51 st International Mathematical Olympiad

4. Let P be a point inside triangle ABC. The lines AP, BP, and CP intersect the circumcircle of triangle ABC again at the points K , L , and M , respectively. The tangent to at C intersects the line AB at S. Suppose that SC = SP. Prove that MK = ML. 5. In each of six boxes B1, B2, B3, B4, B5, B6 there is initially one coin. There are two types of operation allowed: Type 1: Choose a nonempty box B j with 1 ≤ j ≤ 5. Remove one coin from B j and add two coins to B j+1. Type 2: Choose a nonempty box Bk with 1 ≤ k ≤ 4. Remove one coin from Bk and exchange the contents of (possibly empty) boxes Bk+1 and Bk+2.