53rd UNSW Mathematics Competition (1 Viewer)

Kevink98 · Jun 11, 2014

Anyone here do the UNSW maths comp today? (the 3 hour one with 6 questions)

How did you find the questions? How do you think you went?

rumbleroar · Jun 11, 2014

i have heaps of friends doing it and saw the maths paper
wtf LOL

Kurosaki · Jun 11, 2014

I got 2 questions out, the two easy ones lol (the inequalities one and the combinatorics question).
I misread the circle geometry question but apparently it was easy if you transposed it onto the coordinate axes and applied coordinate geometry.

Most of my friends got 3-4 out of the 6 out, and one of them managed to do 5 of them!

Time to quit maths.

Kevink98 · Jun 11, 2014

How'd you do the inequalities??
I tried that for about 20 minutes then gave up...

I got n + k - 1 for the first one, combinatorics question as well using mod, and I had to rush question 5... did it shiftily though using binary... By any chance do you know the answer for question 5b? I got 171 but it's probably wrong... Didn't even touch the geometry haha.

Futuremedstudent · Jun 11, 2014

yeah i did it, everyone in my class had to. 20 minutes after the test, we had a topic test on calculus.
The maths comp was hard for most people, i managed to get 3-4 q's.

Kevink98 · Jun 11, 2014

Wow everyone in your class did it? Which questions did you get done, and how did you do them?

Kurosaki · Jun 11, 2014

I think that the combinatorics one is 301.
The inequality was alright, so, um
$$R.T.P$ \\ \frac{1}{a_1}+....+\frac{1}{a_n}\geq n^2 \\ $ Given a_1+....+a_n=1 $ \\ $ Multiply the inequality we wish to prove by (a_1+....+a_n) $ $which is valid as it is equal to 1 $ \\ LHS=(a_1+....+a_n)(\frac{1}{a_1}+......+\frac{1}{a_n}) \\ = (1+\frac{a_1}{a_2}+.....\frac{a_1}{a_n}) + (1+\frac{a_2}{a_1}+....\frac{a_2}{a_n})+....(1+\frac{a_n}{a_1}+.....+\frac{a_n}{a_{n-1}})=n + (\frac{a_1}{a_2}+\frac{a_2}{a_1})+(\frac{a_1}{a_3}+\frac{3}{1})+.......$

For each pair of $\frac{a_i}{a_j}+\frac{a_j}{a_i}$ , apply the AM-GM inequality for two variables. This makes $\frac{a_i}{a_j}+\frac{a_j}{a_i}\geq 2$ . Now, there are $\binom{n}{2}=\frac{n(n-1)}{2}$ pairs (we need to choose 2 distinct integers for these pairs, and the number of ways we do this corresponds to the number of pairs.
So then

$LHS \geq n(n-1) + n \\ =n^2$
As required.

Futuremedstudent · Jun 11, 2014

Kevink98 said:
Wow everyone in your class did it? Which questions did you get done, and how did you do them?

cos i'm accelerated, so they make everyone do it! I attempted up to 3-4, before it got too tough.

Kurosaki · Jun 11, 2014

Futurmedstudent said:
cos i'm accelerated, so they make everyone do it! I attempted up to 3-4, before it got too tough.

Ah nice, 3/4U?

Futuremedstudent · Jun 11, 2014

Kurosaki said:
Ah nice, 3/4U?

i'm in year 10 atm. but this year we do 2 unit prelim, in year 11 we do 3u prelim and 2u hsc and in year 12 we do both 3u and 4u exams.

Kevink98 · Jun 11, 2014

Kurosaki said:
I think that the combinatorics one is 301.
The inequality was alright, so, um
$$R.T.P$ \\ \frac{1}{a_1}+....+\frac{1}{a_n}\geq n^2 \\ $ Given a_1+....+a_n=1 $ \\ $ Multiply the inequality we wish to prove by (a_1+....+a_n) $ $which is valid as it is equal to 1 $ \\ LHS=(a_1+....+a_n)(\frac{1}{a_1}+......+\frac{1}{a_n}) \\ = (1+\frac{a_1}{a_2}+.....\frac{a_1}{a_n}) + (1+\frac{a_2}{a_1}+....\frac{a_2}{a_n})+....(1+\frac{a_n}{a_1}+.....+\frac{a_n}{a_{n-1}})=n + (\frac{a_1}{a_2}+\frac{a_2}{a_1})+(\frac{a_1}{a_3}+\frac{3}{1})+.......$

For each pair of $\frac{a_i}{a_j}+\frac{a_j}{a_i}$ , apply the AM-GM inequality for two variables. This makes $\frac{a_i}{a_j}+\frac{a_j}{a_i}\geq 2$ . Now, there are $\binom{n}{2}=\frac{n(n-1)}{2}$ pairs (we need to choose 2 distinct integers for these pairs, and the number of ways we do this corresponds to the number of pairs.
So then

$LHS \geq n(n-1) + n \\ =n^2$
As required.

Damnnnnn, that's a nice proof...

Did anyone get question 6 (senior)? The guy next to me had a whole heap of triangles on his page..

mirachael · Jun 11, 2014

I managed to get 5 out

Not sure how rigorous my proofs were though
For the first 1 I got n
Second was just leave the other person with a multiple of 3 counters
Third was as Kurosaki said (there was a similar question in Cambridge 4U harder 3U)
Fourth I just used coordinate geometry letting the first circle be x^2+y^2=1 and the second circle be (x-1)^2+y^2=1 and went from there
Fifth was simply perms and combs, got 301 for the 2nd part but not sure if I'm right
And for the last question I have no idea. Apparently, it was in the junior paper as well. I'd really like to know how to do it.

erckle999 · Jun 12, 2014

Anyone have a copy of the paper they might be kind enough to upload. Would like to have a go.

Kevink98 · Jun 12, 2014

go_cadel · Jun 15, 2014

I didn't actually do the comp (exams), but here's another way to do the inequality.

Screen Shot 2014-06-15 at 4.28.22 PM.png

53rd UNSW Mathematics Competition (1 Viewer)

Kevink98

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rumbleroar

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Kurosaki

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Kevink98

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Futuremedstudent

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Kurosaki

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Kurosaki

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Kevink98

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