IMO Shortlist 2005 problem N5
Kvaliteta:
Avg: 0,0Težina:
Avg: 8,0 Denote by
the number of divisors of the positive integer
. A positive integer
is called highly divisible if
for all positive integers
.
Two highly divisible integers
and
with
are called consecutive if there exists no highly divisible integer
satisfying
.
(a) Show that there are only finitely many pairs of consecutive highly divisible
integers of the form
with
.
(b) Show that for every prime number
there exist infinitely many positive highly divisible integers
such that
is also highly divisible.
![d(n)](/media/m/8/b/5/8b5ba2b86903af1640ec9f08b90773b6.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![d(n) > d(m)](/media/m/4/4/1/44189d116175cf2679b73a05efc572f4.png)
![m < n](/media/m/7/8/e/78efa867bae4b295622ec1d67daf002d.png)
Two highly divisible integers
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![m < n](/media/m/7/8/e/78efa867bae4b295622ec1d67daf002d.png)
![s](/media/m/9/0/8/908014cbadb69e42261a56b450a375b9.png)
![m < s < n](/media/m/d/6/0/d6088248935c3c7a445676a1b66e30dd.png)
(a) Show that there are only finitely many pairs of consecutive highly divisible
integers of the form
![(a, b)](/media/m/0/5/9/059101434c13a86d8297575c7ee676ea.png)
![a\mid b](/media/m/e/3/c/e3cedf8012f017366aa16aa65bde12aa.png)
(b) Show that for every prime number
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
![pr](/media/m/6/2/2/6223a946fab6923c089d139b5a6740bb.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2005