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Difference between revisions of "2023 IOQM/Problem 1"

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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]] in the numbers: <math>4n+1, 4n+2 .... 4n+1000.</math> There are 1000 numbers here.  
 
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]] in the numbers: <math>4n+1, 4n+2 .... 4n+1000.</math> 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.  
+
The idea is that for the same range of numbers, the no. of [[perfect squares]] becomes rarer when the numbers become larger.  
  
 
'''For example, there are 3 [[perfect squares]] between 1 and 10 but none between 50 and 60.'''   
 
'''For example, there are 3 [[perfect squares]] between 1 and 10 but none between 50 and 60.'''   

Revision as of 07:26, 27 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 in the numbers: $4n+1, 4n+2 .... 4n+1000.$ There are 1000 numbers here.

The idea is that for the same range of numbers, the no. of perfect squares becomes rarer when the numbers become larger.

For example, there are 3 perfect squares between 1 and 10 but none between 50 and 60.

⇒ The 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 So, the numbers are 5,6...1004. there are 29 perfect squares here, so $a$ = $max.$($M_n$)= $29$

Maximum value of $n$ = 1000 So, the numbers are 4001,4002...5000. there are 7 perfect squares here, so $b$ = $min.$($M_n$)= $7$

⇒ $a-b= 29-7 = \boxed{22}$

~SANSGANKRSNGUPTA

Video Solutions

Video solution by cheetna: https://www.youtube.com/watch?v=kfEyX5yBdJo

Video solution by Unacademy Olympiad Corner: https://www.youtube.com/watch?v=Mm6mXjwU9bY

Video solution by Vedantu Olympiad School: https://www.youtube.com/watch?v=4DJXtR4VHEA

Video solution by Olympiad Wallah: https://www.youtube.com/watch?v=4HSjmY7d3nA

Video solution by : Motion Olympiad Foundation Class 5th - 10th: https://www.youtube.com/watch?v=oVaeHceHXsQ

Please note that above videos solutions are in Hindi, some in English and some in mixed(Hindi + English).

~SANSGANKRSNGUPTA

See Also

IOQM

Mathematics competitions

Please note that all problems on this page are copyrighted by THE | MTA(I)

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