PUMaC 2019 · 代数（B 组） · 第 5 题

PUMaC 2019 — Algebra (Division B) — Problem 5

Let Q be a quadratic polynomial. If the sum of the roots of Q ( x ) (where Q ( x ) is defined 1 i i − 1 by Q ( x ) = Q ( x ) , Q ( x ) = Q ( Q ( x )) for integers i ≥ 2) is 8 and the sum of the roots of Q is S , compute | log ( S ) | . 2

解析

Let Q be a quadratic polynomial. If the sum of the roots of Q ( x ) (where Q ( x ) is defined 1 i i − 1 by Q ( x ) = Q ( x ) , Q ( x ) = Q ( Q ( x )) for integers i ≥ 2) is 8 and the sum of the roots of Q is S , compute | log ( S ) | . 2 1 Answer: 96 Proposed by: Matthew Kendall j Let the sum of the roots of Q ( x ) be S for j = 1 , . . . , 2019. Our claim is S = 2 S . Let j j +1 j Q ( x ) = a ( x − r )( x − s ), where r and s are the roots of Q . Note that j +1 j j Q ( x ) = a ( Q ( x ) − r )( Q ( x ) − s ) , j +1 j j so the solutions to Q ( x ) = 0 are the solutions to Q ( x ) = r and Q ( x ) = s . Since the j j j degree of Q is at least 1, the sum of the roots to Q ( x ) = r and Q ( x ) = s are both S , so j S = S + S = 2 S . j +1 j j j 99 8 From our recursion we get S = 2 S . Therefore, S = and | log ( S ) | = 96 . 100 1 1 99 2 2