Problem B. 4375. (September 2011)
B. 4375. Let a and b be the legs of a right-angled triangle, and let m be the height drawn to the hypotenuse c. Which line segment is longer, a+b or m+c?
Suggested by P. Székely, Budapest
(3 pont)
Deadline expired on October 10, 2011.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. A háromszög területét \(\displaystyle {1 \over2} ab\) és \(\displaystyle {1 \over 2}cm\) alakban is felírhatjuk. Így a Pithagorasz-tétel alapján
\(\displaystyle (a+b)^2=a^2+b^2+2ab=c^2+2cm<c^2+2cm+m^2=(c+m)^2,\)
vagyis \(\displaystyle a+b<m+c\).
Statistics:
272 students sent a solution. 3 points: 237 students. 2 points: 6 students. 1 point: 10 students. 0 point: 10 students. Unfair, not evaluated: 9 solutionss.
Problems in Mathematics of KöMaL, September 2011