Problem C. 1721. (May 2022)

C. 1721. Bonnie listed \(\displaystyle 2022\) numbers such that the ratio of the second number divided by the first number equals the third number on the list, and so on, for example, the seventh number equals the ratio of the sixth number divided by the fifth. What is the last number on Bonnie's list if the first number is \(\displaystyle 20\), and the second number is \(\displaystyle 22\)?

(5 pont)

Deadline expired on June 10, 2022.

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

Megoldás. A harmadik szám a \(\displaystyle \displaystyle{\frac{22}{20}=\frac{11}{10}}\), a negyedik a \(\displaystyle \displaystyle{\frac{11}{10}: 22=\frac{1}{20}}\), az ötödik az \(\displaystyle \displaystyle{\frac{1}{20}: \frac{11}{10}=\frac{1}{22}}\). A hatodik szám az \(\displaystyle \displaystyle{\frac{1}{22}: \frac{1}{20}=\frac{10}{11}}\), a hetedik a \(\displaystyle \displaystyle{\frac{10}{11}: \frac{1}{22}=20}\), a nyolcadik a \(\displaystyle \displaystyle{20: \frac{10}{11}=22}\). Látjuk, hogy a hetedik szám egyenlő az elsővel, és a nyolcadik a másodikkal, ezért innentől hatosával ismétlődnek a számok. Mivel \(\displaystyle 2022=6 \cdot 337\), így a \(\displaystyle 2022.\) szám egyenlő a hatodikkal, azaz Boglárka a \(\displaystyle \displaystyle{\frac{10}{11}}\)-et írta le utoljára.

Statistics:

71 students sent a solution.
5 points:	55 students.
4 points:	5 students.
1 point:	1 student.
Not shown because of missing birth date or parental permission:	1 solutions.

Problems in Mathematics of KöMaL, May 2022