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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
3 years, 1 month ago

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I took the Junior paper though, and it was easier than I had thought previously, to be honest. I solved all 5 questions :) · 3 years ago

Good luck to all who took the SMO! Wish you all the best! :D · 3 years ago

5 was pretty obvious though... For 2 I used another method. · 3 years ago

How does Question 4 pop out without much effort at all · 3 years ago

Its the hardest question · 3 years ago

Wut! For 4 you have to prove that a,b and c HAVE to be equal man. · 3 years ago

Okay okay fine. I'm removing that with all this debate going on... D: · 3 years ago

Q3 just spam lah hahah · 3 years ago

Answer to second is $$(0,0,00$$ and $$(4,4,4)$$ · 3 years, 1 month ago

Question asked for only positive reals. · 3 years, 1 month ago

Sorry, Then only $$(4,4,4)$$ · 3 years ago

Whats the proof? · 3 years 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. · 3 years ago

I think that's it @Victor Loh did you do that to? · 3 years ago

There is also a solution with AM-GM i think · 3 years 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 · 3 years ago

right... and how did you prove that the huge polynomial has no positive real roots? · 3 years ago

Pure evil. · 3 years ago

If anyone requires, the SMO Junior Round 2 2014 Qns can be found here · 3 years, 1 month ago