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Difference between revisions of "2023 IOQM/Problem 1"

(Created page with "==Problem== Let <math>n</math> be a positive integer such that <math>1 \leq n \leq 1000</math>. Let <math>M_n</math> be the number of integers in the set <math>...")
 
Line 8: Line 8:
 
Find <math>a - b</math>.
 
Find <math>a - b</math>.
  
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==Solution==
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==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
 +
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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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
 +
5, 9... 1004 there are

Revision as of 11:32, 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 $/sqrt(n)$ 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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