HMMT 十一月 2010 · 冲刺赛 · 第 26 题

HMMT November 2010 — Guts Round — Problem 26

[ 14 ] w, x, y, z are real numbers such that w + x + y + z = 5 2 w + 4 x + 8 y + 16 z = 7 3 w + 9 x + 27 y + 81 z = 11 4 w + 16 x + 64 y + 256 z = 1 What is the value of 5 w + 25 x + 125 y + 625 z ? 2 2010

解析

[ 14 ] w, x, y, z are real numbers such that w + x + y + z = 5 2 w + 4 x + 8 y + 16 z = 7 3 w + 9 x + 27 y + 81 z = 11 4 w + 16 x + 64 y + 256 z = 1 What is the value of 5 w + 25 x + 125 y + 625 z ? Answer: − 60 We note this system of equations is equivalent to evaluating the polynomial (in a ) 2 3 4 P ( a ) = wa + xa + ya + za at 1, 2, 3, and 4. We know that P (0) = 0, P (1) = 5, P (2) = 7, P (3) = 11, P (4) = 1. The finite difference of a polynomial f is f ( n + 1) − f ( n ), which is a polynomial with degree one less than the degree of f . The second, third, etc finite differences come from applying this operation repeatedly. The fourth finite difference of this polynomial is constant because this is a fourth degree polynomial. Repeatedly applying finite differences, we get 0 5 7 11 1 5 2 4 − 10 − 3 2 − 14 5 − 16 − 21 and we see that the fourth finite difference is − 21. We can extend this table, knowing that the fourth finite difference is always -21, and we find that that P (5) = − 60. The complete table is Guts Round 0 5 7 11 1 − 60 5 2 4 − 10 − 61 − 3 2 − 14 − 51 5 − 16 − 37 − 21 − 21 2 2010