2023 IOQM/Problem 1

Problem

Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set

$X_n = \left\{\sqrt{4n + 1}, \sqrt{4n + 2}, \ldots, \sqrt{4n + 1000}\right\}$ . Let $a = \max\{M_n : 1 \leq n \leq 1000\}$ , and $b = \min\{M_n : 1 \leq n \leq 1000\}$ .

Find $a - b$ .

Solution 1(Spacing of squares)

If for any integer n $/sqrt(n)$ is an integer this means $n$ 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 $M_n$ occurs when n is minimum and the minimum value of $M_n$ occurs when n is maximum. Minimum n = 1 so no.s are 5, 9... 1004 there are

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2023 IOQM/Problem 1

Problem

Solution 1(Spacing of squares)