s − b , c s − c Proposed by Marian Urs˘arescu Romania JP.123. Solve for real numbers: loga(b x + a − b) = logb (a x +1. Let ABC be a triangle with inradius r and circumradius R. Prove that: X sin3 A + sin3 B sin5 A + sin5 B ≤ R r 2 The sum is over all cyclic permutation of (A, B, C). Proposed by George Apostolopoulos Messolonghi Greece JP.122. Prove that in ∆ABC the following relationship holds: min a s − a , b s − b , c s − c ≤ 2 R r − 1 ≤ max a s − a , b s − b , c s − c Proposed by Marian Urs˘arescu Romania JP.123. Solve for real numbers: loga(b x + a − b) = logb (a x +1. Let ABC be a triangle with inradius r and circumradius R. Prove that: X sin3 A + sin3 B sin5 A + sin5 B ≤ R r 2 The sum is over all cyclic permutation of (A, B, C). Proposed by George Apostolopoulos Messolonghi Greece JP.122. Prove that in ∆ABC the following relationship holds: min a s − a , b s − b , c s − c ≤ 2 R r − 1 ≤ max a s − a , b s − b , c s − c Proposed by Marian Urs˘arescu Romania JP.123. Solve for real numbers: loga(b x + a − b) = logb (a x +