Problem C. 867.

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C. 867. Starting from the origin in the usual Cartesian coordinate system, a broken line segment is drawn. We arrive back at the y-axis in every fourth step, see the figure.

Using a certain ball-pen and a coordinate system with unit length of 0.5 cm, we draw broken line segments of length 8000 meters as in the figure. Count how many times we arrive back at the y-axis.

(5 points)

Deadline expired.

Sorry, the solution is published in Hungarian only.

Megoldás.

$[(1+3+\ldots+(2n-1))\sqrt2+(5+11+\ldots+(6n-1))]\cdot\frac{1}{2}\leq800\,000,$

$n^2\sqrt2+3n^2+2n\leq1600\,000,$

$(3+\sqrt2)n^2+2n\leq1\,600\,000.$

Másodfokú egyenlőtlenség megoldása nélkül is kitalálható, hogy melyik a legnagyobb n egész szám, amire teljesül: n=601.

Statistics on problem C. 867.

459 students sent a solution.

5 points: 235 students.

4 points: 21 students.

3 points: 19 students.

2 points: 27 students.

1 point: 72 students.

0 point: 66 students.

Unfair, not evaluated: 19 solutions.

Problems in Mathematics of KöMaL, October 2006

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