Math problems from competitions

Hello! I would like to ask some maths problems here.:-)

1) How to prove that any triangle can be cut into six similar triangles? (Your proof must be valid for all triangles and not just a specific type. )

2) Given that real number $a, b, c$ where $abc=-1, a+b+c=4$

$\frac {a}{(a^2-3a-1)}+\frac {a}{(b^2-3b-1)}+\frac {a}{(c^2-3c-1)}=\frac {4}{9}$

Thus, find $a^2+b^2+c^2=$

These problems may be easy for you, but I hope you can explain the working/solution as a reply below:-D. Thanks!

There's still some maths problems...maybe in the next note!

*See part 2 here

#Equations #SimilarFigures #Proofs #MathProblem #Pleasehelp

Easy Math Editor

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

Hi, Plz are you sure of ur second question,,, Seems there needs to be cyclic terms in numerator, Plz can u edit it

Dinesh Chavan - 7 years ago

But...it is from the competition question...but can you tell me what's wrong in that question and I'm not so sure about cyclic terms...

How about the first question? I tried it, but I still cannot find a way to proof it...

A Former Brilliant Member - 7 years ago

I Mean see the terms in the inequality,, a's on 1st term, b's on second term, c's on 3rd one. I mean to say, normally these inequalities are symetric type. I think u might have mae a mistake while copying . U might have not seen numerator clearly. Just in case, plz clarify it.. While, I will try now :P

Dinesh Chavan - 7 years ago

@Dinesh Chavan – Ok.The question is really printed like that...:-(

A Former Brilliant Member - 7 years ago

I've solved #2 but I'm having a hard time with the first one can anyone help me?

brittany gutknecht - 6 years, 7 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}`	$\boxed{123}$