Problem C. 1455. (January 2018)

C. 1455. The currency used on a distant island consists of coins of unusual denominations. The basic units are three different one-digit numbers, and there are their multiples, too: ten times, a hundred times, and also a thousand times their value. The price of one kilo of coconut may be paid with two identical coins plus a third coin of different value. In order to pay for a kilo of passion fruit, which costs twice as much, the third coin needs to be replaced by the coin with 10 times its value. Given that no coin has a denomination of 1 and the largest denomination is 7000, what other coins are used on the island?

(5 pont)

Deadline expired on February 12, 2018.

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

Megoldás. Legyenek az alapegységek \(\displaystyle 1<a<b<c<10\). A legértékesebb érme: \(\displaystyle 7000=1000c\), vagyis \(\displaystyle c=7\).

További információk két pénzérméről: \(\displaystyle K=2x+y\) és \(\displaystyle M=2K=2x+10y\), amiből \(\displaystyle 4x+2y=2x+10y\).

Ezt rendezve: \(\displaystyle x=4y\), tehát \(\displaystyle x\) páros. Alapegységekkel nem teljesülhet az egyenlet, mert \(\displaystyle 2≤y\), viszont \(\displaystyle x≤7\). Ezért \(\displaystyle 4y\)-nak \(\displaystyle 0\)-ra kell végződnie. Ez úgy lehetséges, ha \(\displaystyle y=5\), (vagy \(\displaystyle 50\), vagy \(\displaystyle 500\)), ekkor \(\displaystyle x=20\) (vagy \(\displaystyle 200\), vagy \(\displaystyle 2000\)).

Ezekből következik, hogy \(\displaystyle a=2\), \(\displaystyle b=5\) és \(\displaystyle c=7\). Ekkor a kókusz \(\displaystyle K=2\cdot20+5=45\) és a maracuja \(\displaystyle M=2\cdot20+10\cdot5=90=2K\) (vagy ezeknek \(\displaystyle 10\) vagy \(\displaystyle 100\) szorosai).

Tehát a következő érmék vannak forgalomban: \(\displaystyle 2\), \(\displaystyle 20\), \(\displaystyle 200\), \(\displaystyle 2000\), \(\displaystyle 5\), \(\displaystyle 50\), \(\displaystyle 500\), \(\displaystyle 5000\), \(\displaystyle 7\), \(\displaystyle 70\), \(\displaystyle 700\), \(\displaystyle 7000\).

Statistics:

117 students sent a solution.
5 points:	82 students.
4 points:	11 students.
3 points:	4 students.
2 points:	11 students.
1 point:	4 students.
0 point:	5 students.

Problems in Mathematics of KöMaL, January 2018