2023 IOQM/Problem 1
Revision as of 11:32, 26 September 2023 by Sansgankrsngupta (talk | contribs)
Problem
Let be a positive integer such that . Let be the number of integers in the set
. Let , and .
Find .
Solution 1(Spacing of squares)
If for any integer n is an integer this means is a perfect square. Now the problem reduces to finding the minimum and maximum no. of perfect squares between 4n+1, 4n+2 .... 4n+1000. There are 1000 numbers here.
The idea is for the same range of no.s the no. of perfect squares becomes less when the numbers become larger for example, there are 3 perfect squares between 1 and 10 but 0 between 50 and 60.
So maximum value of occurs when n is minimum and the minimum value of occurs when n is maximum. Minimum n = 1 so no.s are 5, 9... 1004 there are