Australian Maths Competition (1 Viewer)

Mongoose528

Member
Joined
Jun 30, 2016
Messages
72
Location
WA
Gender
Male
HSC
2019
How would you solve this algebraically: A nude number is a natural number of whose digits are a factor of the number. Find all 3 digit nude numbers where no digits are repeated.

Can't seem to make in roads :/

I can solve it case by case, but I want to know a quicker way of how to do it.
Bump.
 

jathu123

Active Member
Joined
Apr 21, 2015
Messages
356
Location
Sydney
Gender
Male
HSC
2017
Re: Westpac Maths Comp marathon

Hi can anyone solve question 27 of the 2013 Senior AMC paper. I have tried many different methods but just cannot get it to work. I've tried finding the ratio of the areas/solving the areas simultaneously and all that. Please someone help as I've been trying to figure it out for a fortnight now. Show Working
Refer to the previous posts
Here's a solution to one of them:

View attachment 34157
 

Mongoose528

Member
Joined
Jun 30, 2016
Messages
72
Location
WA
Gender
Male
HSC
2019
A bit of help on this question please, this is a bit of practice for the AIMO.

11.JPG

This is all the working I've managed to do

working.JPG
 
Last edited:

Ashaaz

New Member
Joined
Mar 25, 2018
Messages
12
Gender
Male
HSC
2022
Australian Mathematics Competition

I was just wondering if anyone has past papers and answers for this competition as I would like to make an start to practicing for the test. Year 8
 

tywebb

dangerman
Joined
Dec 7, 2003
Messages
1,712
Gender
Undisclosed
HSC
N/A
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)
 
Last edited:

tywebb

dangerman
Joined
Dec 7, 2003
Messages
1,712
Gender
Undisclosed
HSC
N/A
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:

 
Last edited:

LookingGood2

New Member
Joined
Jul 30, 2023
Messages
7
Gender
Undisclosed
HSC
2019
The AMC past papers are available for free download at <Had to remove the original link to post reply> 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 <Had to remove the original link to post reply> 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 <Had to remove the link to post reply>

The solutions can also be purchased at <Had to remove the link to post reply> 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)
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.
 

tywebb

dangerman
Joined
Dec 7, 2003
Messages
1,712
Gender
Undisclosed
HSC
N/A
Here is 25.

25.png

Here is my solution.

25sol.png

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.
 

Attachments

Last edited:

tywebb

dangerman
Joined
Dec 7, 2003
Messages
1,712
Gender
Undisclosed
HSC
N/A
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.
 
Last edited:

tywebb

dangerman
Joined
Dec 7, 2003
Messages
1,712
Gender
Undisclosed
HSC
N/A
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.
 
Last edited:

tywebb

dangerman
Joined
Dec 7, 2003
Messages
1,712
Gender
Undisclosed
HSC
N/A
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

int30.png

Here is what I put for the solution:
amc2023int30.png
I also gave the answer key to aops so they can check that way too that my one is the correct snswer.
 
Last edited:

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top