×

AM-GM Struggle (2)!

I have struggled on this proof. Please a hint, especially related to AM-GM (Cauchy-Schwarz, Jensen are also okay). For $$a,b,c,d \in \mathbb{R}^{+}$$, proof that $\frac{a^2}{b} +\frac{b^2}{c} +\frac{c^2}{d} +\frac{d^2}{a} \geq a+b+c+d$ Thank you very much!

Note by Figel Ilham
2 years, 8 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

Check out Applying AM-GM inequality wiki, and in particular Rearranging creatively.

Staff - 2 years, 8 months ago

Does the proof true? Consider $$\frac{a^2}{b}+\frac{a^2}{b} +\frac{b^2}{c}+c$$ Applying AM-GM, we have $\frac{a^2}{b} +\frac{a^2}{b} +\frac{b^2}{c}+c \geq 4\sqrt[4]{\frac{a^2}{b} \frac{a^2}{b} \frac{b^2}{c} c} = 4a$ $2\frac{a^2}{b} +\frac{b}{c} +c \geq 4a$ Apply also to $2\frac{b^2}{c} +\frac{c}{d} +d \geq 4b$ $2\frac{c^2}{d} +\frac{d^2}{a} +a \geq 4c$ $2\frac{d^2}{a} +\frac{a^2}{b} +b \geq 4d$

Adding those, we have $3(\frac{a^2}{b} + \frac{b^2}{c} +\frac{c^2}{d} +\frac{d^2}{a}) +(a+b+c+d) \geq 4(a+b+c+d)$ $3(\frac{a^2}{b} + \frac{b^2}{c} +\frac{c^2}{d} +\frac{d^2}{a}) \geq 3(a+b+c+d)$ $\frac{a^2}{b} + \frac{b^2}{c} +\frac{c^2}{d} +\frac{d^2}{a} ) \geq (a+b+c+d)$

- 2 years, 8 months ago

Perfect! Well done :)

Could you add this as an example to the wiki page?

Staff - 2 years, 8 months ago

I'll try

- 2 years, 8 months ago

I'll try first

- 2 years, 8 months ago

×