MathDB

9

Part of 2016 LMT

Problems(2)

2016 LMT Individual #9

Source:

4/10/2016
An acute triangle has area 8484 and perimeter 4242, with each side being at least 1010 units long. Let SS be the set of points that are within 55 units of some vertex of the triangle. What fraction of the area of SS lies outside the triangle?
Proposed by Nathan Ramesh
2016 LMT Theme #9

Source:

4/11/2016
A function f:{1,2,3,⋯ ,2016}→{1,2,3,⋯ ,2016}f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\} is called good if the function g(n)=∣f(n)−n∣g(n)=|f(n)-n| is injective. Furthermore, a good function ff is called excellent if there exists another good function f′f' such that f(n)−f′(n)f(n)-f'(n) is nonzero for exactly one value of nn. