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HSC 2015 MX2 Marathon (archive) (1 Viewer)

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Drsoccerball

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Re: HSC 2015 4U Marathon

A relatively nice complex numbers Q for you all -


x, y and z are complex numbers. If 0, x, y and z are concyclic, prove that 1/x, 1/y and 1/z are collinear!
Can't we just say that x=-1/x and it a cyclic quad forms with x then it will also form for 1/x ... Or is that what were trying to prove LOL
 

InteGrand

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Re: HSC 2015 4U Marathon

Here is a question I thought up while staring at a clock waiting for the time to pass:








Are you sure it's p possible straight times for part α? I'm getting (p – 1) straight times. (Might have screwed up somewhere.)
 

Sy123

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Re: HSC 2015 4U Marathon

Are you sure it's p possible straight times for part α? I'm getting (p – 1) straight times. (Might have screwed up somewhere.)
I got a straight time for every hour, so p straight times

PM me your working out
 

InteGrand

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Re: HSC 2015 4U Marathon

I got a straight time for every hour, so p straight times

PM me your working out
It can't be a straight time for 0 o'clock (i.e. p o'clock) though can it? Since the minute hand and hour hand should be both pointing up then, rather than in opposite directions. Edit: wait I get what you mean now. I'll have another look.
 
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InteGrand

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Re: HSC 2015 4U Marathon

I tried to show that theres only one value for n such that the angles of the triangles added up equal to pi but i guess i excluded the minutes as you pointed out.
What do you mean angles though? We can't place the hands of the clock arbitrarily on the clock, since their movement is governed by how clocks work. E.g. it is impossible to have the hour hand at 2 and the minute hand at 5 at the same instant.
 

Drsoccerball

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Re: HSC 2015 4U Marathon

What do you mean angles though? We can't place the hands of the clock arbitrarily on the clock, since their movement is governed by how clocks work. E.g. it is impossible to have the hour hand at 2 and the minute hand at 5 at the same instant.
LOOOOOOOOOOOOOOOOOOOOOOOOOL my bad.
 

Paradoxica

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Re: HSC 2015 4U Marathon

11 straight times can only occur if the clock is continuous (established already). The clock, however is not continuous.
The straight times on a continuous clock occur in equal elapses of time (Provable, although I don't see how to do it yet). This means that on a 12-clock, the straight lines occur every 1440/11 minutes. Ten of these cannot be integer times, and only one can be integral. Since we have established that 6:00 is a straight line, it is the only straight line on the clock.
 

Sy123

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Re: HSC 2015 4U Marathon

InteGrand has pointed out an error in the question, for part ii.a there are (p-1) straight times, not p straight times, edited

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