Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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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.

Statistics:

 73 students sent a solution. 3 points: 55 students. 2 points: 14 students. 1 point: 4 students.

Problems in Mathematics of KöMaL, April 2011