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Difference between revisions of "2023 IOQM/Problem 1"
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==Solution 1(Spacing of squares)==
 
==Solution 1(Spacing of squares)==
  
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If for any integer n <math>/sqrt(n)</math> is an integer this means <math>n</math> is a perfect square. Now the problem reduces to finding the minimum and
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If for any [[integer]] <math>n</math>, if <math>\sqrt{n}</math> is an [[integer]] this means <math>n</math> is a [[perfect square]]. Now the problem reduces to finding the difference between maximum and minimum no. of [[perfect squares]] between <math>4n+1, 4n+2 .... 4n+1000.</math> There are 1000 numbers here.  
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maximum no. of perfect squares between 4n+1, 4n+2 .... 4n+1000. There are 1000 numbers here.  
 
  
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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.     
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The idea is that for the same range of numbers, the no. of [[perfect squares]] becomes less when the numbers become larger for example, there are 3 [[perfect squares]] between 1 and 10 but none between 50 and 60.     
  
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So maximum value of <math>M_n</math> occurs when n is minimum and the minimum value of <math>M_n</math> occurs when n is maximum. Minimum n = 1 so no.s are
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So maximum value of <math>M_n</math> occurs when <math>n</math> is minimum and the minimum value of <math>M_n</math> occurs when <math>n</math> is maximum.  
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5, 9... 1004 there are
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Minimum value of <math>n</math> = 1

Revision as of 11:45, 26 September 2023

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 $n$, if $\sqrt{n}$ is an integer this means $n$ is a perfect square. Now the problem reduces to finding the difference between maximum and minimum no. of perfect squares between $4n+1, 4n+2 .... 4n+1000.$ There are 1000 numbers here.

The idea is that for the same range of numbers, the no. of perfect squares becomes less when the numbers become larger for example, there are 3 perfect squares between 1 and 10 but none 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 value of $n$ = 1

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