Impressing Inequality

Algebra Level 4

$\large x^2 + 2y^2 + \dfrac12 \leq x(2y + 1 )$

Find all real numbers $x$ and $y$ satisfying the inequality above. Submit your answer as the sum of all distinct possible value(s) of $2xy$ .

1 solution

Lakshya Singh
May 29, 2016

x^2+2y^2+1/2 is less than equal to 2xy+x

Multiplying both sides by 2

2x^2+4y^2+1 is less than equal to 4xy +2x

2x^2+4y^2-4xy-2x+1 is less than equal to 0.

(x-2y)^2+(x-1)^2 is less than equal to 0.

Case 1

(x-2y)^2+(x-1)^2<0 Not possible

Case 2

(x-2y)^2+(x-1)^2=0

x=1,y=1/2

Therefore 2xy=1

Did the same . I think the problem is a bit overrated.

Aditya Kumar - 5 years ago