Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem K. 573. (January 2018)

K. 573. Kate, Alex and Steve went to the sweet shop. Kate bought 9 identical boxes of sweets for Christmas, but she only had $\displaystyle 11\,000$ forints (Hungarian currency) on her, so she borrowed all the change that Alex had. With that, she just had the right amount of money to pay for the sweets. Then Alex also thought that these sweets would make nice Christmas presents so he decided to buy 13 boxes of the same kind. Since he only had $\displaystyle 15\,000$ forints left now, he borrowed all the change that Steve had on him. Thus he just had the right amount of money to pay for his sweets. Given that the price of a box of sweets ends in 0 and the amounts borrowed by Kate and by Alex were both less than 1000 forints, how much does Kate owe Alex, and how much does Alex owe Steve?

(6 pont)

Deadline expired on February 12, 2018.

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

Megoldás. Jelölje egy doboz bonbon árát $\displaystyle x$. Kati vásárlása alapján $\displaystyle 11\,000 < 9x < 12\,000$, azaz $\displaystyle 1222 < x < 1334$, Sanyi vásárlása alapján $\displaystyle 15\,000 < 13x < 16\,000$, azaz $\displaystyle 1153 < x < 1231$. Az $\displaystyle x$-re kapott összefüggések figyelembevételével csak az $\displaystyle 1230$ jöhet szóba megoldásként. Kati vásárlása $\displaystyle 9\cdot1230 = 11070$, tehát Kati $\displaystyle 70$ Ft-tal tartozik Sanyinak. Sanyi vásárlása $\displaystyle 13\cdot1230 = 15\,990$, tehát Sanyi $\displaystyle 990$ Ft-tal tartozik Pistinek.

### Statistics:

 113 students sent a solution. 6 points: 84 students. 5 points: 8 students. 4 points: 9 students. 3 points: 6 students. 2 points: 3 students. 1 point: 2 students. 0 point: 1 student.

Problems in Mathematics of KöMaL, January 2018