8(abcd+1)>(a+1)(b+1)(c+1)(d+1) prove it

if a,b,c,d,all are greater than 1

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

Math	Appears 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}`	$\boxed{123}$

Comments

What have you tried? What do you know?

As a hint : If $x, y >1$ , then $(x-1)(y-1) > 0$ so $xy + 1 > x + y$ . Use this many times.

Calvin Lin Staff - 7 years, 12 months ago

this question was from the beginning section of the book in inequalities so it does not require any prerequisities so i tried so solve this without am gm inequalitiy or any other but cannot solve it

Aditya Kumar - 7 years, 11 months ago

As a hint : If $x,y>1$ , then $(x-1)(y-1)>0$ so $xy+1>x+y$ .

This looks very similar to the inequality, especially on the LHS. Use this many times.

Calvin Lin Staff - 7 years, 11 months ago

Just expand it , and try to put in square form , its very simple

Shivang Jindal - 7 years, 11 months ago

I think, its from Challenge and thrill . right?

Shivang Jindal - 7 years, 11 months ago

I agree with just expand it. I'm not so sure what you mean by square form, though there is a way to write it as the sum of several positive (non-negative) terms.

Calvin Lin Staff - 7 years, 11 months ago

thanks i have now solved it

Aditya Kumar - 7 years, 11 months ago

Can someone explain it once more??...

Tanmay Dokania - 3 years, 8 months ago

Try to substitute $x=a-1, y=b-1, z=c-1, w=d-1$ .

Shourya Pandey - 7 years, 11 months ago

Don't you mean $x=a+1,y=b+1$ , etcetera?

Tim Vermeulen - 7 years, 11 months ago

Shourya Pandey - 7 years, 11 months ago

Try doing it, it becomes very easy

Shourya Pandey - 7 years, 11 months ago

Math	Appears 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}`	$\boxed{123}$