HMMT 十一月 2010 · GEN1 赛 · 第 5 题

英文解析

5. [ 5 ] A polynomial P is of the form ± x ± x ± x ± x ± x ± x ± 1. Given that P (2) = 27, what is
   P (3)?
   Answer: 439 We use the following lemma:
   n n − 1 n
   Lemma. The sign of ± 2 ± 2 ± · · · ± 2 ± 1 is the same as the sign of the 2 term.
   n n n − 1
   Proof. Without loss of generality, let 2 be positive. (We can flip all signs.) Notice that 2 ± 2 ±
   n − 2 n n − 1 n − 2
   2 ± · · · 2 ± 1 ≥ 2 − 2 − 2 − · · · − 2 − 1 = 1, which is positive.
   We can use this lemma to uniquely determine the signs of P . Since our desired sum, 27, is positive, the
   6 5 4
   coefficient of x must be positive. Subtracting 64, we now have that ± 2 ± 2 ± . . . ± 2 ± 1 = − 37, so the
   5 6 5 4 3 2
   sign of 2 must be negative. Continuing in this manner, we find that P ( x ) = x − x − x + x + x − x +1,
   6 5 4 3 2
   so P (3) = 3 − 3 − 3 + 3 + 3 − 3 + 1 = 439. 2