Sub-setting II

[dtweah]

Let S be the set {1, 2, 3, . . . , n} consisting of the first n positive integers.
What is the maximum value of n for which every 100-element subset of
S contains two integers which differ by 25 ?

(a) 171
(b) 172
(c) 173
(d) 174
(e) 175

[francopiccolo]

For n = 100 we have that 75 numbers which differ in 25. From 26 to 100.
If we add 101 we could substract the number 76, eliminating one pair of numbers that differ in 25 and not adding any pair of numbers. Then if we can substract one pair of number every time we add a number we need to add 25, therefore the maximum value for n is 175.

Took me mooore than 2 minutes, he.