Problem K. 273. (December 2010)

K. 273. There is a mouse in a long straight tube, at 3/8 of the length. A cat is sitting at a point along the extension of the line of the tube, closer to the end of the tube that the mouse is also closer to. The cat notices the mouse, and starts to run towards the tube. At the same instant, the mouse also starts to run towards one of the ends of the tube. (Both of them run at uniform speeds.) However, the mouse cannot escape: No matter which end of the tube he chooses, the cat will just catch him at the endpoint of the tube. By what factor does the cat run faster than the mouse?

(6 pont)

Deadline expired on January 10, 2011.

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

Megoldás. Használjuk a következő jelöléseket: a Macska \(\displaystyle l\) távolságra van a cső végétől és \(\displaystyle v\) sebességgel fut. Az Egér a \(\displaystyle d\) hosszú cső belsejében ül és \(\displaystyle u\) sebességgel menekül. Az üldözés időtartama akkor, ha egymás felé kezdenek el futni \(\displaystyle \frac lv =\frac{3 \over 8 d}u\), ha az Egér a távolabbi csővég felé kezd menekülni, akkor \(\displaystyle \frac{l+d}v=\frac{5 \over 8 d}u\). Az elsőből \(\displaystyle l=\frac vu \cdot \frac 38 d\), amit a második egyenlőségbe beírva \(\displaystyle \frac vu \cdot \frac 38 d +d=\frac vu \cdot \frac 58 d\), ahonnan (\(\displaystyle d\ne 0\)) \(\displaystyle \frac vu=4\). A Macska négyszer olyan gyorsan fut, mint az Egér.

Statistics:

222 students sent a solution.
6 points:	176 students.
5 points:	6 students.
4 points:	7 students.
3 points:	6 students.
2 points:	3 students.
1 point:	10 students.
0 point:	8 students.
Unfair, not evaluated:	6 solutionss.

Problems in Mathematics of KöMaL, December 2010