MathDB

Purple Comet 2009 HS Problem 25

Source:

September 3, 2011

algebrapolynomialfunctionnumber theoryrelatively primerational function

Problem Statement

The polynomial

P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9

has the property that

P(\tfrac{1}{k})=\tfrac{1}{k}

for

k=1,2,3,4,5,6,7,8,9

. There are relatively prime positive integers

m

and

n

such that

P(\tfrac{1}{10})=\tfrac{m}{n}

. Find

n-10m