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 solutionss.
Problems in Mathematics of KöMaL, November 2010