PUMaC 2020 · 代数（B 组） · 第 6 题

PUMaC 2020 — Algebra (Division B) — Problem 6

Let P be a 10-degree monic polynomial with roots r , r , . . . , r 6 = 0 and let Q be a 45- 1 2 10 1 1 1 degree monic polynomial with roots + − where i < j and i, j ∈ { 1 , . . . , 10 } . If r r r r i j i j a P (0) = Q (1) = 2, then log ( | P (1) | ) can be written as for relatively prime integers a, b . Find 2 b a + b .

解析

Let P be a 10-degree monic polynomial with roots r , r , . . . , r 6 = 0 and let Q be a 45- 1 2 10 1 1 1 degree monic polynomial with roots + − where i < j and i, j ∈ { 1 , . . . , 10 } . If r r r r i j i j a P (0) = Q (1) = 2, then log ( | P (1) | ) can be written as for relatively prime integers a, b . Find 2 b a + b . Proposed by: Matthew Kendall Answer: 19 We can factor Q as a product of its roots: ( ) ∏ 1 1 1 Q ( x ) = x − − + . r r r r i j i j i<j Then we see ( ) ∏ ∏ 1 1 1 1 1 9 Q (1) = 1 − − + = (1 − r )(1 − r ) = P (1) . i j 9 r r r r r r ( r r · · · r ) i j i j i j 1 2 10 i<j i<j 10 1 9 9 Hence | P (1) | = 2, so | P (1) | = 2 , giving an answer of 19 . 9 2 2