calculus help

Question

Question 1:
1. Given a cycle of radius 2, we put inside it 13 squares with all fours sides being 1 Show that there are at least 2 squares that share a point. Remake The area of a cycle of radius r is π*r2.
2. Given 12 different 2 digits numbers show that you can choose two numbers a, b out of them so that a – b ≥ 10 and the two digits of a - b are equal (you may use the fact that there are two numbers a,b so that a mod 11 = b mod 11).

Answer 1

1.

 

The total area of the squares is 13.

 

The area of the disk is 4π < 4 × 3.25 = 13.

 

Thus clearly, some squares must overlap

 

2. This is actually a Number Theory problem.

 

If we choose 12 different two digit numbers then by the Pigeonhole Principle, there are two numbers which are congruent modulo 11. If we take their difference then it will be a multiple of 11 and that multiple will be less than 100. All such multiples of 11 are two digits numbers with a pair of identical digits.