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!

#Algebra #Combinatorics #Proofs #MathCompetitions #SMO

Easy Math Editor

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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

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

John Ashley Capellan - 6 years, 11 months ago

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

Ryan Wong Jun Hui - 6 years, 11 months ago

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

Dinesh Chavan - 6 years, 11 months ago

Question asked for only positive reals.

mietantei conan - 6 years, 11 months ago

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

Dinesh Chavan - 6 years, 11 months ago

@Dinesh Chavan – Whats the proof?

Happy Melodies - 6 years, 11 months ago

@Happy Melodies – 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.

Fengyu Seah - 6 years, 11 months ago

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

Fengyu Seah - 6 years, 11 months ago

@Happy Melodies – 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

Dinesh Chavan - 6 years, 11 months ago

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

mathh mathh - 6 years, 11 months ago

@Mathh Mathh – Pure evil.

Joshua Ong - 6 years, 11 months ago

@Happy Melodies – There is also a solution with AM-GM i think

Daniel Remo - 6 years, 11 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 :)

Yuxuan Seah - 6 years, 11 months ago

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

Yuxuan Seah - 6 years, 11 months ago

How does Question 4 pop out without much effort at all

Victor Loh - 6 years, 11 months ago

@Victor Loh – Its the hardest question

Victor Loh - 6 years, 11 months ago

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

Joshua Ong - 6 years, 11 months ago

Q3 just spam lah hahah

Fengyu Seah - 6 years, 11 months ago

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

Joshua Ong - 6 years, 11 months ago

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

Yuxuan Seah - 6 years, 11 months ago

@Yuxuan Seah – How about your first comment

Victor Loh - 6 years, 11 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}`	$\boxed{123}$