Help Me!

Find $ab + bc + ca$ given that:

$a^{2} + ab + b^{2} = 2$

$b^{2} + bc + c^{2} = 1$

$c^{2} + ca + a^{2} = 3$

Giveb that a, b, and c are positive numbers.

#Algebra

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

Is the question correct? There don't exist positive $a,b,c$ for which these equations are true.

Siddhartha Srivastava - 5 years, 8 months ago

Yes sir, there is. :)

Christian Daang - 5 years, 8 months ago

Christian Daang has posted the same problem recently.

Here

Ankit Kumar Jain - 4 years, 3 months ago

Yah sir, after a year and a half, I figure it out how this can be solve. :)

Christian Daang - 4 years, 3 months ago

Its absolutely brilliant that you have devised your own method...:) :) :)

Please don't call me sir , I am your fellow BRILLIANTIAN and a FRIEND.

Ankit Kumar Jain - 4 years, 3 months ago

@Ankit Kumar Jain – I saw someone who figure it out how this can be solve. :) I just somewhat get it. XD And somewhat make it more elaborative.

~ Do you think there are other ways to solve this except for the geometric method?

Christian Daang - 4 years, 3 months ago

@Christian Daang – At present , I can't think of any other way.

Ankit Kumar Jain - 4 years, 3 months ago

By the way , you post some really interesting questions ...I love them.

Even you can try my sets out

My Problems and THRILLER

Ankit Kumar Jain - 4 years, 3 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}`	$\boxed{123}$