Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem C. 819. (September 2005)

C. 819. A fly is sitting at the centre K of a regular hexagon ABCDEF, there is another fly at vertex B and a spider sitting at vertex A. The flies start clawling from B towards C and from K towards E simultaneously at the same speed. (The spider stays in placed.) Show that the three of them form a regular triangle at every time instant.

(5 pont)

Deadline expired on October 17, 2005.

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

Megoldás. Legyen egy adott pillanatban az egyik légy a BC szakaszon a P pontban, a másik légy a KE szakaszon a Q pontban. Bizonyítandó, hogy APQ szabályos háromszög. Mivel AB=AK, BP=KQ, és ABP\angle=AKQ\angle=120o, ezért ABP_{\triangle}\cong AKQ_{\triangle}, vagyis AP=AQ. Az egybevágóság miatt BAP\angle=KAQ\angle, így KAB\angle=QAP\angle=60o. Az AP=AQ és a QAP\angle=60o igazolja, hogy APQ szabályos háromszög.


546 students sent a solution.
5 points:317 students.
4 points:38 students.
3 points:21 students.
2 points:13 students.
1 point:14 students.
0 point:132 students.
Unfair, not evaluated:11 solutions.

Problems in Mathematics of KöMaL, September 2005