<s>[INST] Given $x^2 + y^2 + z^2 = 3$ prove that $x/\sqrt{x^2+y+z} + y/\sqrt{y^2+x+z} + z/\sqrt{z^2+x+z} \le \sqrt3$ Given $x^2 + y^2 + z^2 = 3$
Then prove that $${x\over\sqrt{x^2+y+z}} + {y\over\sqrt{y^2+x+z}} + {z\over\sqrt{z^2+x+y}} \le \sqrt 3$$
I tried using the Cauchy-Schwarz inequality but the inequality is coming in opposite direction.
 [/INST] Use $(a+b+c)^2 \leq 3(a^2+b^2+c^2)$.  Then multiply out the denominators and solve.
Alternatively, maximize the left hand side subject to the constraint.
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