Problem K. 247. (March 2010)

K. 247. During the first three years of her life, Sleeping Beauty slept an average of 14 hours a day. Then, until she was 16 years old she slept 8 hours a day, and then from her 16th birthday onwards, when she wounded her finger with the reel, she slept 24 hours a day for 100 years. A filbert mouse hibernates for 5 months of the year (from 1 November to 31 March), and in the rest of the year it spends a daily average of 8 hours awake (during the nighttime hours). After how many years of sleep should the prince have woken up Sleeping Beauty so that her daily average of sleep from her birth to her waking up equals that of a filbert mouse? (For simplicity, ignore leap years, that is, assume that February is always 28 days long.)

(6 pont)

Deadline expired on April 12, 2010.

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

Megoldás. A mogyorós pele napi alvásátlaga: \(\displaystyle (151\cdot 24+(365-151) \cdot 16) : 365 = \frac{7048}{365} \approx 19,31\). Csipkerózsikával kapcsolatban a következőt írhatjuk fel: \(\displaystyle (3\cdot 14\cdot 365 + 13\cdot 8\cdot 365 + x\cdot 24\cdot 365) : ((16+x) \cdot 365) = \frac{7048}{365}\). Ebből rendezéssel az \(\displaystyle 53290+8760x = 112768+7048x\) egyenletet kapjuk, ahonnan \(\displaystyle x = \frac{29739}{856}\approx 34,74\). Tehát 34,74 év, azaz kb. 34 év és 270 napnyi alvás után kell felébreszteni.

Statistics:

107 students sent a solution.
6 points:	74 students.
5 points:	15 students.
4 points:	3 students.
3 points:	6 students.
2 points:	4 students.
1 point:	2 students.
Unfair, not evaluated:	3 solutionss.

Problems in Mathematics of KöMaL, March 2010