PUMaC 2020 · 代数（B 组） · 第 3 题

PUMaC 2020 — Algebra (Division B) — Problem 3

Let f ( x ) = satisfy f ( f ( f ( x ))) = x for real numbers a , b . If the maximum value of a is , x + b q where p, q are relatively prime integers, what is | p | + | q | ? x ( x +1)(2 x +1) 2 1

解析

Let f ( x ) = satisfy f ( f ( f ( x ))) = x for real numbers a , b . If the maximum value of a is , x + b q where p, q are relatively prime integers, what is | p | + | q | ? Proposed by: Henry Erdman Answer: 7 2 2 (1+2 a + ab ) x +( a + a + ab + ab ) Substituting in f ( x ) for x in f ( x ) twice yields that f ( f ( f ( x ))) = . 2 3 (1+ a + b + b ) x +( a +2 ab + b ) We note that the coefficient of x in the denominator must be zero, and thus we have that 1 2 a = − b − b − 1. This parabola opens down and has its vertex at b = − , giving an upper limit 2 3 3 1 on a of − . We now need to verify that ( a, b ) = ( − , − ) satisfies the rest of the problem. We 4 4 2 3 3 1 3 9 3 3 have 1+2 a + ab = 1 − + = − as the coefficient of x in the numerator, − + + − = 0 2 8 8 4 16 8 16 3 3 1 1 as the constant in the numerator, and − + − = − as the constant in the denominator. 4 4 8 8 Thus, we do indeed have a solution, and it is the greatest possible value of a . So, our answer is | − 3 | + | 4 | = 7 . x ( x +1)(2 x +1) 2 1