• Best of luck to the class of 2019 for their HSC exams. You got this!
    Let us know your thoughts on the HSC exams here
  • Help the next generation of students in the community for the new syllabus!
    Share your notes and exam papers on our Notes & Resources page

Extracurricular Elementary Mathematics Marathon (1 Viewer)

Paradoxica

-insert title here-
Joined
Jun 19, 2014
Messages
2,479
Location
Outside reality
Gender
Male
HSC
2016
Rules are as per the other marathons. Difficulty should be reasonable, and hints should be provided as deemed necessary. Use any "elementary" techniques, provided you state them, and if it's fairly advanced, outline a proof.

I'll start off simple.



Other things:

For the purposes of this thread, the set of all natural numbers excludes zero.

Vectors and elementary functions of any kind are allowed, but little to no calculus. (this one due to leehuan)

Define terms that the average person following this thread probably wouldn't know.

State any theorems/techniques that may help in solving the problem.
 
Last edited:

InteGrand

Well-Known Member
Joined
Dec 11, 2014
Messages
6,104
Gender
Male
HSC
N/A
Rules are as per the other marathons. Difficulty should be reasonable, and hints should be provided as deemed necessary. Use any "elementary" techniques, provided you state them, and if it's fairly advanced, outline a proof.

I'll start off simple.





















 

leehuan

Well-Known Member
Joined
May 31, 2014
Messages
5,807
Gender
Male
HSC
2015
Isn't this just the Advanced X2 Marathon ?
If this were my thread I would allow hyperbolic functions and elementary vector notation now, however probably a bit of guidance as to how they work as we haven't attended a lecture yet.
 
Last edited:

Paradoxica

-insert title here-
Joined
Jun 19, 2014
Messages
2,479
Location
Outside reality
Gender
Male
HSC
2016
If this were my thread I would allow hyperbolic functions and elementary vector notation now, however probably a bit of guidance as to how they work as we haven't attended a lecture yet.
Olympiad allows for that, but I've never seen it in good use. Except for vectors, those things are really useful for a good bunch of problems.
 

seanieg89

Well-Known Member
Joined
Aug 8, 2006
Messages
2,671
Gender
Male
HSC
2007


Hint: Use Fermat's Little Theorem.
p-1 = mn + r for non-negative integers m, r with r < n.

Then (a^r)(a^n)^m=a^(p-1)
=> a^r=1 (using FLT)

From the minimality of n, we must conclude r=0.

That is n|p-1.
 

seanieg89

Well-Known Member
Joined
Aug 8, 2006
Messages
2,671
Gender
Male
HSC
2007
Solve the equation x^2+y^2+1=xyz over the positive integers.

Hint: First concentrate on determining what possibilities there are for z.
 
Last edited:

GoldyOrNugget

Señor Member
Joined
Jul 14, 2012
Messages
583
Gender
Male
HSC
2012
Trivial for n=1.

Suppose the property holds for 1..n-1.

For general n, if their sum is divisible by n, then their sum mod n is 0. Take all elements mod n. If all elements mod n are 0, pick any subset. Otherwise, pick out an element x mod N which is nonzero mod n. 0 <= x < n. Of the remaining elements, take any n-x > 0 elements. These n-x elements have a subset divisible by n-x, so if we then add x to the subset, we have a subset with sum (n-x) + x = 0 mod N, so the subset is divisible by n.
 

seanieg89

Well-Known Member
Joined
Aug 8, 2006
Messages
2,671
Gender
Male
HSC
2007
Nice :). I actually forgot a term on the LHS from the diophantine equation I was trying to remember though lol, try the edited problem too.

It is slightly harder, but not greatly so. The original hint still stands.

Repost for visibility:

Solve the equation:



over the positive integers.

Hint: Try to determine what z can be first.
 
Last edited:

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

Top