Question

let f(x)=|2{x}-1| where {x} denotes the fractional part of x. The number n is the smallest positive interger such that the equation nf(xf(x))=x has at least 2012 real solutions. What is n? NOTE:the fractional part of x is a real number y={x} such that 0<=y<1 and x-y is an integer.
(a)30 (b)31 (c)32 (d)62 (e)4

Answer 1

don't know...wish I could help...(;

Answer 2

thanks anyway~

Answer 3

Problem source?

Answer 4

AMC12 2012

Answer 5

no solution yet. I'm preparing for amc12 test on 2.23.(cuz i'm Chinese, we have that test on 2.23)

Answer 6

So f(x) is easy to visualize: it's a symmetric sawtooth function with period 1/2, oscillating between 0 and 1. xf(x) scales that up and hence f(xf(x)) is the saw tooth pattern speeded up, except for a small part around the value x=0 which is quadratic, and needn't bother us.

Then of course g(x) = x/n is a straight line which reaches 1 when x = n.

Hence to find the number of solutions for which nf(xf(x)) = x, we need to find the number of times the graph of the straight line g(x) = x/n intersects f(xf(x)), for positive x, which is exactly the same as asking: how many times does the function f(xf(x)) oscillate in a given interval [0,n] and multiply that number by two (because the function goes up and down).

Hence the first step is make more precise the period of each oscillation in f(xf(x)) and then count them. This isn't too hard.

The first step, as always, is to try and sketch a graph of f(x), xf(x) and f(xf(x)) for a small interval around x = 0 to see how this function is behaving.

Answer 7

2012 intersects ？It takes times...

Answer 8

correction: can't have two first steps! The zeroth step is the draw a graph. You need to begin to get a feel for what this function looks like.

Answer 9

wait i guess i can solve that using computer programing

Answer 10

still can't help___the graph is not that accuate

Answer 11

The purpose of the graph isn't to use it explicitly to solve the problem. The purpose of the graph is to strengthen your intuition so you can count precisely.

Answer 12

got it I'm trying