# Extracurricular Elementary Mathematics Marathon (1 Viewer)

##### -insert title here-
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.

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• leehuan

#### InteGrand

##### Well-Known Member
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.

• VBN2470

16

#### InteGrand

Correct!

##### -insert title here-
The answer, has already been posted, but here, we will prove uniqueness.

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Hint: Use Fermat's Little Theorem.

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#### InteGrand

##### Well-Known Member
The answer, has already been posted, but here, we will prove uniqueness.

• VBN2470

##### -insert title here-
That doesn't prove uniqueness of b, it just restricts the value range of b to the solution interval without proof...

#### Drsoccerball

##### Well-Known Member
Isn't this just the Advanced X2 Marathon ?

#### InteGrand

##### Well-Known Member
That doesn't prove uniqueness of b, it just restricts the value range of b to the solution interval without proof...

• VBN2470

##### -insert title here-
Isn't this just the Advanced X2 Marathon ?
No, it is unrestricted from the arbitrary bounds placed on it. I can't just post any Olympiad problem on there, but I can do so here.

#### leehuan

##### Well-Known Member
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.

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##### -insert title here-
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

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
Solve the equation x^2+y^2+1=xyz over the positive integers.

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

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##### -insert title here-
Solve the equation x^2+y^2=xyz over the positive integers.

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

#### GoldyOrNugget

##### Señor Member
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
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.

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