Practice for Q7-8 (1 Viewer)

irvine · Jun 29, 2010

Okay, so- occasionally when I attempt a past paper, I can almost get full marks in the first 5-6 questions, but when I get to the Q7/Q8 (Harder 3U) I don't even have a clue where to start. It's not that I find the questions difficult to do, I simply don't have a clue.

Aside from simple practice, what are some other ways to improve proficiency in these open-ended/undergraduate math questions?

Thankyou

Carrotsticks · Jun 29, 2010

For question 7, you should be aiming to get about 60%. For question 8, just aim to get about 50%.

Question 7 is usually slightly harder applications of some randie 4u topic.

Question 8 is usually some rape ambiguous question.

For Q7, I just practise the harder questions which involve the use of things people tend to forget (like simpsons rule or cosine rule)

For Q8, I just persevere and write any shit I can deduce, then hope to grab some marks. I don't think many people can get the whole Q8 done.

Aquawhite · Jun 29, 2010

I think I get about a total of 3 marks in Q8. It's near on impossible.

sifANDREW · Jun 29, 2010

Wow you guys start past papers already? Does that mean you finished everything? i think im gonna start but im still missing out some topics =S

kcqn93 · Jun 29, 2010

sifANDREW said:
Wow you guys start past papers already? Does that mean you finished everything? i think im gonna start but im still missing out some topics =S

lol most schools should have finished the MX2 course by like half way through this term

Trebla · Jun 30, 2010

kcqn93 said:
lol most schools should have finished the MX2 course by like half way through this term

No way. Most selective schools haven't finished the Ext2 course yet. Few schools ever finish the entire course before Term 3. At this stage, most schools should be up to Mechanics.

The best way to succeed in these questions is to perservere. There will obviously be questions which have really complicated answers that few people ever come up with, but most of them are actually quite simple if you sit there and have a good long hard think about it.

irvine · Jun 30, 2010

kcqn93 said:
lol most schools should have finished the MX2 course by like half way through this term

Our class still has volumes and mechanics to go, the other class in my grade just mechanics. That said, we've done all of the harder extension one which I know some schools haven't done...

buchanan · Jul 1, 2010

Try these:

$\textbf{1.}\text{ Let } m\text{ be a positive integer. Prove that }\sum\limits_{k=m}^{m^2-m+1}{{{m^2-2m+1}\choose{k-m}}\over{k{m^2\choose k}}}={1\over{m{2m-1\choose m}}}.$

$\textbf{2.}\text{ For a triangle } ABC,\text{ let } f_A\text{ denote the distance from } A\text{ to the intersection of the line bisecting}\\ \text{angle } BAC\text{ with edge } BC,\text{ and define } f_B\text{ and } f_C\text{ similarly. Prove that } ABC\text{ is equilateral if and only if }\\ f_A=f_B=f_C.$

$\textbf{3.}\text{ Let } N \text{ be a nonnegative integer. For } x\ge0,\text{ prove that }\\ \sum\limits_{m=0}^N{1\over m!}\Bigg(\sum\limits_{k=1}^{N-m+1}{x^k\over k}\Bigg)^m\ge1+x+\cdots+x^N.$

$\textbf{4.}\text{ For a triangle with area } F,\text{ semiperimeter } s,\text{ inradius } r,\text{ circumradius } R,\text{ and heights } h_a,\ h_b,\text{ and } h_c,\\ \text{ show that }5(h_a+h_b+h_c)\ge{2sF\over Rr}+18r\ge{{10r(5R-r)}\over R}.$

$\textbf{5.}\text{ Let } k \text{ be a positive integer, and let }a_1,\dots,a_n\text{ be positive numbers such that }\\ \sum_{i=1}^na_i^k=1.\text{ Show that }\sum\limits_{i=1}^na_i+{1\over\prod_{i=1}^na_i}\ge n^{1-1/k}+n^{n/k}.$

$\textbf{6.}\text{ Find a closed-form expression for } \sum\limits_{n=1}^\infty4^n\sin^4(2^{-n}\theta).$

Affinity · Jul 1, 2010

in case someone's not too familiar with the terms:
semiperimeter --> 1/2 the perimeter
inradius ---> the radius of the circle with the 2 edges as tangents
circumradius --> radius of the circle passing through the 3 corners
heights -> perpendicular distance from a corner to the opposite edge

fullonoob · Jul 2, 2010

schools should be up to harder 3u now, but most people should've tried the 2001-2009 papers for the heck of it. i just death stare at the question 8 :x

seanieg89 · Jul 5, 2010

buchanan · Jul 5, 2010

Correct.

