<s>[INST] Why is $ \frac{a^2}{a+b}+\frac{d^2}{a+d}+\frac{b^2}{b+c}+\frac{c^2}{c+d} \geq 0.5 $ with $a+b+c+d = 1$? For positive real numbers $a,b,c,d>0$ it seems to be true that:
if $$a+b+c+d = 1$$
then 
$$  \frac{a^2}{a+b}+\frac{d^2}{a+d}+\frac{b^2}{b+c}+\frac{c^2}{c+d} \geq 0.5  $$
I can't think of a way to prove this statement.
Any help will be appreciated!
 [/INST] Use Cauchy-Schwarz:
$$  \frac{a^2}{a+b}+\frac{d^2}{a+d}+\frac{b^2}{b+c}+\frac{c^2}{c+d}=\left(\frac{a^2}{a+b}+\frac{d^2}{a+d}+\frac{b^2}{b+c}+\frac{c^2}{c+d}\right)\cdot\left((a+b)+(a+d)+(b+c)+(c+d)\right)\cdot 0.5\geq (a+b+c+d)^2\cdot 0.5= 0.5  $$
</s>