Intereting Posts

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Proving $\sqrt 3$ is irrational.
Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?

I’m interested why this is true:

$$ \text{Considering }\forall (x,y,z) \in (1,\infty) $$

The following holds:

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- {Thinking}: Why equivalent percentage increase of A and decrease of B is not the same end result?
- Prove $e^{i \pi} = -1$

$$\log_xy^z+\log_x{z^y}+log_y{z^x} \geq \frac{3}{2}$$

This is taken from a high school textbook of mine. I tried finding a meaningful manipulation by using AM-GM, but that got pretty messy. I’d like to avoid Lagrange multipliers since this is meant to be a pretty basic problem.

Any progress would be appreciated.

- Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
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- A gamma function inequality
- Proving $ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n}$
- Solving an inequality involving factorials and an exponent
- In most geometry courses, we learn that there's no such thing as “SSA Congruence”.
- Is it always possible to factorize $(a+b)^p - a^p - b^p$ this way?
- Why is the Riemann sum less than the value of the integral?
- A WolframAlpha error?

CW answer to remove it from unanswered queue:

For $(x,y,z)=(1.1, 1.01, 1.001)$, the expression is clearly $\simeq 0.22<\frac{3}{2}$, contradicting the inequality at hand.

**Hint**

Set

$$f(x,y,z)=z\frac{\ln y}{\ln x}+y\frac{\ln z}{\ln x}+x\frac{\ln z}{\ln y} $$

look at the stationary point and conclude.

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