Westpac Mathematics Competition (3 Viewers)

[bassguy]

Re: Westpac Australian Maths Comp Senior Question !!

Hey All,
I just did the Westpac maths comp today and found it fairly challanging. One question in particular racked my brain, so ill post it here and see if any of you know how to do it.

Question 27

the sum of three numbers is 4, the sum of their squares is 10 and the sum of their cubes is 22. What is the sum of their fourth powers ?

So basically your given that
a + b + c = 4
a^2 + b^2 + c^2 = 10
a^3 + b^3 + c^3 = 22
find what a^4 + b^4 + c^4 equals

I am doing three unit maths and still couldnt do it !
Best of luck to all of you

haha
im doing extension as well
but i was just annoyed by that point
so i was like '10 is 2x4 + 2, and 22 is 2x10 +2, so logically the number is 2x22 +2'
good times
i know its not right
but really, can anyone honestly say they do well in these tests?
say, %90 +
because if you do, and are in my year
then i shall cry

 

[modezero]

Did anyone solve q30?
I remember seeing the problem before, its a number theory problem disguised as geometry, but I've forgotten how to do it.

 

[short!ez]

lol question 30.. the one with the four circles, radius 3, 2, 1 and p/q yeah?

i guessed that one lol. ran outta time. i put in 6. my friend got that too, apparently she worked it out. we're not spesh students.. YET ~!! go the general specialist class consisting of 9 students!

what'd everyone else get =]

 

[SoulSearcher]

Guessed it, no idea how to do it

 

[short!ez]

hmm.. i can't seem to figure it out either, without using trig (inclusive of sine, cosine and tangent) or pi. that's quite troubling lol. but i'm sure it had something to do with the fact that when all the radii met, creating a triangle, it was a right angled triangle. i tried it in pythagora's theorem, it works lol. question now is, then what? haha

 

[Sober]

Did anyone solve q30?
I remember seeing the problem before, its a number theory problem disguised as geometry, but I've forgotten how to do it.

My solution to question 30:

The red dot indicates the center of the small circle and a is its radius.

Find expressions for h and w in terms of a: (I used herons triangle formula but you should be able to find them using trig as well).

h = √(3a(4+a)) / 2

w = 2√(2a(3+a)) / 3

Using pythagoras:

h² + w² = (1+a)²

Boils down to the quadratic:

23a² + 132a - 36 = 0

With only one positive solution a = 6/23

Final answer 6+23 = 29.

 

[short!ez]

lol the clouds have parted and all is clear..

ALL HAIL MATHS MASTER SOBER ~!! *mexican wave of hailing*

 

[SoulSearcher]

My solution to question 30:

The red dot indicates the center of the small circle and a is its radius.

Find expressions for h and w in terms of a: (I used herons triangle formula but you should be able to find them using trig as well).

h = √(3a(4+a)) / 2

w = 2√(2a(3+a)) / 3

Using pythagoras:

h² + w² = (1+a)²

Boils down to the quadratic:

23a² + 132a - 36 = 0

With only one positive solution a = 6/23

Final answer 6+23 = 29.

Now why couldn't I think of that in the last minute of the test

 

[humphdogg]

Anyone get the results for this?

Got mine on Tuesday. 82/120 = 100th percentile = HD

10/10 for first section
7/10 for second section
4/10 for third section

 

[SoulSearcher]

I apparently got a HD, but I haven't got any evidence of this yet.

 

[Sober]

HD, but 99th percentile, didn't know there was a one hundredth.

 

[milton]

normally, the first 10 questions are completely trivial, then q11-20 are a bit harder, but still easy, and the last 10 actually require a bit of time and for you to think. just make sure you get the first 20 right and do 5 of the last 10 you it should be enough to get you a prize, maybe even a medal.

HD is top 2%, but prize is top 0.3%, cutoff for prize is normally about 23/24 questions

 

[airie]

HD is top 2%, but prize is top 0.3%, cutoff for prize is normally about 23/24 questions

I thought they changed the percentages for HD and D for the Senior division, starting from this year?

 

[vulgarfraction]

normally, the first 10 questions are completely trivial, then q11-20 are a bit harder, but still easy, and the last 10 actually require a bit of time and for you to think. just make sure you get the first 20 right and do 5 of the last 10 you it should be enough to get you a prize, maybe even a medal.

HD is top 2%, but prize is top 0.3%, cutoff for prize is normally about 23/24 questions

And don't be fricking cocky and rush through the first 20 questions in 15 minutes to give yourself more time for the last ten, do only okay the last ten anyway, and then get stuff wrong in both the first and second sections.

It sort of makes you not get a prize.

 

[SoulSearcher]

Just own the first 20 then try to get as many as you can in the last 10 right

 

[Boxxxhead]

HD is top 5% and D is top 25% in the senior division.

Note: It should also be noted that awards up to the level of cash prizes are decided by comparison only within the region and year level of the student. For example a student in a certain year in an Australian state or another country will only be compared for the purpose of a certificate or prize with other students in the same group. As a result there will be different cutoffs for awards between Australian states (which each have their own curriculum) and various countries.

PRIZE
A prize is awarded generally to no more that 1 student for every 300 students within their region and year group, the value of a prize depending on issues such as sponsorship money available.

HIGH DISTINCTION
A High Distinction Certificate is awarded to a student who has not received a higher award but is in the top 2% of their year and region (top 5% for Senior).

DISTINCTION
A Distinction Certificate is awarded to a student who has not received a higher award but is in the top 15% (25% for Senior) of their year and region.

CREDIT
A Credit Certificate is awarded to a student who has not received a higher award but is in the top 50% (60% for Senior) of their year and region.

PARTICIPATION
A Participation Certificate is awarded to a student who has not received a higher award but has participated in the AMC.

PRUDENCE AWARD
This will be awarded to the student in each school with the highest number of consecutive correct responses starting from question 1. If there is a tie more than one such prize can be awarded in a school. In order for the prize to be awarded the school needs to have an entry of at least 50 students and the winner must have at least the first five answers correct.


For further info see here.

I got top 5% and got a mere Distinction... damn lol

 

[jyu]

For your future reference

Solutions for the last three Westpac Mathematics Competitions are available from http://itute.com/mathline/exams_solutions.html

 

[Forbidden.]

Not doing Maths Competition ever again.

 

[kony]

just make sure you get the first 20 right and do 5 of the last 10 you it should be enough to get you a prize, maybe even a medal.

this is really old, but i think getting a medal needs even more than 25/30. i got 27/30 last yr (2005) when i was in year 10, and it was still only prize.

 

[modezero]

this year you only needed 25/30 to get a medal (for yr12)