Problem B. 4357. (April 2011)
B. 4357. Let n>1 be a positive integer. Show that n3-n2 is a factor of the binomial coefficient .
(Suggested by G. Holló, Budapest)
(3 pont)
Deadline expired on May 10, 2011.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. A szóban forgó oszthatóság leolvasható az
\(\displaystyle \binom{n^2}{n+1}= \frac{n^2(n^2-1)(n^2-2)\cdots(n^2-n)}{(n+1)n(n-1)\cdots1}= \)
\(\displaystyle =\frac{n^2(n^2-1)(n^2-n)}{(n+1)n(n-1)}\cdot \frac{(n^2-2)\cdots(n^2-n+1)}{(n-2)\cdots1} =(n^3-n^2)\dbinom{n^2-2}{n-2}\)
átalakításról.
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73 students sent a solution. 3 points: 55 students. 2 points: 14 students. 1 point: 4 students.
Problems in Mathematics of KöMaL, April 2011