2012 Year 9 &10 Mathematics Marathon (1 Viewer)

[mirakon]

Re: 2012 Year 9 &10 Mathematics Marathon

here is a question from the AIMO a few years back, you adept year 10s should find it quite a fun one to solve

Let ∆ABC be an equilateral triangle with AB = x. On the extension of BC, we define points A' (on the same side as B) and A" (on the same side as C) such that A'B = CA" = y. Similarly, on the extension of side CA, we define B' (on the same side as C) and B" (on the same side as A) such that B'C = AB" = y, while on the extension of side AB, we define C' (on the same side as A) and C" (on the same side as B) such that C'A = BC" = y.
(a) Prove that the points A', B", C', A", B' and C" all lie on a circle.
(b) If x and y are positive integers, determine the smallest integer value for R2, where R is the radius of that circle.

 

[Demento1]

Re: 2012 Year 9 &10 Mathematics Marathon

Instead of "plus", it should be a "times"

LOL, no doubt. Didn't read my working out properly. Apologies for the careless error.

 

[twinklegal19]

Re: 2012 Year 9 &10 Mathematics Marathon

LOL, no doubt. Didn't read my working out properly. Apologies for the careless error.

dw, it happens to all of us

 

[Demento1]

Re: 2012 Year 9 &10 Mathematics Marathon

here is a question from the AIMO a few years back, you adept year 10s should find it quite a fun one to solve

Let ∆ABC be an equilateral triangle with AB = x. On the extension of BC, we define points A' (on the same side as B) and A" (on the same side as C) such that A'B = CA" = y. Similarly, on the extension of side CA, we define B' (on the same side as C) and B" (on the same side as A) such that B'C = AB" = y, while on the extension of side AB, we define C' (on the same side as A) and C" (on the same side as B) such that C'A = BC" = y.
(a) Prove that the points A', B", C', A", B' and C" all lie on a circle.
(b) If x and y are positive integers, determine the smallest integer value for R2, where R is the radius of that circle.

Familiar with this Q. Should I let ymcaec do it? I've seen it before. It's a good question from the 2009 paper I believe?

 

[mirakon]

Re: 2012 Year 9 &10 Mathematics Marathon

Familiar with this Q. Should I let ymcaec do it? I've seen it before. It's a good question from the 2009 paper I believe?

can't recall whether it was 2008 or 2009. I remember sitting the AIMO that year and finding it quite a nice problem

 

[Demento1]

Re: 2012 Year 9 &10 Mathematics Marathon

can't recall whether it was 2008 or 2009. I remember sitting the AIMO that year and finding it quite a nice problem

Ahh, I think I know what method is used now that I recall it. Doesn't it require the intersecting chord theorem (not sure if that's the name for it)?

Edit: I better look up this theorem to make sure I do not have any incorrect ideas.

 

[ymcaec]

Re: 2012 Year 9 &10 Mathematics Marathon

Ahh, I think I know what method is used now that I recall it. Doesn't it require the intersecting chord theorem (not sure if that's the name for it)?

Edit: I better look up this theorem to make sure I do not have any incorrect ideas.

am i supposed to know it ? cuz i dont...

 

[twinklegal19]

Re: 2012 Year 9 &10 Mathematics Marathon

am i supposed to know it ? cuz i dont...

you learn circle geo either at the end of the year 10 course with the optional topics or in the MX1 course (either in yr 11 or 12)

 

[Demento1]

Re: 2012 Year 9 &10 Mathematics Marathon

you learn circle geo either at the end of the year 10 course with the optional topics or in the MX1 course (either in yr 11 or 12)

am i supposed to know it ? cuz i dont...

That is the method I would use to solve the Q. Not sure if there is simply another way. Look it up on google and it should make a little more sense. I have not done circle geometry (not accelerated at school) although I know this from reading some interesting math theorems in my spare time.

It's basically, where two chords intersect each other in a circle. The product of two segments in one line is equal to equal the product of the two segments in the 2nd line.

 

[twinklegal19]

Re: 2012 Year 9 &10 Mathematics Marathon

That is the method I would use to solve the Q. Not sure if there is simply another way. Look it up on google and it should make a little more sense. I have not done circle geometry (not accelerated at school) although I know this from reading some interesting math theorems in my spare time.

It's basically, where two chords intersect each other in a circle. The product of two segments in one line is equal to equal the product of the two segments in the 2nd line.

Circle geo is pretty cool, I have a feeling you will like the questions and proofs Plus I believe the theorems are in one of those small Australian Mathematics Trust books, can't remember which one.

Also, do you know the proof for that intersecting chords theorem? It's quite simple really

 

[RealiseNothing]

Re: 2012 Year 9 &10 Mathematics Marathon

Circle geo is pretty cool, I have a feeling you will like the questions and proofs Plus I believe the theorems are in one of those small Australian Mathematics Trust books, can't remember which one.

Also, do you know the proof for that intersecting chords theorem? It's quite simple really

How about make this a question?

Prove the intersecting chords theorem.

 

[twinklegal19]

Re: 2012 Year 9 &10 Mathematics Marathon

How about make this a question?

Prove the intersecting chords theorem.

Yes, this.

Let AB and PQ be chords of a circle intersecting at M, prove that AM x MB = PM x MQ

 

[RealiseNothing]

Re: 2012 Year 9 &10 Mathematics Marathon

Yes, this.

Let AB and PQ be chords of a circle intersecting at M, prove that AM x MB = PM x MQ

m8

 

[twinklegal19]

Re: 2012 Year 9 &10 Mathematics Marathon

Oops, on second thoughts though, you need to know the angles in the same segment theorem. Basically those two triangles are similar and since their sides are in the same proportion, that's how you can get that result.

 

[Carrotsticks]

Re: 2012 Year 9 &10 Mathematics Marathon

Fix'd

 

[twinklegal19]

Re: 2012 Year 9 &10 Mathematics Marathon

A teddy bear is shot out of a stunt cannon and follows a parabolic path given by the equation , where x and y are measured in metres. A ramp with a gradient of 0·5 begins at the cannon and extends underneath the path of flight.

i. How high off the ground is the teddy bear at the highest point of its flight?
ii. The teddy bear lands on the ramp. How high off the ground is the teddy
bear when it lands?

 

[Demento1]

Re: 2012 Year 9 &10 Mathematics Marathon

Will definitely solve late tomorrow if possible unless someone solves it earlier. I am on my iPod at the moment I'm afraid.

 

[twinklegal19]

Re: 2012 Year 9 &10 Mathematics Marathon

Will definitely solve late tomorrow if possible unless someone solves it earlier. I am on my iPod at the moment I'm afraid.

Who says you can't LaTex on an iPod?


lol jks, the more variety of people answer, the better

 

[kazemagic]

Re: 2012 Year 9 &10 Mathematics Marathon

A teddy bear is shot out of a stunt cannon and follows a parabolic path given by the equation , where x and y are measured in metres. A ramp with a gradient of 0·5 begins at the cannon and extends underneath the path of flight.

i. How high off the ground is the teddy bear at the highest point of its flight?
ii. The teddy bear lands on the ramp. How high off the ground is the teddy
bear when it lands?

for the first question, you find the y coordinate of vertex right?
and for the 2nd one are u supposed to use simultaneous equation?
lol prob too good to be true

 

[twinklegal19]

Re: 2012 Year 9 &10 Mathematics Marathon

for the first question, you find the y coordinate of vertex right?
and for the 2nd one are u supposed to use simultaneous equation?

yes that's right, go try it