Problem C. 1097. (November 2011)

C. 1097. A motorcycle has brand new tyres on both wheels. A tyre is considered worn if it has run 15 000 km on the rear wheel or 25 000 km on the front wheel. What is the maximum possible number of kilometres that the motorcycle can run until the tyres become worn if the front and rear tyres are interchanged at the appropriate time?

(5 pont)

Deadline expired on December 12, 2011.

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

Megoldás. Akkor kopik el egyszerre a két kerék, ha ugyanakkora utat futnak elöl és hátul. Legyen ez az út \(\displaystyle \frac{s}{2}\).

Ha az első kerék kopása 1 km alatt \(\displaystyle 1/25000\), akkor \(\displaystyle \frac{s}{2}\) km alatt \(\displaystyle \frac{s}{2}\cdot\frac{1}{25000}\). A hátsó kerék kopása 1 km alatt \(\displaystyle 1/15000\), így \(\displaystyle \frac{s}{2}\) km alatt \(\displaystyle \frac{s}{2}\cdot\frac{1}{15000}\).

A gumi kopására felírható a következő egyenlet:

\(\displaystyle \frac{s}{2}\cdot\frac{1}{25000}+\frac{s}{2}\cdot\frac{1}{15000}=1.\)

Ebből:

\(\displaystyle 3s+5s=150000,\)

\(\displaystyle s=18750.\)

Tehát legfeljebb 18750 km-t futhat a motor.

Kedves Máté (Mohács, Kisfaludy Károly Gimn., 11. o. t.)

Statistics:

311 students sent a solution.
5 points:	197 students.
4 points:	67 students.
3 points:	14 students.
2 points:	7 students.
1 point:	11 students.
0 point:	12 students.
Unfair, not evaluated:	3 solutionss.

Problems in Mathematics of KöMaL, November 2011