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Minimum value of x/z - Iran First Round 2018, P19

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March 7, 2021
inequalitiesalgebra

Problem Statement

Let x≥y≥zx \geq y \geq z be positive real numbers such that \begin{align*}x^2+y^2+z^2 \geq 2xy+2yz+2zx.\end{align*} What is the minimum value of xz\frac{x}{z}?
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> 4<span class='latex-bold'>(A)</span>\ 1\qquad<span class='latex-bold'>(B)</span>\ \sqrt 2\qquad<span class='latex-bold'>(C)</span>\ \sqrt 3\qquad<span class='latex-bold'>(D)</span>\ 2\qquad<span class='latex-bold'>(E)</span>\ 4
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