Problem C. 1592. (February 2020)

C. 1592. In England, a man lost his wedding ring, and set out to search for it with the help of a friend and a metal detector. They did not find the ring, but they did find some gold coins from the time of King Henry VIII, delivering \(\displaystyle 100\,000\) pounds to the two friends. The one-pound coins were preserved in very good condition, and during the 500 years elapsed, their average annual increase in value was between 1.42% and 1.43%. How many coins may they have found?

(5 pont)

Deadline expired on March 10, 2020.

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

Megoldás. Jelölje \(\displaystyle n\) a talált érmék számát. Ekkor a feladat feltételei alapján:

\(\displaystyle n \left( 1+ \frac{1,42}{100} \right)^{500} \leq 100\,000 \leq n \left( 1+ \frac{1,43}{100}\right)^{500},\)

hiszen egyetlen érme értéke \(\displaystyle \left( 1+ \frac{1,42}{100} \right)^{500}\) és \(\displaystyle \left( 1+ \frac{1,43}{100} \right)^{500}\) közé esik.

Tehát azon \(\displaystyle n\) értékek lehetségesek, melyekre

\(\displaystyle \frac{100\,000}{\left( 1+ \frac{1,43}{100} \right)^{500}} \leq n \leq \frac{100\,000}{\left( 1+ \frac{1,42}{100} \right)^{500}}.\)

A hatványok körülbelüli értékét kiszámolva kapjuk, hogy

\(\displaystyle 1152,93<\left( 1+ \frac{1,42}{100} \right)^{500}<1152,94\)

és

\(\displaystyle 1211,19<\left( 1+ \frac{1,43}{100} \right)^{500}<1211,20.\)

Így

\(\displaystyle 82<\frac{100\,000}{\left( 1+ \frac{1,43}{100} \right)^{500}}<83\)

és

\(\displaystyle 86<\frac{100\,000}{\left( 1+ \frac{1,42}{100} \right)^{500}}<87.\)

Azaz a lehetséges \(\displaystyle n\) értékek: \(\displaystyle n\in\{ 83, 84, 85, 86\}.\)

Tehát 83, 84, 85 vagy 86 érmét találhattak.

Statistics:

231 students sent a solution.
5 points:	157 students.
4 points:	42 students.
3 points:	11 students.
2 points:	4 students.
1 point:	4 students.
0 point:	13 students.

Problems in Mathematics of KöMaL, February 2020