Problem K. 293. (March 2011)

K. 293. Fred's cows are grazing on a field. Each cow eats the same daily amount of grass, independently of the actual number of cows on the field. One day, Fred took 6 cows to the field. It took the cows 3 days to eat all the grass on the field, so at the end of the third day Fred had to withdraw his cows from the field to let the grass grow back. When the original amount of grass was restored, Fred took 3 cows to the field. He was surprised to observe that this time it took 7 days for the cows to eat up all the grass. He was so puzzled that he asked Ben who was good at maths. Ben reminded him that he had forgotten about something while watching the cows graze and counting the days. What did Fred forget, and how long would it take a single cow to eat all the grass in the field?

(6 pont)

Deadline expired on April 11, 2011.

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

Megoldás. Frédi arról feledkezett meg, hogy a fű folyamatosan nő.

A tehenek naponta \(\displaystyle k\) kazal füvet esznek, miközben a fű \(\displaystyle v\) kazalnyival nő, a legelőn \(\displaystyle N\) kazalnyi fű van a legelés kezdetén. Ezért \(\displaystyle N-3\cdot 6k+3v=0\) összefüggés adható a hat tehén legelésére, míg \(\displaystyle N-7\cdot 3k+7v=0\) összefüggés, amikor csak három tehén legelt. E kettő különbségéből \(\displaystyle 3k-4v=0\), azaz \(\displaystyle v=\frac34 k\). Ez azt is jelenti, hogy a legelőn \(\displaystyle N=\frac{63}4 k\) fű van legelés előtt. Ha egy tehén legeli le \(\displaystyle n\) nap alatt a legelőt, akkor \(\displaystyle N-nk+nv=\frac{63}4 k-nk+n(\frac34 k)=0\), ahonnan \(\displaystyle n=63\). Egy tehén 63 nap alatt legeli le a füvet egyedül.

Statistics:

105 students sent a solution.
6 points:	Antalicz Balázs, Arnold Balázs, Árvay Júlia, Balogh Tamás, Csibi Levente, Csóti Annamária, Déri Tamás, Francsics Fanny, Gosztonyi Dorottya, Gömbös Patrik, H. Kovács Bálint, Helfrich Réka, Imre Nóra, Jónás Judit, Kecskeméti Ádám, Kerner Bálint, Kling József, Kolláth Mihály Gábor, Kóródi Brúnó Zoltán, Kovács Bence, Kovács Norbert Krisztián, Kulcsár Ildikó, Lévai Botond Miklós, Makk László, Mihály 930 Ákos, Móricz Tamás, Németh Klára Anna, Pajor Péter, Rácz 413 Bence, Rikker Bálint, Rovó Judit, Sárvári Péter, Sipos 320 László, Somogyvári Kristóf, Suhai Kristóf, Szabó Zsófia Judit, Székely Ádám, Szendrei Judit, Szöllős Kristóf, Tarnay Mátyás, Temesvári Fanni, Tihanyi Dániel, Tóth Kristóf, Tragor Viktória, Várkonyi Balázs, Vető Bálint, Xie Wu Di.
5 points:	Vörös Zoltán János.
4 points:	5 students.
3 points:	1 student.
2 points:	2 students.
1 point:	28 students.
0 point:	7 students.
Unfair, not evaluated:	14 solutionss.

Problems in Mathematics of KöMaL, March 2011