Problem A. 785. (October 2020)

A. 785. Let \(\displaystyle k\ge t\ge 2\) positive integers. For integers \(\displaystyle n\ge k\) let \(\displaystyle p_n\) be the probability that if we choose \(\displaystyle k\) from the first \(\displaystyle n\) positive integers randomly, any \(\displaystyle t\) of the \(\displaystyle k\) chosen integers have greatest common divisor 1. Let \(\displaystyle q_n\) be the probability that if we choose \(\displaystyle k-t+1\) from the first \(\displaystyle n\) positive integers the product is not divisible by a perfect \(\displaystyle t\)-th power that is greater then 1.

Prove that sequences \(\displaystyle p_n\) and \(\displaystyle q_n\) converge to the same value.

Submitted by Dávid Matolcsi, Budapest

(7 pont)

Deadline expired on November 10, 2020.

Statistics:

4 students sent a solution.
6 points:	Fleiner Zsigmond, Füredi Erik Benjámin, Seres-Szabó Márton, Sztranyák Gabriella.

Problems in Mathematics of KöMaL, October 2020