Problem K. 242. (February 2010)

K. 242. The maximum score on a test is 100 points. The scores of the students are recorded in a computer. When a new score is entered, the program immediately calculates the average of the scores entered so far. While entering the scores of the first five students, the teacher observed that the average increased by 3 points with every score entered. By how many points did the fifth student score more than the first one?

(6 pont)

Deadline expired on March 10, 2010.

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

Megoldás. Az első ember pontszáma legyen \(\displaystyle a\), így pontjait beírva a kijelzett átlag is \(\displaystyle a\) lesz. A negyedik ember után az átlag \(\displaystyle a+9\), az ötödik ember után az átlag \(\displaystyle a+12\). Az első négy ember összpontszáma tehát \(\displaystyle 4 \cdot (a+9)=4a+36\), mind az öt ember összpontszáma \(\displaystyle 5 \cdot (a+9)=5a+60\). Az utolsó ember pontszáma a két összpontszám különbsége, azaz \(\displaystyle a+24\). Tehát az ötödik embernek 24 ponttal volt több, mint az elsőnek.

Statistics:

161 students sent a solution.
6 points:	79 students.
5 points:	19 students.
4 points:	11 students.
3 points:	13 students.
2 points:	10 students.
1 point:	12 students.
0 point:	9 students.
Unfair, not evaluated:	8 solutionss.

Problems in Mathematics of KöMaL, February 2010