Problem K. 709. (December 2021)

K. 709. A family are planning to grow a special type of onion in their garden. They want to eat 400 onion bulbs each year. The onion is grown from a seed. Each onion plant can produce 51 seeds every year if we let it. However, once an onion plant has gone into seeds, its bulb will dissolve during growing the seeds. What is the minimum number of seeds the family need to buy in order to launch the onion project so that they can have the desired quantity of onion bulbs to eat, and they never need to buy seeds any more?

(5 pont)

Deadline expired on January 10, 2022.

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

Megoldás. Ha \(\displaystyle x\) db magot vettek, akkor ebből 400 darabból a fogyasztásra szánt hagymát, a fennmaradó \(\displaystyle x-400\) darabból pedig a magokat hozó növényeket növesztik ki. Az önfenntartáshoz az kell, hogy a magnövények összesen legalább \(\displaystyle x\) magot hozzanak. A legkisebb \(\displaystyle x\) értéket tehát akkor kapjuk, ha \(\displaystyle (x-400)\cdot51=x\). Az egyenletet rendezve \(\displaystyle 51x-20\;400=x\), vagyis \(\displaystyle x = \frac{20\;400}{50}=408\). Ellenőrzés: 408 magból 400 evésre termett hagymát hoz, a maradék 8-ból pedig \(\displaystyle 8\cdot51=408\) új mag keletkezik.

Statistics:

85 students sent a solution.
5 points:	78 students.
3 points:	1 student.
Unfair, not evaluated:	5 solutionss.
Not shown because of missing birth date or parental permission:	1 solutions.

Problems in Mathematics of KöMaL, December 2021