Mongoose528
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Australian Maths Competition (1 Viewer)
- Thread starter maths94
- Start date
Mongoose528
Member
A bit of help on this question please, this is a bit of practice for the AIMO.
This is all the working I've managed to do
This is all the working I've managed to do
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The AMC past papers are available for free download at https://shop.amt.edu.au/collections/amc-past-papers/products/amc-past-papers-pdf and the new ones usually come out mid-January the following year.
For some reason the 2023 ones are not there yet.
There is a Thailand dude Dr. Chatawut Chanvanichskul who has a site for past AMC papers at https://drive.google.com/drive/folders/1yi32YSWs8N6k9jZBwIuyh4Sp86OqKC06 and the 2023 ones were there last year but the 2023 disappeared for several months.
A few days ago Dr. Chatawut Chanvanichskul put the 2023 back up at https://drive.google.com/drive/folders/1ULBYBaisIgHj74_MhYMH-DN3usv14eWO
The solutions can also be purchased at https://shop.amt.edu.au/collections/amc-solutions but again they seem to be very late getting the 2023 up this year - so the answer key I put up would have to suffice for now. (Note: Dr. Chatawut Chanvanichskul doesn't seem to have the 2023 answer key)
For some reason the 2023 ones are not there yet.
There is a Thailand dude Dr. Chatawut Chanvanichskul who has a site for past AMC papers at https://drive.google.com/drive/folders/1yi32YSWs8N6k9jZBwIuyh4Sp86OqKC06 and the 2023 ones were there last year but the 2023 disappeared for several months.
A few days ago Dr. Chatawut Chanvanichskul put the 2023 back up at https://drive.google.com/drive/folders/1ULBYBaisIgHj74_MhYMH-DN3usv14eWO
The solutions can also be purchased at https://shop.amt.edu.au/collections/amc-solutions but again they seem to be very late getting the 2023 up this year - so the answer key I put up would have to suffice for now. (Note: Dr. Chatawut Chanvanichskul doesn't seem to have the 2023 answer key)
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With AMT dragging their feet getting solution up on their website which should have been there by now (in the past it was there mid-January), I put solutions together for the Senior 2023 AMC paper:
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LookingGood2
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Thanks Tywebb.
From your post first saw that Dr. Chatawut Chanvanichskul has made the 2023 AMC papers available, but he has set them to view only. Cannot even download, however he's doing better than AMT for interested people, with his putting up last year's papers.
Saw you put up solutions for the Senior Paper too. Super.
Here is 25.
Here is my solution.
That's a rather sophisticated solution, one which I think very few students would use.
It can also be done by other far less efficient ways but I believe my way is the most efficient way.
The theorem I used was proved first in 1967 by Daniel Pedoe:
D. Pedoe, On a theorem in geometry, Amer. Math. Monthly 74 (1967), 627–640.
and then again in 1968 by Harold Scott MacDonald Coxeter:
H. S. M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae 1 (1968), 104–121.
I have attached their proofs.
Here is my solution.
That's a rather sophisticated solution, one which I think very few students would use.
It can also be done by other far less efficient ways but I believe my way is the most efficient way.
The theorem I used was proved first in 1967 by Daniel Pedoe:
D. Pedoe, On a theorem in geometry, Amer. Math. Monthly 74 (1967), 627–640.
and then again in 1968 by Harold Scott MacDonald Coxeter:
H. S. M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae 1 (1968), 104–121.
I have attached their proofs.
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This theorem had it's origins in poetry.
On July 17, 1936 John Herbert de Paz Thorold Gosset first enunciated the generalisation of Descartes' Theorem in a private letter to Frederick Soddy and subsequently published in 1937. This added a fourth verse to Soddy's poem The Kiss Precise which was published in 1936.
F. Soddy, The Kiss Precise. Nature (June 20, 1936), p. 1021.
J. H. d. P. T. Gosset, The Kiss Precise, Nature 139, 62 (1937)
Here are all 4 verses put together (first 3 by Soddy and the last by Gosset):
The Kiss Precise
FOR pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the centre.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.
To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.
