# BoS Maths Trials 2019 (1 Viewer)

#### InteGrand

##### Well-Known Member
Thanks, but i think this is way over the top for 4unit.
I saw a way a teacher did it through binary. The analogy goes something like this:

1 0 1 1 1 1 0 0 0 is a binary string of length 10. In a binary string of length 50, how many ways are there to have a string with exactly nine lots of ones and that no ones are next to each other?

I fail to see how these two are the same question/where is the one-to-one correspondence.
Maybe trebla can help?
I got 42C9 using the method I posted above, so I assumed there was a simpler method.

For the equivalence between the method your teacher used and the original question, it is as follows:

The bit string of length 50 corresponds to your choice of whether you pick each number from 1-50 or not. A bit of 0 in the string means you did NOT pick that number, and a bit of 1 means you picked that number.

For example, (assume you're picking 3 numbers from a 1-10 for simplicity here): 1001000100 has a 1 in positions 1, 4, 8, so it corresponds to picking 1, 4 and 8.

The original question is then like counting bit strings of length 50 (since we pick from numbers 1-50) with 9 1's (since we pick a total of 9 numbers), and no two 1's can be adjacent (since that represents picking two consecutive numbers).

#### blyatman

##### Well-Known Member
I have a feeling this is HscBuzman on an alt acc or something. What do you guys think? @blyatman @jazz519 .
Dunno, doubt it based on account creation date. Also the response post to the question doesn't really make any sense, sounds more like a joke so I don't think it was intended to be rude. Either way, I'll give them the benefit of the doubt. and assume we've all moved past it. Besides, too busy to think about stuff like this lol.

#### aa180

##### Member
So obviously the answer is 42C9
My formula gives 42C9 exactly, but you originally told me that the answer was 42C7

#### InteGrand

##### Well-Known Member
So obviously the answer is 42C9
Why is it obviously 42C9?

$\bg_white \noindent This answer follows from noting that we need to insert at least one 0 between each 1. If there are K 1's and N-K 0's, there are K-1 gaps between 1's, so after putting one 0 in each gap, there are now N-K-(K-1)=N-2K+1 0's left to place. There are K+1 gaps now these can be placed in (either between two 1's or on an endpoint), and using Stars and Bars, the no. of ways to place these is \binom{N-2K+1 +K}{K} = \binom{N-K+1}{K}, which is the answer. In your question, K=9 and N=50, giving \binom{42}{9}.$

I assume the intended solution was simple.

#### worldno17

##### Active Member
As a quick warm up for tomorrow, can anyone answer with reasons
Given 50 cards with the integers 1, 2, 3, ... 50 printed on them, how many ways are there to select 9 distinct cards, such that no two cards have consecutive numbers printed on them?
Yikes...this question was in my school's trial and it was worth one mark. :')

#### Trebla

It’s exam day!

Forgot to mention please turn up around 5-10 mins before actual start time (at latest). See you there!

And we’re off!

#### TheOnePheeph

##### Active Member
fascinating problems and I really enjoyed it xD thanks in advance
What was q16 this year?

#### sharky564

##### Member
As a quick warm up for tomorrow, can anyone answer with reasons
Given 50 cards with the integers 1, 2, 3, ... 50 printed on them, how many ways are there to select 9 distinct cards, such that no two cards have consecutive numbers printed on them?
In general, the number of ways of choosing $\bg_white a$ cards from $\bg_white b$ consecutive cards such that there are at least $\bg_white c$ cards between any pair of cards is $\bg_white \binom{b+c-ac}{a}$, which isn't too difficult to prove using double-counting.

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#### Drdusk

##### π
Moderator
How was it guys!? What was question 16?

#### sharky564

##### Member
How was it guys!? What was question 16?
16a was proving the Dandelin spheres property
16b was proving the eccentricity of an ellipse is equal to the ratio of the sines of the angle the plane cuts through the cone to the sine of the angle of the cone
16c was determining the probability a game (in which players A and B flip a coin till either 3 heads or 3 tails are flipped in a row) is won by Player A, depending on which face is flipped ifrst

#### CellerySticks

##### -insert title here-
Thank you to everyone who showed up! I'm sorry for the typos (although nobody noticed that Trebla wrote "focii" instead of "foci"

Here are the papers, edited and refined (and a little bit of visual changes as well to both)

#### Attachments

• 298 KB Views: 65
• 518.2 KB Views: 59

#### Drdusk

##### π
Moderator
Thank you to everyone who showed up! I'm sorry for the typos (although nobody noticed that Trebla wrote "focii" instead of "foci"

Here are the papers, edited and refined (and a little bit of visual changes as well to both)
HOLY CRAP THE 4U TRIAL LOOKS HARD AS. MY UNI FINAL ARE EASIER GUYS.

This is much harder than the one I sat last year!!!

#### Drdusk

##### π
Moderator
That is defamatory. It is not me....
Well sorry mate. I'll delete the message if you think that's the case

#### Trebla

Thanks to everyone that attended on the day! Hopefully you guys found some challenging and interesting questions to practice on.

Massive thanks to the rest of the team who helped put the questions and paper together (Paradoxica (esp for the diagrams!), iso1234, jjlim7, Carrotsticks, sharky564, Kingom, RealiseNothing to name a few). Also big thanks to the staff at UTS and iStudy for helping to organise the venue.

We will begin the process of solution writing and marking in the next few weeks and will upload these in due course (before your first maths exam of course!).

In the meantime, if you want any hints/guides on specific questions right now or just want to discuss the paper feel free to post in this thread. Keen to hear your thoughts on both the papers. Were there any questions you particularly liked? How was the difficulty in general and in comparison to previous years (if you attempted those)?

##### -insert title here-
Thank you to everyone who showed up! I'm sorry for the typos (although nobody noticed that Trebla wrote "focii" instead of "foci"

Here are the papers, edited and refined (and a little bit of visual changes as well to both)
I have reuploaded the file (into the post) since this was posted, as it was noticed that the slice diagram for the volumes question was missing (I accidentally commented out the line of code for that image, my bad)

Here are the files anyway to avoid confusion.

#### Attachments

• 298 KB Views: 22
• 518.2 KB Views: 19

#### integral95

##### Well-Known Member
Wow this is definitely harder than the previous years. I see that Carrotsticks having extra assistance from former BOS members (who are also math whizzes ) definitely revoluntionised the content.

However, with the new syllabus being implemented, I wonder what mathematical theorem do you plan to use that could be solved using strictly high-school math.