IMO Problem 2

Algebra Level 4

For what real values of x x is

x + 2 x − 1 + x − 2 x − 1 = A \sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = A ,

given A = 2 A=2 , where only non-negative real numbers are admitted for square roots?

This problem is from the IMO.This problem is part of this set .


The answer is 1.5.

View wiki
A Former Brilliant Member by A Former Brilliant Member

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

Saad Mukarram
Mar 21, 2015

x + 2 x − 1 + x − 2 x − 1 = 2 u = 2 x − 1 S q u a r i n g b o t h s i d e s : x + u + x − u + 2 x 2 − u 2 = 4 2 x + 2 x 2 − u 2 = 4 F a c t o r i n g 2 f r o m L S a n d d i v i d i n g R S b y i t : x + x 2 − u 2 = 2 x + x 2 − u = 2 S u b b i n g i n e q u a t i o n f o r u : x + ( x − 1 ) 2 = 2 x + x − 1 = 2 2 x = 3 x = 1.5 \sqrt { x+\sqrt { 2x-1 } } +\sqrt { x-\sqrt { 2x-1 } } =2\\ u=2x-1\\ Squaring\quad both\quad sides\quad :\\ x+\sqrt { u } +x-\sqrt { u } +2\sqrt { x^{ 2 }-\sqrt { u^{ 2 } } } =4\\ 2x+2\sqrt { x^{ 2 }-\sqrt { u^{ 2 } } } =4\\ Factoring\quad 2\quad from\quad LS\quad and\quad dividing\quad RS\quad by\quad it\quad :\\ x+\sqrt { x^{ 2 }-\sqrt { u^{ 2 } } } =2\\ x+\sqrt { x^{ 2 }-u } =2\\ Subbing\quad in\quad equation\quad for\quad u:\\ x+\sqrt { (x-1)^{ 2 } } =2\\ x+x-1=2\\ 2x=3\\ x=1.5

12 Helpful 0 Interesting 0 Brilliant 0 Confused
Jake Lai
Mar 20, 2015

Sometimes, don't be afraid to square ;)

8 Helpful 0 Interesting 0 Brilliant 0 Confused

x + 2 x − 1 + x − 2 x − 1 = 2. S q u a r i n g : − x + 2 x − 1 + x − 2 x − 1 + 2 ∗ x + 2 x − 1 ∗ x − 2 x − 1 = 4. 2 x + 2 ∗ x 2 − ( 2 x − 1 ) 2 = 4. 2 x + 2 ∗ x 2 − 2 x + 1 = 4 2 x + 2 ( x − 1 ) = 4. ∴ x = 1.5. \sqrt { x+\sqrt { 2x-1 } } +\sqrt { x-\sqrt { 2x-1 } } =2 .\\ Squaring:-\\ x+\sqrt { 2x-1 } +x-\sqrt { 2x-1 }+2*\sqrt { x+\sqrt { 2x-1 }}*\sqrt { x-\sqrt { 2x-1 }}=4.\\ 2x+2*\sqrt{x^2-( \sqrt { 2x-1 })^2}=4.\\ 2x+2*\sqrt{x^2-2x+1}=4 \\ 2x+2(x-1)=4. \\ \therefore~x=\Large~~~\color{#D61F06}{1.5}.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...