Problem K. 689. (March 2021)

K. 689. In the 6th, 7th, 8th and 9th games of the season, a basketball player scored 23, 14, 11 and 20 points, respectively. His points average was higher after the 9th game than after the 5th game. With the 10th game, his average rose above 18. What is the lowest possible number of points that he may have scored in the 10th game?

(6 pont)

Deadline expired on April 12, 2021.

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

Megoldás: Ha a kosárlabdázó átlaga az első 5 mérkőzés után \(\displaystyle x\) volt, akkor a 9 mérkőzés után az átlaga \(\displaystyle \frac{5x+68}{9}\) lett. Mivel \(\displaystyle x<\frac{5x+68}{9}\), ezért \(\displaystyle x<17\). A 10. mérkőzés után az átlaga 18 fölé ment, tehát ha a 10. mérkőzésen \(\displaystyle y\) pontot szerzett, akkor az átlagára \(\displaystyle 18<\frac{5x+68+y}{10}\) teljesül. Ebből \(\displaystyle 112<5x+y\). Mivel \(\displaystyle 5x<85\), és \(\displaystyle 5x\) egész, ezért \(\displaystyle 5x\leq84\). Tehát a feltétel teljesüléséhez \(\displaystyle 112-84=28<y\) szükséges. Tehát legalább \(\displaystyle 29\) pontot kellett szereznie a 10. mérkőzésen.

Statistics:

102 students sent a solution.
6 points:	64 students.
5 points:	6 students.
4 points:	9 students.
3 points:	6 students.
2 points:	9 students.
1 point:	5 students.
0 point:	1 student.
Not shown because of missing birth date or parental permission:	2 solutions.

Problems in Mathematics of KöMaL, March 2021