Problem K. 367. (February 2013)

K. 367. A school organized an ice cream building competition for the students. The participants built their ice cream towers on 10-cm tall cones, by placing the scoops one by one on top of each other. Initially, each scoop was a spherical ball of 4-cm diameter, but they were compressed under the weight of the overlying balls. The height of a ball decreased by 1 mm owing to each ball on top of it. The prize winning ice cream tower was 47.5 cm tall, measured from the bottom of the cone to the top of the uppermost ball, and one third of the height of the lowermost ball was inside the cone. How many balls were used to build this tower?

(6 pont)

Deadline expired on March 11, 2013.

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

Megoldás. A legfelső fagyigombóc magassága 4 cm, az alatta levőé 3,9 cm, az alatta levőé 3,8 cm, és így tovább. Ha az első 10 ilyen számot összeadjuk (4-től 3,1 cm-ig), akkor ezek összege 35,5 cm, ami még a 10 cm-es tölcsér teljes magasságával is 2 cm-rel kisebb, mint a győztes fagyi. Ha azonban az első 11-et vesszük figyelembe, akkor a 11. gombóc magassága 3 cm, ennek harmada van a tölcsérben, tehát 2 cm-rel lóg fölé, így pont megfelel a feltételeknek. Tehát a győztes fagyi 11 gombócból épült fel.

Statistics:

129 students sent a solution.
6 points:	60 students.
5 points:	15 students.
4 points:	22 students.
3 points:	11 students.
2 points:	6 students.
1 point:	5 students.
0 point:	10 students.

Problems in Mathematics of KöMaL, February 2013