Rezen · Jul 6, 2010

For question 5, so far i can only prove

$\frac{1}{\prod_{i=1}^{n}a_i}\geq n^\frac{n}{k}$ .

But im having difficulty in showing that,

$\sum_{i=1}^{n}a_i \geq n^{1-1/k}$

Infact, i think it could be the opposite. ): Buchanan, any hints on how to show it?
Also, where do these questions come from?

buchanan · Jul 6, 2010

No solutions are published yet.

seanieg89 is the first person to have provided a correct answer to one of these.

Authorships of the questions are as follows:

1. William Stanford, University of Illinois-Chicago, Chicago, IL.
2. Retkes Zoltan, Szeged, Hungary.
3. Finbarr Holland, University College Cork, Cork, Ireland.
4. Pal Peter Dalyay, Szeged, Hungary.
5. Mihaly Bencze, Brazov, Romania.
6. Estelle L. Basor, American Institute of Mathematics, Palo Alto, CA, Steven N. Evans, University of California, Berkeley, CA, and Kent E. Morrison, California Polytechnic State University, San Luis Obispo, CA.

shaon0 · Jul 6, 2010

Rezen said:
For question 5, so far i can only prove

$\frac{1}{\prod_{i=1}^{n}a_i}\geq n^\frac{n}{k}$ .

But im having difficulty in showing that,

$\sum_{i=1}^{n}a_i \geq n^{1-1/k}$

Infact, i think it could be the opposite. ): Buchanan, any hints on how to show it?
Also, where do these questions come from?

$(\sum_{i=1}^{n} a_{i})^n\geq n^{n}\prod_{i=1}^{n} a_{i} \geq n^{n}n^{-n/k}=n^{n(1-1/k)} \\ \therefore \sum_{i=1}^{n} a_{i}\geq n^{1-1/k}$

buchanan · Jul 6, 2010

Incidently, if you want your solutions to be published and authorship acknowledged, solutions should be submitted in duplicate by October 31, 2010 to:

Doug Hensley, Monthly Problems
Department of Mathematics
Texas A&M University
3368 TAMU
College Station, TX 77843-3368
USA

In lieu of duplicate hardcopy, authors may submit pdfs to monthlyproblems@math.tamu.edu

If you make such a submission, use this question number map from my post's question numbers to the Monthly Problems question numbers:

Q1 is Q11509
Q2 is Q11511
Q3 is Q11512
Q4 is Q11513
Q5 is Q11514
Q6 is Q11515

beardedwoman · Jul 6, 2010

just wondering but does anyone else here actually have difficulty with majority of the Qs in the CSSA trial papers? I'm about to kill myself..........

Affinity · Jul 6, 2010

shaon0 said:
$(\sum_{i=1}^{n} a_{i})^n\geq n^{n}\prod_{i=1}^{n} a_{i} \geq n^{n}n^{-n/k}=n^{n(1-1/k)} \\ \therefore \sum_{i=1}^{n} a_{i}\geq n^{1-1/k}$

doesn't work.

nat_doc · Jul 6, 2010

fullonoob said:
schools should be up to harder 3u now, but most people should've tried the 2001-2009 papers for the heck of it. i just death stare at the question 8 :x

i try that as well

shaon0 · Jul 6, 2010

Affinity said:
doesn't work.

It works if:
$\prod_{i=1}^{n} a_{i} \geq \left ( \prod_{i=1}^{n} a_{i} \right )^{-1}$

Practice for Q7-8 (1 Viewer)

USYD 2011 :D

Retired

Retiring

New Member

Member

Administrator

USYD 2011 :D

.

Active Member

fail engrish? unpossible!

Well-Known Member

.

Member

.

...

.

Member

Active Member

Member

...

Users Who Are Viewing This Thread (Users: 0, Guests: 1)