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C. 941. There are two lamps on the ceiling of a rectangular classroom of length 10 m. They emit conical light beam of apex angles of 90o. One is situated in the middle of the ceiling, and illuminates a circle of diameter 6 m on the floor. The lampshade of the other lamp is turned so that the illuminated region on the floor is long enough to draw a line segment of length 10 m on it in the direction parallel to the length of the room, but no light falls on the walls at the ends of the room. Find the distance between the two lamps.

(5 points)

Deadline expired on 15 May 2008.

Google Translation (Sorry, the solution is published in Hungarian only.)

Megoldás. Az AGB háromszögben az AB oldallal szemközti szög, AGB=90o, így Thalesz tétele szerint .

Tudjuk, hogy CE=ED=3. Mivel a CFD háromszögben CD-vel szemben 90o-os szög található, ezért szintén a Thalesz-tétel miatt EF=3.

Végül az EFG háromszögre Pitagorasz tételét felírva kapjuk, hogy FG2=EG2-FE2=52-32=16, ahonnan FG=4.

Statistics on problem C. 941.
 145 students sent a solution. 5 points: 123 students. 4 points: 11 students. 3 points: 4 students. 1 point: 3 students. 0 point: 1 student. Unfair, not evaluated: 3 solutions.

• Problems in Mathematics of KöMaL, April 2008

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