Which is greater?

Number Theory Level 1

Let x x and y y be distinct positive integers.

Which is greater?

A . x 2 − y 2 x − y B . x 2 + y 2 x + y \LARGE {A.\frac{x^2-y^2}{x-y}\\ B.\frac{x^2+y^2}{x+y}}

Two are equal B A

Md Mehedi Hasan by Md Mehedi Hasan , 22, Bangladesh

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2 solutions

Md Mehedi Hasan
Nov 5, 2017

x 2 − y 2 x − y = ( x − y ) ( x + y ) x − y = x + y \frac{x^2-y^2}{x-y}=\frac{(x-y)(x+y)}{x-y}=x+y

Again, x 2 + y 2 x + y = ( x + y ) 2 − 2 x y x + y = ( x + y ) − 2 x y x + y \frac{x^2+y^2}{x+y}=\frac{(x+y)^2-2xy}{x+y}=(x+y)-\frac{2xy}{x+y}

Here, 2 x y x + y > 0 x , y ϵ N \frac{2xy}{x+y}>0\quad \boxed{x,y\epsilon N}

Then, x + y > ( x + y ) − 2 x y x + y x+y>(x+y)-\frac{2xy}{x+y}

So, the right ans is A \boxed{A}

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Saksham Jain
Nov 10, 2017

Simplify A A is x+y Add and subtract 2xy in numerator then simplify B is x+y-2xy/(x+y) Therefore A>B

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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