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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

Note by Peng Ying Tan
3 years, 7 months ago

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

- 3 years, 2 months ago

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

- 3 years, 7 months 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...

- 3 years, 7 months 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

- 3 years, 7 months ago

Ok.The question is really printed like that...:-(

- 3 years, 7 months ago