International Mathematical Olympiad $2019$ , Day $2$ , Problem $4$ of $6$

Algebra Level 2

Note - Supposed to set this $3$ weeks ago but never mind...

Question:

Find all pairs $(k, n)$ of positive integers such that:

$k! = (2^n - 1)(2^n - 2)(2^n - 4)$ ... $(2^n - 2^{n - 1})$

Country that gave the Question: El Salvador

Give your answer as the number of pairs $(k, n)$ of positive integers that satisfies the question.

The person that answers this correctly and gives the official solution first - there is $2$ - go to International Mathematical Olympiad (IMO) Hall of Fame .

The answer is 2.

3 solutions

A Former Brilliant Member
Jun 15, 2020

Since for $k\geq 4$ all the odd numbers will not be present in the R. H. S. of the identity for any value of $n$ , this side can never be expressed as the factorial of a number. Hence $0\leq k\leq 3$ . In this range, only two pairs, namely $(k, n)=(1,1)$ and $(3,2)$ exist. So the total number of such pairs is $\boxed 2$

Good solution! Not the official one though... @Alak Bhattacharya .

P.S. I have the official solutions...

A Former Brilliant Member - 12 months ago

A Former Brilliant Member
Jun 16, 2020

Do you have the official solutions? @Hamza Anushath

A Former Brilliant Member - 12 months ago

Not exactly, but I know the format of how to submit the answers

Why @Yajat Shamji ?

A Former Brilliant Member - 12 months ago

Oh, just that if anybody (apart from me) has the official solutions and uses it, that's cheating.

Luckily, you don't have them, otherwise...

A Former Brilliant Member - 12 months ago

@A Former Brilliant Member – Hahaha, if I had them, I would have used them for every question you posted.

Anyway, from where did you get the text from? I thought the solution were a big book, not a PDF?

A Former Brilliant Member - 12 months ago

@A Former Brilliant Member – It's a PDF...

A Former Brilliant Member - 12 months ago

I didn't know how to type some symbols in latex, so I typed them somewhere else and took a screenshot

A Former Brilliant Member - 12 months ago

Ok. I will change the date then in the Hall of Fame...

A Former Brilliant Member - 12 months ago

A Former Brilliant Member
Jun 16, 2020

Solution $1$ : Solution $2$ :

Ok. I wanted submit the solutions and not the number of solutions.

A Former Brilliant Member - 12 months ago

There is $2$ , that's why I put $2$ ...

A Former Brilliant Member - 12 months ago

Oh god, I was just going to post a solution @Yajat Shamji

A Former Brilliant Member - 12 months ago

Did you not see the Hall of Fame? Last time of submission for solutions for recognition is $23:58$ pm on the day of submitting the problem.

A Former Brilliant Member - 12 months ago

That's true @Yajat Shamji , yet trial is necessary, isn't it?

A Former Brilliant Member - 12 months ago

And wich timezone?

A Former Brilliant Member - 12 months ago