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Question

Cho x, y, z là độ dài 3 cạnh tam giác có chu vi bằng 2p thỏa mãn: 15yz + 10zx + 1964xy= 2023xyz

Tìm GTNN: $K=\frac{1974}{p-x}+\frac{1979}{p-y}+\frac{25}{p-z}$

tran85295 · Answer 1 · 6/28/2016 2:57:32 PM

$K=1964\bigg(\frac{1}{p-x}+\frac{1}{p-y}\bigg)+10\bigg(\frac{1}{p-x}+\frac{1}{p-z}\bigg)+15\bigg(\frac{1}{p-y}+\frac{1}{p-z}\bigg)\\ \ge\frac{1964.4}{2p-(x+y)}+\frac{10.4}{2p-(x+z)}+\frac{15.4}{2p-(y+z)} \\= 4\left( \frac{1964}{z}+\frac{10}{y}+\frac{15}{x}\right)=4.2023=8092 \\ \mathbf{KL}\min K=8092\Leftrightarrow x=y=z=\frac{117}{119}$

Trả lời 28-06-16 02:57 PM