2023 IOQM/Problem 1
Problem
Let be a positive integer such that
. Let
be the number of integers in the set
. Let
, and
.
Find .
Solution 1(Spacing of squares)
If for any integer , if
is an integer this means
is a perfect square. Now the problem reduces to finding the difference between maximum and minimum no. of perfect squares between
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 occurs when
is minimum and the minimum value of
occurs when
is maximum.
Minimum value of = 1 So, the numbers are 5,6...1004. there are 29 perfect squares here, so
=
(
)=
Maximum value of = 1000 So, the numbers are 4001,4002...5000. there are 7 perfect squares here, so
=
(
)=
⇒
~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