Mathematical and Physical Journal
for High Schools
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Problem K. 410. (February 2014)

K. 410. In a regular triangular lattice, a regular triangle is drawn with its sides lying on lattice lines. There are 5995 lattice points in the interior of the triangle. How many lattice points were touched by the pencil drawing the triangle?

(6 pont)

Deadline expired on March 10, 2014.


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

Megoldás. Az 1++n=5995 egyenlet megoldására van szükségünk: \frac{(n+1)n}{2}=5995, ebből n2+n-11990=0, vagyis n=\frac{-1\pm\sqrt{1+47960}}{2}=\frac{-1\pm219}{2}. A negatív megoldás nem jó, tehát n=109. A háromszög egy oldalán így 109+3=112 rácspont van. A 112+112+112 összegben a csúcsokban levőket kétszer számoljuk, így a határvonalon összesen 333 rácspont van.


Statistics:

134 students sent a solution.
6 points:83 students.
5 points:17 students.
4 points:7 students.
3 points:12 students.
2 points:3 students.
1 point:2 students.
0 point:2 students.
Unfair, not evaluated:8 solutions.

Problems in Mathematics of KöMaL, February 2014