G.I.F. Problem!

Algebra Level 5

Let $n\geq 2$ be a positive integer. In terms of $n$ , find the number of roots of $x^2 - \lfloor x^2 \rfloor = ( x- \lfloor x \rfloor )^2$ where $1\leq x \leq n$ .

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .

n

n^3-n^2-1

n^3-n^2+1

n^2+n+1

3 1

n^2-n+1

1 solution

Naitik Sanghavi
May 21, 2017

{x²}={x}²

Let , x=I+f . (Where I is integral and f is fractional part of x respectively)

{I²+2If+f²}={f²}

Therefore, if 2If is an integer then the equation satisfies,

Now let's count all x which are not integers (we will count them later.).

For 1<x<2 I=1 So, 2f should be an integer. So there is only one solution.

Similarly for 2<x<3 there are 3 solution.

For 3<x<4 the will be 5 solutions.

And so on..

So for 1<x<n. (Excluding all integers)

There will be (n-1)² solutions.

Also every integer value of x satisfies the equation so the are n solutions for 1≤x≤n.(only integers)

So total no. of solutions for 1≤x≤n are

(n-1)²+n=n²-n+1

G.I.F. Problem!

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