PUMaC 2015 · 代数（B 组） · 第 1 题

PUMaC 2015 — Algebra (Division B) — Problem 1

[ 3 ] Roy is starting a baking company and decides that he will sell cupcakes. He sells n cupcakes for ( n + 20)( n + 15) cents. A man walks in and buys $10 . 50 worth of cupcakes. Roy bakes cupcakes at a rate of 10 cupcakes an hour. How many minutes will it take Roy to complete the order?

解析

- 7( x + x + 1). We check that ( x − 5 x − 4 x − 5 x + 1) = ( x + x + 1)( x − 6 x + 1). Thus 2 2 2 2 4 3 2 P ( x ) = ( x + x + 1)( x ( x − 6 x + 1) + 7) = ( x + x + 1)( x − 6 x + x + 7). There are still no convenient roots of the quartic, thus we again use standard guess and check with quadratics 2 2 4 2 2 of constant terms 7 and 1 to find ( x + x + 1)( x − 7 x + 7) = x − 6 x + x + 7 = ⇒ P ( x ) = 2 2 2 2 ( x + x + 1) ( x − 7 x + 7). The non repeated roots are those of ( x − 7 x + 7), and thus by Vieta’s, the sum of the non repeated roots is 7 . Author: Christopher Zhang 2 na n − 1