Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem C. 1117. (March 2012)

C. 1117. We have drawn a rectangle on squared paper, such that (its sides are lattice lines and) it consists of n small lattice squares. Prove that if the half of the number of lattice points on the boundary of the rectangle is added to the number of lattice points in its interior, and 1 is subtracted from the sum then the result will be n.

(5 pont)

Deadline expired on April 10, 2012.


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

Megoldás. Legyen a rácstéglalap oldalainak hossza \(\displaystyle a\) és \(\displaystyle b\) (\(\displaystyle a\), \(\displaystyle b\) pozitív egészek): a téglalap \(\displaystyle a\cdot b=n\) kis rácsnégyzetből áll, a határvonalán található rácspontok száma \(\displaystyle 4+2(a-1)+2(b-1)=2a+2b\), a belsejébe eső pontok száma pedig \(\displaystyle (a-1)(b-1)\). Tehát \(\displaystyle (a-1)(b-1)+\frac{2a+2b}{2}-1\) összeget kell kiszámolnunk, ami \(\displaystyle (ab-a-b+1)+(a+b)-1=ab=n\).


Statistics:

217 students sent a solution.
5 points:172 students.
4 points:28 students.
3 points:1 student.
2 points:5 students.
1 point:4 students.
0 point:3 students.
Unfair, not evaluated:4 solutions.

Problems in Mathematics of KöMaL, March 2012