Mathematical and Physical Journal
for High Schools
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Problem C. 1159. (February 2013)

C. 1159. Consider all non-congruent rectangles whose side lengths are two distinct elements selected from the set \{1; 2; \ldots; 100\}. What is the total area of all such rectangles?

(5 pont)

Deadline expired on March 11, 2013.


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

Megoldás. A feladatot tetszőleges \(\displaystyle n\) pozitív egész esetére oldjuk meg. A területösszegben az 1, 2, ..., \(\displaystyle n\) számok egymással vett szorzatai szerepelnek, kivéve a négyzetszámokat. Vagyis a keresett összeg:

\(\displaystyle \frac{(1+2+\dots+n)^2-(1^2+2^2+\dots+n^2)}{2}=\frac12\cdot\left(\left(\frac{(n+1)n}{2}\right)^2-\frac{n(n+1)(2n+1)}{6}\right)=\)

\(\displaystyle =\frac{(n+1)^2n^2}{8}-\frac{n(n+1)(2n+1)}{12}.\)

Ha \(\displaystyle n\) helyébe behelyettesítjük a 100-at, akkor az eredmény: 12582075.


Statistics:

164 students sent a solution.
5 points:75 students.
4 points:22 students.
3 points:23 students.
2 points:15 students.
1 point:12 students.
0 point:7 students.
Unfair, not evaluated:10 solutions.

Problems in Mathematics of KöMaL, February 2013