 # Understand the problem

If $a,b$ are positive reals such that $a+b<2$ ,then prove that  $$\displaystyle \frac {1}{1+a^2} + \frac {1}{1+b^2} \leq \frac {2}{1+ab}$$
##### Source of the problem
Mathematical Circles
##### Topic
Inequality involving AM-GM
Medium
##### Suggested Book
Mathematical Circles

Do you really need a hint? Try it first!

Assume that the given inequality is true for $a>0 , b>0$ and $a+b<2$ . Then proceed .
In order to simplyfy the given inequality multiply both the sides by $(1+a^2)(1+b^2)(1+ab)$ (as its a positive quantity and it is directly coming from   $a>0 , b>0$  ) .
Come up with the simplest form of inequality  i.e. $(a-b)^2 (1-ab) \geq 0$ .
$(a-b)^2 \geq 0$ as $a,b \in{R}$ . And to get $(1-ab)>0$ use the well known inequality for positive reals i.e. $AM \geq GM$ and the still unused inequality i.e $a+b <2$ also . $$\displaystyle a>0 , b>0 \Rightarrow \sqrt{ab}>0 \Rightarrow( 1+ \sqrt{ab})>0 \\ a>0 , b>0 , a+b <2 \Rightarrow 1 > \frac{a+b}{2} \geq \sqrt {ab} \\ \Rightarrow 1 > \sqrt{ab} \\ \Rightarrow ( 1 – \sqrt{ab}) >0 \\ \Rightarrow (1 – \sqrt{ab}) (1+ \sqrt{ab}) >0 \\ \Rightarrow (1 – ab)>0$$

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