Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 41. (March 2005)

K. 41. The half of the square of an integer and one sixth of the cube of the same integer are added to one third of the same integer. Prove that the result is an integer.

(6 pont)

Deadline expired on April 11, 2005.


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

Megoldás. Jelöljük az egész számot a-val. Ekkor {a\over3}+{a^2\over2}+{a^3\over6}={a^3+3a^2+2a\over6}={a(a^2+3a+2)\over6}={a(a+1)(a+2)\over6}. Mivel a számláló három egymást követő egész szám szorzata, ami mindig osztható 6-tal, ezért a tört értéke egész szám.


Statistics:

108 students sent a solution.
6 points:91 students.
5 points:4 students.
3 points:2 students.
2 points:2 students.
0 point:8 students.
Unfair, not evaluated:1 solution.

Problems in Mathematics of KöMaL, March 2005