What is the minimum value of the expression $$xy+yz+zx+\frac 1x+\frac 2y+\frac 5z$$where $x, y, z$ are positive real numbers?
Problem
Source: Turkey TST 2022 P7 Day 3
Tags: inequalities, algebra
13.03.2022 14:49
13.03.2022 16:22
20.03.2022 23:08
I have a bit longer but different proof.
Attachments:
Turkey TST P7.docx (18kb)
21.03.2022 03:30
Let $x, y, z$ are positive real numbers. Prove that$$ xy+yz+ zx+\frac{1}{x}+\frac{3}{y}+\frac{3}{z} \geq 6\sqrt[3]{4}$$$$ xy+2yz+ zx+\frac{1}{x}+\frac{3}{y}+\frac{3}{z} \geq \frac{15\sqrt[3]{3}}{2}$$$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+ab+bc+ca \geq \frac{9}{\sqrt[3]{4}}$$h h CahitArf wrote: I have a bit longer but different proof.
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30.05.2022 09:22
Let $x, y, z$ are positive real numbers. Prove that $$ xy+2yz+2zx+\frac{2}{x}+\frac{2}{y}+\frac{1}{z} \geq 9$$$$ xy+zx+\frac{5}{x}+\frac{2}{y}+\frac{2}{z} \geq 6\sqrt[3]{5}$$$$ x+3y+3z+\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx} \geq 6\sqrt[3]{4}$$ BarisKoyuncu wrote: What is the minimum value of the expression $$xy+yz+zx+\frac 1x+\frac 2y+\frac 5z$$where $x, y, z$ are positive real numbers? $$xy+yz+zx+\frac 1x+\frac 2y+\frac 5z\geq 3\sqrt[3]{36}$$$$ xy+yz+ zx+\frac{1}{x}+\frac{3}{y}+\frac{3}{z} \geq 6\sqrt[3]{4}$$
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31.10.2023 22:54
This problem has a good general one however I will give the minimum.
31.10.2023 22:55
Generalization 1 Let $x,y,z,\lambda,a_{1},b_{1}$ be positive real numbers. Then prove that $$\lambda \left(xy+yz+xz\right) + \dfrac{a_{1}\left(b_{1}+1\right)}{b_{1}x}+\dfrac{b_{1}\left(a_{1}+1\right)}{a_{1}y}+\dfrac{a_{1}+b_{1}}{a_{1}b_{1}z} \geq 3\sqrt[3]{\lambda a_{1}b_{1}}\left(\dfrac{a_{1}+b_{1}+a_{1}b_{1}}{a_{1}b_{1}}\right)$$
05.11.2023 21:22
The original problem and Generalization 1 are special cases of the following generalization where $$n=3; \lambda =1, a_{1}=\dfrac{1}{3}, a_{2}=b_{1}=\dfrac{1}{2}$$
05.11.2023 21:23
Generalization 2 Let $x_{1},x_{2},\cdots,x_{n},\lambda ,a_{1},a_{2},\cdots,a_{n-1}$ be positive reals ($n\geq 3$). Then prove that $$\sum_{j=1}^{n-1}{\left(\dfrac{a_{j}\left(1+\sum_{p=1}^{n-1}{\left(\dfrac{1}{a_{p}}\right)-\dfrac{1}{a_{j}}}\right)}{x_{j}}\right)}+\dfrac{\sum_{k=1}^{n-1}{\left(\dfrac{1}{a_{k}}\right)}}{x_{n}}\geq n\sqrt[n]{\lambda \prod{a_{1}}}\left(1+\sum_{p=1}^{n-1}{\dfrac{1}{a_{p}}}\right)$$