Problem K. 562. (November 2017)
K. 562. Alice went shopping. She only had 10-forint coins (HUF, Hungarian currency) and 1000-forint notes on her, at least one of each. When she had spent half her money, she noticed that she only had 10-forint coins and 1000-forint notes again. She had as many 10-forint coins as the number of 1000-forint notes she had set out with, and she had half as many 1000-forint notes as the initial number of her 10-forint coins. Given that she had spent the least amount of money that meets the given conditions, how many forints had she spent?
(6 pont)
Deadline expired on December 11, 2017.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. Jelölje \(\displaystyle x\) a \(\displaystyle 10\) forintosok kezdeti számát, \(\displaystyle y\) pedig az \(\displaystyle 1000\) forintosokét. Alíznak így a végén \(\displaystyle y\) db \(\displaystyle 10\) forintosa, és \(\displaystyle \frac x2\) db \(\displaystyle 1000\) forintosa volt. A pénze megfeleződött, tehát \(\displaystyle (10x+1000y)\cdot\frac12=10y+1000\cdot\frac x2\). Az egyenletet rendezve a \(\displaystyle 98y=99x\) összefüggést kapjuk. Mivel \(\displaystyle 98\) és \(\displaystyle 99\) relatív prímek, ezért \(\displaystyle x\) osztható \(\displaystyle 98\)-cal, \(\displaystyle y\) pedig \(\displaystyle 99\)-cel, a lehető legkisebb pozitív \(\displaystyle x\) és \(\displaystyle y\) tehát \(\displaystyle x = 98\), \(\displaystyle y = 99\). Alíz tehát \(\displaystyle 10\cdot98+1000\cdot99=99\,980\) Ft-tal indult el vásárolni, és \(\displaystyle 49\,990\) Ft-tal (\(\displaystyle 49\) db \(\displaystyle 1000\)-es és \(\displaystyle 99\) db \(\displaystyle 10\)-es) végzett, ami valóban az eredeti összeg fele. Tehát Alíz \(\displaystyle 49\,990\) Ft-ot költött el.
Statistics:
104 students sent a solution. 6 points: 84 students. 5 points: 5 students. 4 points: 4 students. 3 points: 4 students. 1 point: 3 students. 0 point: 2 students. Unfair, not evaluated: 2 solutionss.
Problems in Mathematics of KöMaL, November 2017