Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 93. (October 2006)

K. 93. Sherlock Holmes was investigating a crime in the Musgrave property famous for a beautiful elm tree. Looking for the hidden treasure, he was to ``walk ten and ten steps to the north, five and five steps to the east, two and two steps to the south, and one and one step to the west.'' Holmes found out, however, that one of the directions was wrong, one should walk in the opposite direction instead (for example, south instead of north). Since he did not know which was wrong, he tried all possibilities. Each walk ended within the park of the Musgraves. Given that Holmes' steps are 80 cm long and the park is a rectangle with sides in the north-south and east-west directions, find the minimum possible area of the park.

(6 pont)

Deadline expired on November 10, 2006.

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

Megoldás. A kiindulási pontból észak felé akkor megy a legtöbbet, amikor a déli lépéseket is északra teszi meg, ez 24 lépést jelent észak felé. Déli irányba akkor megy a legtöbbet, ha az északi lépéseket is dél felé teszi meg, ez szintén 24 lépést jelent, dél felé. Tehát a legnagyobb észak-déli eltérés a végpontok között 48 lépés, azaz 38,4 méter. A legnagyobb kelet-nyugat irányú eltérést hasonlóan kapjuk, ez 24 lépés, azaz 19,2 méter. Az angolpark területe tehát legalább 737,28 m2.


247 students sent a solution.
6 points:74 students.
5 points:71 students.
4 points:24 students.
3 points:22 students.
2 points:23 students.
1 point:14 students.
0 point:16 students.
Unfair, not evaluated:3 solutionss.

Problems in Mathematics of KöMaL, October 2006