Problem C. 1051. (November 2010)
C. 1051. A natural number n is chosen between two consecutive square numbers. The smaller square is obtained by subtracting k from n, and the larger one is obtained by adding l to n. Prove that the number n-kl is a perfect square.
(5 pont)
Deadline expired on December 10, 2010.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. A két egymást követő négyzetszám legyen \(\displaystyle x^2\) és \(\displaystyle (x+1)^2\). Így a feladat szerint \(\displaystyle n, k, l\)-re \(\displaystyle x^2+k=n\) és \(\displaystyle n+l=(x+1)^2\). Kifejezve \(\displaystyle k\)-t és \(\displaystyle l\)-t \(\displaystyle k=n-x^2\) és \(\displaystyle l=(x+1)^2-n\) a vizsgálandó kifejezés \(\displaystyle n-kl=n-(n-x^2)((x+1)^2-n)=n-(n(x+1)^2)-n^2-x^2(x+1)^2+nx^2)=n-n(2x^2+2x+1)+n^2+(x(x+1))^2=(x(x+1)-n)^2\), ami valóban négyzetszám.
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210 students sent a solution. 5 points: 168 students. 4 points: 22 students. 3 points: 4 students. 2 points: 4 students. 1 point: 7 students. 0 point: 2 students. Unfair, not evaluated: 3 solutionss.
Problems in Mathematics of KöMaL, November 2010