HMMT 二月 2011 · TEAM1 赛 · 第 5 题

HMMT February 2011 — TEAM1 Round — Problem 5

[ 20 ] Let a and b be positive real numbers. Define two sequences of real numbers { a } and { b } for all n n n positive integers n by ( a + bi ) = a + b i . Prove that n n 2 2 | a | + | b | a + b n +1 n +1 ≥ | a | + | b | a + b n n for all positive integers n . Page 1 of 3 Coin Flipping [75] In a one-player game, the player begins with 4 m fair coins. On each of m turns, the player takes 4 unused coins, flips 3 of them randomly to heads or tails, and then selects whether the 4th one is heads or tails (these four coins are then considered used). After m turns, when the sides of all 4 m coins have been determined, if half the coins are heads and half are tails, the player wins; otherwise, the player loses.

解析

[ 20 ] Let a and b be positive real numbers. Define two sequences of real numbers { a } and { b } for all n n n positive integers n by ( a + bi ) = a + b i . Prove that n n 2 2 | a | + | b | a + b n +1 n +1 ≥ | a | + | b | a + b n n for all positive integers n . Solution: Let z = a + bi . It is easy to see that what we are asked to show is equivalent to n +1 n +1 n +1 n +1 | z + ¯ z | + | z − z ¯ | 2 z z ¯ ≥ n n n n | z + ¯ z | + | z − z ¯ | | z + ¯ z | + | z − z ¯ | Cross-multiplying, we see that it suffices to show n +2 n +1 n +1 n +2 n +2 n +1 n +1 n +2 | z + z z ¯ + z z ¯ + ¯ z | + | z − z z ¯ + z z ¯ − z ¯ | n +2 n +1 n +1 n +2 n +2 n +1 n +1 n +2

| z + z z ¯ − z z ¯ − z ¯ | + | z − z z ¯ − z z ¯ + ¯ z | n +1 n +1 n +1 n +1 ≥ 2 | z z ¯ + z z ¯ | + 2 | z z ¯ − z z ¯ | However, by the triangle inequality, n +2 n +1 n +1 n +2 n +2 n +1 n +1 n +2 n +1 n +1 | z + z z ¯ + z z ¯ + ¯ z | + | z − z z ¯ − z z ¯ + ¯ z | ≥ 2 | z z ¯ + z z ¯ | Team Round A and n +2 n +1 n +1 n +2 n +2 n +1 n +1 n +2 n +1 n +1 | z − z z ¯ + z z ¯ − z ¯ | + | z + z z ¯ − z z ¯ − z ¯ | ≥ 2 | z z ¯ − z z ¯ | This completes the proof. Remark: more computationally intensive trigonometric solutions are also possible by reducing the problem to maximizing and minimizing the values of the sine and cosine functions. Coin Flipping [75] In a one-player game, the player begins with 4 m fair coins. On each of m turns, the player takes 4 unused coins, flips 3 of them randomly to heads or tails, and then selects whether the 4th one is heads or tails (these four coins are then considered used). After m turns, when the sides of all 4 m coins have been determined, if half the coins are heads and half are tails, the player wins; otherwise, the player loses.