Mathematical and Physical Journal
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Problem B. 4304. (November 2010)

B. 4304. Is there a positive integer k, such that \big(\dots\big((4\underbrace{!)!\big)!\dots\big)!}_{k} >

(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

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


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

Problems in Mathematics of KöMaL, November 2010