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