Questions on proving and manipulating inequalities.

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Prove this inequality with $xyz\le 1$

if $x,y,z>0$ and $\color{red}{xyz\le 1}$, show that $$\color{blue}{\dfrac{x^2-x+1}{x^2+y^2+1}+\dfrac{y^2-y+1}{y^2+z^2+1} +\dfrac{z^2-z+1}{z^2+x^2+1}\ge 1}$$ Sorry,This is not 2015 TST ...
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How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$

Zhautykov Olympiad 2015 problem 6 This links discuss olympiad problem none of student solve it,therefore, meaning this problem is so hard. Question: The area of a convex pentagon $ABCDE$ is $S$, ...
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Cauchy-Schwarz Hermitian Inner Product Remainder

A couple weeks ago, someone showed me a proof of Cauchy-Schwarz where he ended up deriving something of the form $$\langle a,b\rangle^2=\langle a,a\rangle^2\langle b,b\rangle^2 +f(a,b)$$ Where ...