Problem K. 400. (December 2013)

K. 400. Steve and Kate are running around the football field, and measuring their times. Steve's stopwatch is set to lap times, that is, whenever he presses the button the watch will store the actual time reading and start the measurement from zero again. Kate's stopwatch is set differently: whenever she presses the button, the watch stores the actual time reading, but it will then continue the measurement without resetting zero. The two of them ran four laps together. Unfortunately, Kate forgot to press the button at the ends of the first and third laps. Thus she only had two readings at the end: 164 seconds and 340 seconds. Based on his own watch, Steve told her that they had run the first lap 5 seconds faster than the mean of the four lap times, and that the mean of the second and third lap times was 4 seconds more than the fourth lap time. How long did they take to cover the fourth lap?

(6 pont)

Deadline expired on January 10, 2014.

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

Megoldás. A négy köridő összesen 340 másodperc, így a köridők átlaga 85 másodperc. Az első kör 80 másodperc, a második pedig \(\displaystyle 164-80=84\) másodperc lett. A harmadik és negyedik köridő összege \(\displaystyle 340-80-84=176\) másodperc. Ha a harmadik köridőt \(\displaystyle x\)-szel jelöljük, akkor a \(\displaystyle \frac{84+x}{2}=176-x+4\) egyenlet írható fel. Rendezve \(\displaystyle 84+x = 360 – 2x\), innen \(\displaystyle x = 92\), tehát a negyedik köridő \(\displaystyle 176-92=84\) másodperc. Ellenőrizhető, hogy a 80, 84, 92, 84 másodperces körök eleget is tesznek a feltételeknek.

Statistics:

196 students sent a solution.
6 points:	154 students.
5 points:	7 students.
4 points:	14 students.
3 points:	3 students.
2 points:	6 students.
1 point:	1 student.
0 point:	6 students.
Unfair, not evaluated:	5 solutionss.

Problems in Mathematics of KöMaL, December 2013