And let us not confine our cares
To simple circles, planes and spheres,
But rise to hyper flats and bends
Where kissing multiple appears.
In n-ic space the kissing pairs
Are hyperspheres, and Truth declares -
As n + 2 such osculate
Each with an n + 1 fold mate
The square of the sum of all the bends
Is n times the sum of their squares.
On July 17, 1936 John Herbert de Paz Thorold Gosset first enunciated the generalisation of Descartes' Theorem in a private letter to Frederick Soddy and subsequently published in 1937. This added a fourth verse to Soddy's poem The Kiss Precise which was published in 1936.
F. Soddy, The Kiss Precise. Nature (June 20, 1936), p. 1021.
J. H. d. P. T. Gosset, The Kiss Precise, Nature 139, 62 (1937)
Here are all 4 verses put together (first 3 by Soddy and the last by Gosset):
The Kiss Precise
FOR pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the centre.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.
To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.
And let us not confine our cares
To simple circles, planes and spheres,
But rise to hyper flats and bends
Where kissing multiple appears.
In n-ic space the kissing pairs
Are hyperspheres, and Truth declares -
As n + 2 such osculate
Each with an n + 1 fold mate
The square of the sum of all the bends
Is n times the sum of their squares.
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I have made an edit to the solutions.
In 25 I previously said it was by Descartes' theorem, but I replaced that with Soddy-Gosset theorem.
The Soddy-Gosset one is a generalisation of Descartes' theorem so it is better to call it Soddy-Gosset.
So I put it at https://4unitmaths.com/amc-2023-sen-sol.pdf in case I edit again instead of an attachment.
In 25 I previously said it was by Descartes' theorem, but I replaced that with Soddy-Gosset theorem.
The Soddy-Gosset one is a generalisation of Descartes' theorem so it is better to call it Soddy-Gosset.
So I put it at https://4unitmaths.com/amc-2023-sen-sol.pdf in case I edit again instead of an attachment.
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I put together the Intermediate 2023 solutions here: https://4unitmaths.com/amc-2023-int-sol.pdf
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Since August 21, 2023 there has been an incorrect solution by lpieleanu to intermediate q30 on aops at https://artofproblemsolving.com/community/c4h3127970_australian_math_competition_2023 and despite having 120 views since then nobody corrected it.
Even without access to the answer key, and without making any attempt at all to solve this question one can easily check the oeis at https://oeis.org/A134438 and that gives answer 170.
aops is usually held in high regard, but this time I am rather dispirited about it. So I put a message there putting them on the right track.
Here is the question
Here is what I put for the solution:
I also gave the answer key to aops so they can check that way too that my one is the correct snswer.
Even without access to the answer key, and without making any attempt at all to solve this question one can easily check the oeis at https://oeis.org/A134438 and that gives answer 170.
aops is usually held in high regard, but this time I am rather dispirited about it. So I put a message there putting them on the right track.
Here is the question
Here is what I put for the solution:
I also gave the answer key to aops so they can check that way too that my one is the correct snswer.
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I put my 2023 AMC Junior solutions here: https://4unitmaths.com/amc-2023-jun-sol.pdf
The AMT still haven't published their solutions.
However I think my solutions concur with the answers in the answer key
The AMT still haven't published their solutions.
However I think my solutions concur with the answers in the answer key
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There was a typo in 28.
- should have been x
Correct version is up at same link if you got it 18 minutes ago, just use same link or refresh page.
I'll check again tomorrow for any other typos and maybe start on the up and mp papers.
- should have been x
Correct version is up at same link if you got it 18 minutes ago, just use same link or refresh page.
I'll check again tomorrow for any other typos and maybe start on the up and mp papers.
I don't think year 12 have a significant advantage over year 11 for the senior paper, or year 10 have a significant advantage over year 9 for the intermediate paper.
But year 8 probably do have an advantage over year 7 for the junior paper. What do you think?
But year 8 probably do have an advantage over year 7 for the junior paper. What do you think?
amt have finally published the 2023 solutions: https://shop.amt.edu.au/products/amc-2023-solutions-pdf
It’s about 4 months later than before when they used to come out in January
It’s about 4 months later than before when they used to come out in January
