Mathematical and Physical Journal
for High Schools
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Problem C. 1176. (September 2013)

C. 1176. a, b, c, d, e are five consecutive integers in increasing order. The dimensions of a cuboid are a, bc. For what values will the diagonal of the cuboid be the hypotenuse of a right-angled triangle with legs d and e?

(5 pont)

Deadline expired on October 10, 2013.


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

Megoldás. Az egymást követő öt egész szám legyen \(\displaystyle n-2\), \(\displaystyle n-1\), \(\displaystyle n\), \(\displaystyle n+1\) és \(\displaystyle n+2\). A téglatest testátlójának mérőszáma: \(\displaystyle \sqrt{(n-2)^2+(n-1)^2+n^2}=\sqrt{3n^2-6n+5}\). A szöveg szerint teljesülni kellene, hogy \(\displaystyle 3n^2-6n+5=(n+1)^2+(n+2)^2=2n^2+6n+5\), amiből \(\displaystyle n^2-12n=0\). Ennek az egyenletnek az \(\displaystyle n=0\) és az \(\displaystyle n=12\) a két gyöke, de a feladat feltételeinek csak az \(\displaystyle n=12\) felel meg. A keresett számok: 10, 11, 12, 13, 14.


Statistics:

215 students sent a solution.
5 points:142 students.
4 points:43 students.
3 points:7 students.
2 points:1 student.
1 point:5 students.
0 point:13 students.
Unfair, not evaluated:4 solutions.

Problems in Mathematics of KöMaL, September 2013