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Fie \( x,y,z>0 \) astfel incat \( x+y+z=\frac{\pi}{2} \). Sa se arate ca
\( (1-\sin x)(1-\sin y)(1-\sin z)\geq \sin x\sin y\sin z \).

Shortlist ONM 2004

Deoarece sin e concava pe \( [0,\pi] \)

\( LHS=\prod 2\sin^2(\frac{y+z}{2})\ge8\prod \(\frac{\sin y+\sin z}{2}\)^2\ge8\prod \sin y\sin z \ge RHS \)

intrucat e cunoscut ca intr-un triunghi ABC
\( \prod \sin \frac{A}{2}\le \frac{1}{8} \)