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A teacher must divide 221 apples evenly among 403 students. What is the minimal number of pieces into which she must cut the apples? (A whole uncut apple counts as one piece.) Keun: The teacher does not have to cut all apples into equal number of pieces.

1. In triangle ABC with pedal triangle DEF (such that D, E, F are the feet of the altitudes from A, B, C to the opposite sides respectively), let P be the intersection of AD and EF. Prove that P is the orthocenter of triangle AQR, and that A, Q, D, R are concylic. 2. For a triangle ABC, let D...