Problem K. 565. (December 2017)

K. 565. Arthur Dumpling the fat bird (a popular Hungarian cartoon character) plans to make a new year's resolution on 31 December 2017. Starting with 1 January 2018, he would go on a special slimming diet. On each day, he needs to start by calculating how many bars of chocolate he is allowed to eat that day. He considers the number of the current day in the year 2018. If the day number is even then he may eat as many bars of chocolate as on the day with half the current day number. (For example, on the 26th day of the year he is allowed to eat as many bars as on day 13.) If the day number is odd and greater than 1, then he may eat one fewer as he will be allowed to eat on the following day. The slimming diet ends on 30 December. Given that Arthur will eat 3 bars of chocolate on 9 January, how many bars will he eat on 24 December 2018?

(6 pont)

Deadline expired on January 10, 2018.

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

Megoldás. Mivel az év \(\displaystyle 9.\) napján \(\displaystyle 3\) tábla csokit evett, ezért a \(\displaystyle 10.\) napon \(\displaystyle 4\) tábla csokit evett. Az \(\displaystyle 5\). napon is \(\displaystyle 4\) táblát, azaz a \(\displaystyle 6\). napon \(\displaystyle 5\) táblát, csak úgy, mint a \(\displaystyle 3\). napon. A \(\displaystyle 4\). napon \(\displaystyle 6\) tábla csokit evett és így a \(\displaystyle 2\). és az \(\displaystyle 1\). napon is. \(\displaystyle 2018\). december \(\displaystyle 24\)-e az év \(\displaystyle 365-7=358\). napja.

Ha ezen a napon \(\displaystyle x\) táblával eszik, akkor a \(\displaystyle 179\). napon is, a \(\displaystyle 180\). napon \(\displaystyle x+1\) táblával. A \(\displaystyle 90\)., \(\displaystyle 45\). napon is \(\displaystyle x+1\) táblával. A \(\displaystyle 46\). napon \(\displaystyle x+2\) táblával, a \(\displaystyle 23\). napon is \(\displaystyle x+2\) táblával, a \(\displaystyle 24\). napon pedig \(\displaystyle x+3\) táblával. A \(\displaystyle 12\)., \(\displaystyle 6\). és \(\displaystyle 3\). napon is \(\displaystyle x+3\) táblával.

\(\displaystyle x+3=5\), azaz \(\displaystyle x = 2\).

\(\displaystyle 2018\). december \(\displaystyle 24\)-én \(\displaystyle 2\) tábla csokit eszik.

Statistics:

136 students sent a solution.
6 points:	109 students.
5 points:	11 students.
4 points:	1 student.
3 points:	3 students.
2 points:	5 students.
0 point:	3 students.
Unfair, not evaluated:	4 solutionss.

Problems in Mathematics of KöMaL, December 2017