Mathematical and Physical Journal
for High Schools
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Problem K. 318. (December 2011)

K. 318. Prove that if a, b, c, d are consecutive natural numbers, then d2 is a factor of the sum a+b2+c3.

(6 pont)

Deadline expired on January 10, 2012.

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

Megoldás. Az első három szám legyen $\displaystyle d–3$, $\displaystyle d–2$, $\displaystyle d–1$. Ekkor az összeg: $\displaystyle d–3 + (d–2)^2 + (d–1)^3 = d – 3 + d^2 – 4d + 4 + d^3 – 3d^2 + 3d – 1 = d^3 – 2d^2 = d^2(d–2)$, azaz osztható $\displaystyle d$–vel.

Statistics:

 211 students sent a solution. 6 points: 139 students. 5 points: 17 students. 4 points: 14 students. 3 points: 9 students. 2 points: 6 students. 1 point: 5 students. 0 point: 5 students. Unfair, not evaluated: 16 solutions.

Problems in Mathematics of KöMaL, December 2011