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National and Regional Contests
Iran Contests
Iran MO (1st Round)
2018 Iran MO (1st Round)
19
19
Part of
2018 Iran MO (1st Round)
Problems
(1)
Minimum value of x/z - Iran First Round 2018, P19
Source:
3/7/2021
Let
x
≥
y
≥
z
x \geq y \geq z
x
≥
y
≥
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
x
z
\frac{x}{z}
z
x
?
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1
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2
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3
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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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2
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3
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E
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4
inequalities
algebra