Why always x x and y y ?

If x , y Z + x,y \in {Z}^{+} and

y 3 = x 3 + 8 x 2 6 x + 8 {y}^{3} = {x}^{3} + 8{x}^{2} - 6x + 8

Find the value of x + y x + y .


The answer is 20.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

Shubhendra Singh
Nov 24, 2014

Here is what I did

The equation can be written as

y 3 2 3 = x 3 + 8 x 2 6 x y^{3}-2^{3}=x^{3}+8x^{2}-6x

( y 2 ) ( y 2 + 4 + 2 y ) = x ( x 2 + 8 x 6 ) (y-2)(y^{2}+4+2y)=x(x^{2}+8x-6 )

Now Let y 2 = x . . . . . ( 1 ) y-2=x \ \ \ \ .....(1) y 2 + 4 + 2 y = x 2 + 8 x 6 . . . . . ( 2 ) y^{2}+4+2y =x^{2}+8x-6 \ \ \ \ .....(2)

Substitute y = x + 2 y=x+2 in eq.(2) to get

x 2 + 6 x + 12 = x 2 + 8 x 6 x^{2}+6x+12=x^{2}+8x-6 2 x = 18 2x=18

By this we get x = 9 x=9 and y = 9 + 2 = 11 y=9+2=11

So x + y = 20 x+y=20

Nice Problem !

Shubhendra Singh - 6 years, 6 months ago

Log in to reply

Thanks! BTW, I too did the same! It is an RMO problem

Kartik Sharma - 6 years, 6 months ago

Log in to reply

How can we say that no more solutions are possible? Thanks.

Satvik Golechha - 6 years, 6 months ago

Log in to reply

@Satvik Golechha I included an argument that explained this in my solution.

Patrick Corn - 6 years, 5 months ago

I did the same

Paola Ramírez - 6 years, 4 months ago
Patrick Corn
Jan 5, 2015

Note that y 3 ( x + 1 ) 3 = 5 x 2 9 x + 7 y^3 - (x+1)^3 = 5x^2-9x+7 is always positive (check the discriminant). And y 3 ( x + 3 ) 3 = x 2 33 x 19 y^3 - (x+3)^3 = -x^2-33x-19 is always negative. So the only possibility is that y = x + 2 y = x+2 . Plugging in and solving gives 2 x 2 18 x = 0 2x^2-18x = 0 ; the only positive solution is x = 9 x=9 , so y = 11 y = 11 and the answer is 20 \fbox{20} .

Sourabh Jangid
Dec 1, 2016

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...