Problem C. 1399. (February 2017)

C. 1399. In a 50-metre running race, if Martin gives a 4-metre advantage to Bill, he will just catch up with him at the finish line. If Bill gives 15 metres of advantage to Henry in a 200-metre race, they will finish side by side. How many metres of advantage may Martin give to Henry in a 1000-metre race so that the two of them finish together? (Assume that each of the three runners maintains the same constant running speed throughout the races.)

(5 pont)

Deadline expired on March 10, 2017.

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

Megoldás. Legyen Marci sebessége \(\displaystyle v\).

Mivel Bálint csak 46 métert tesz meg, míg Marci 50-et, ezért Bálint sebessége \(\displaystyle \frac{46}{50}v\).

Henrik 185 métert tesz meg, míg Bálint 200-at, így Henrik sebessége \(\displaystyle \frac{185}{200}\cdot\frac{46}{50} v=\frac{851}{1000} v\).

Marci az 1000 méteres távot \(\displaystyle t=\frac{1000}{v}\) idő alatt teljesíti. Ennyi idő alatt Henrik \(\displaystyle s=\frac{851}{1000} v\cdot\frac{1000}{v}=851\) métert tesz meg, 149 méterrel kevesebbet, mint Marci.

Tehát Marci 149 méter előnyt adhat Henriknek, hogy éppen együtt érjenek célba.

Statistics:

183 students sent a solution.
5 points:	140 students.
4 points:	1 student.
3 points:	6 students.
2 points:	30 students.
1 point:	3 students.
0 point:	2 students.
Unfair, not evaluated:	1 solutions.

Problems in Mathematics of KöMaL, February 2017