K. 318. Prove that if a, b, c, d are consecutive natural numbers, then d2 is a factor of the sum a+b2+c3.
This problem is for grade 9 students only.
Deadline expired on 10 January 2012.
Google Translation (Sorry, the solution is published in Hungarian only.)
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.