Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2023 IOQM/Problem 1"

(→‎Video Solutions)
(→‎Video Solutions)
Line 38: Line 38:
 
Video solution by : Motion Olympiad Foundation Class 5th - 10th: https://www.youtube.com/watch?v=oVaeHceHXsQ
 
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).
+
'''Please note that above videos solutions are in Hindi, some in English and some in mixed(Hindi + English).'''
 +
 
 
~SANSGANKRSNGUPTA
 
~SANSGANKRSNGUPTA
 +
 +
==See Also==

Revision as of 07:10, 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 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.

⇒ 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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_IOQM/Problem_1&oldid=198651"