# Discussion of SMO Senior Round 2 Qns 2014

The link to the Round 2 Paper for this year's Singapore Mathematical Olympiad is here. The Round 2 competition takes place just yesterday (28 June) so it's pretty new.

Here's the place to discuss the answers to these qns. :) So for those interested to try them or discuss them, feel free to do so! Note by Happy Melodies
5 years, 6 months ago

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Still weak at solving symmetric systems... Number 2....

- 5 years, 6 months ago

If anyone requires, the SMO Junior Round 2 2014 Qns can be found here

- 5 years, 6 months ago

Answer to second is $(0,0,00$ and $(4,4,4)$

- 5 years, 6 months ago

Question asked for only positive reals.

- 5 years, 6 months ago

Sorry, Then only $(4,4,4)$

- 5 years, 6 months ago

Whats the proof?

- 5 years, 6 months ago

Something like WLOG $a \geq b \geq c > 0$. Then $a+b \geq a+c$, yet $a\sqrt{b} \geq b\sqrt{c}$, which implies $a+c \geq a+b$ instead, so $b = c$. Similarly, $a = b = c$. And we are done.

- 5 years, 6 months ago

I think that's it @Victor Loh did you do that to?

- 5 years, 6 months ago

An easy way is to substitute $x=\sqrt{a},y=\sqrt{b},z=\sqrt{c}$ . The you get 3 equations. Try to find ab expression involving just one variable. So, we get $x\left( x-2\right)\left(x^{12}-3x^{11}+6x^{10}-14x^{9}+22x^{8}-28x^{7}+37x^{6}-35x^{5}+26x^{4}-21x^{3}+14x^{2}-12x+8\right) =0$. . Now this gives $x=2,x=0$. So, we get $a=4,0$. So, leaving $0$, we get the desired result

- 5 years, 6 months ago

right... and how did you prove that the huge polynomial has no positive real roots?

- 5 years, 6 months ago

Pure evil.

- 5 years, 6 months ago

There is also a solution with AM-GM i think

- 5 years, 6 months ago

I took the Junior paper though, and it was easier than I had thought previously, to be honest. I solved all 5 questions :)

- 5 years, 6 months ago

Good luck to all who took the SMO! Wish you all the best! :D

- 5 years, 6 months ago

How does Question 4 pop out without much effort at all

- 5 years, 6 months ago

Its the hardest question

- 5 years, 6 months ago

5 was pretty obvious though... For 2 I used another method.

- 5 years, 6 months ago

Q3 just spam lah hahah

- 5 years, 6 months ago

Wut! For 4 you have to prove that a,b and c HAVE to be equal man.

- 5 years, 6 months ago

Okay okay fine. I'm removing that with all this debate going on... D:

- 5 years, 6 months ago