Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
 Already signed up? New to KöMaL?

# Problem B. 4304. (November 2010)

B. 4304. Is there a positive integer k, such that ?

(3 pont)

Deadline expired on December 10, 2010.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Nincsen ilyen $\displaystyle k$ szám. Legyen $\displaystyle a_k=(\dots((3\underbrace{!)!)!\dots)!}_{k}$, $\displaystyle b_k=(\dots((4\underbrace{!)!)!\dots)!}_{k}$. A $\displaystyle k=1$ esetben $\displaystyle a_2=(3!)!=6!>4!=b_1$. Ha pedig valamely $\displaystyle k$ pozitív egészre $\displaystyle a_{k+1}>b_k$ pozitív egészek, akkor $\displaystyle a_{k+2}=a_{k+1}!>b_k!=b_{k+1}$ is teljesül. Így a teljes indukció elve szerint minden $\displaystyle k$ pozitív egészre

$\displaystyle (\dots((3\underbrace{!)!)!\dots)!}_{k+1}> (\dots((4\underbrace{!)!)!\dots)!}_{k}\ .$

### Statistics:

 209 students sent a solution. 3 points: 176 students. 2 points: 25 students. 0 point: 1 student. Unfair, not evaluated: 7 solutions.

Problems in Mathematics of KöMaL, November 2010