Mathematical and Physical Journal
for High Schools
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Problem K. 235. (January 2010)

K. 235. The official currency on the planet Kloppar is the tirof, for which the small change is the dimar, and there is an even smaller change called the nepi. One dimar is worth a whole number of nepis, and a tirof is worth a whole number of dimars. Sangi had dinner in a restaurant on the planet. After dinner he received the following bill: Fried vondar: 7 dimars and 2 nepis; Pickled rotins: 10 dimars and 5 nepis; Vundeg bread: 1 dimar and 6 nepis; Carbonated nestaki drink: 6 dimars and 4 nepis; Krabban cake: 4 dimars; Total: 2 tirofs, 8 dimars and 1 nepi. How many dimars are worth a tirof, and how many nepis are worth a dimar?

(6 pont)

Deadline expired on February 10, 2010.

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

Megoldás. Legyen \(\displaystyle x\) dimar 1 tirof és \(\displaystyle y\) nepi 1 dimar. Az étlap árai szerint 1 dimar legalább 7 nepi (\(\displaystyle y>6\)) és 1 tirof legalább 11 dimar (\(\displaystyle x>10\)). Ekkor a számla összesen \(\displaystyle (7y+2)+(10y+5)+(y+6)+(6y+4)+(4y)=28y+17\), másrészről a kifizetett 2 tirof 8 dimar 1 nepi \(\displaystyle 2xy+8y+1\). Az egyenlőségükből \(\displaystyle 2xy=20y+16\), amiből \(\displaystyle x\)-t kifejezve \(\displaystyle x=\frac{10y+8}{y}=10+\frac 8y\). A kezdeti feltételek miatt \(\displaystyle y=8\) és \(\displaystyle x=11\), azaz 1 tirof 11 dimar és 1 dimar 8 nepi.


172 students sent a solution.
6 points:77 students.
5 points:9 students.
4 points:4 students.
3 points:38 students.
2 points:20 students.
1 point:14 students.
0 point:7 students.
Unfair, not evaluated:3 solutionss.

Problems in Mathematics of KöMaL, January 2010