be a convex pentagon with 
and 
For each point 
in the plane, define 
The least possible value of 
can be expressed as 
where 
and 
are positive integers and 
is not divisible by the square of any prime. Find 
be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
.
Y by DeathLlama9, wu2481632, atmchallenge, GoJensenOrGoHome, 62861, Adventure10, Mango247
Hello guys,
I am back with, I hope, a good new. There will be in nearly one month the 5th edition of the "Oliforum contest". It is an individual and telematic contest, with some training problems at national/imo level. Maybe it can be useful to some forumers around here.
Some details below:
1- The contest is made by a unique round, starting in the early afternoon of Tuesday 31 January and ending at 23:59 of Sunday 05 February (Rome meridian +1GTM) (in practice, 5.5 days).
2- Every problem will have some points, from 0 to 7. One additional point could be given for each clear, correct, and original solution.
3- There are no age contraints.
4- Non-olympic arguments could be used, even if it is preferred to not do it.
5- It is enough to send the solutions by mail, at the address that you can find below. Enrollment is not necessary.
6- Time of incoming solutions will be considered only for ex-aequo positions.
7- How to send solutions.
- You need to send solutions to the following mail:
.
- Solutions need to be attached in a unique attached .pdf file with a reasonable size.
- The .pdf file must be written with
or in a way that can be easily understood.
- You have to rename the .pdf file with the nickname that you have here on MathLinks.
- Try to be clear everytime.
Previous editions: { Ed. 2008, round 1, round 2, round 3 } , { Ed. 2009, round 1, round 2 }, { Ed. 2012 round 1 }, { Ed. 2013 round 1 }.
I will add below the list of problems on the 7th February, together with the list with people who sent correctly their solutions. Comments and suggestions are welcome.
Cheers,
Paolo Leonetti
Italian version
I am back with, I hope, a good new. There will be in nearly one month the 5th edition of the "Oliforum contest". It is an individual and telematic contest, with some training problems at national/imo level. Maybe it can be useful to some forumers around here.
Some details below:
1- The contest is made by a unique round, starting in the early afternoon of Tuesday 31 January and ending at 23:59 of Sunday 05 February (Rome meridian +1GTM) (in practice, 5.5 days).
2- Every problem will have some points, from 0 to 7. One additional point could be given for each clear, correct, and original solution.
3- There are no age contraints.
4- Non-olympic arguments could be used, even if it is preferred to not do it.
5- It is enough to send the solutions by mail, at the address that you can find below. Enrollment is not necessary.
6- Time of incoming solutions will be considered only for ex-aequo positions.
7- How to send solutions.
- You need to send solutions to the following mail:
.- Solutions need to be attached in a unique attached .pdf file with a reasonable size.
- The .pdf file must be written with
or in a way that can be easily understood.- You have to rename the .pdf file with the nickname that you have here on MathLinks.
- Try to be clear everytime.
Previous editions: { Ed. 2008, round 1, round 2, round 3 } , { Ed. 2009, round 1, round 2 }, { Ed. 2012 round 1 }, { Ed. 2013 round 1 }.
I will add below the list of problems on the 7th February, together with the list with people who sent correctly their solutions. Comments and suggestions are welcome.
Cheers,
Paolo Leonetti
Italian version
This post has been edited 6 times. Last edited by bboypa, Feb 2, 2017, 5:39 PM
