Mathematical and Physical Journal
for High Schools
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Problem B. 3939. (October 2006)

B. 3939. Find the angle encompassed by the hypotenuse of a right-angled triangle with circumference of 2 units, as seen from a point lying on the inner angle bisector of the right angle at a distance of \sqrt{2} from that vertex.

(4 pont)

Deadline expired on November 15, 2006.

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

Megoldás. Az ábra jelöléseit használva a+b+c=2 és a2+b2=c2, ahonnan a2+b2=(2-a-b)2, vagyis ab=2a+2b-2. Az ACD, illetve BCD háromszögekre a koszinusz-tételt felírva kapjuk, hogy

u2=b2-2b+2,  v2=a2-2a+2.




ahonnan az ADB szög koszinuszára


adódik. A háromszög átfogója tehát 45o-os szög alatt látszik a D pontból.


139 students sent a solution.
4 points:104 students.
3 points:14 students.
2 points:4 students.
1 point:12 students.
0 point:5 students.

Problems in Mathematics of KöMaL, October 2006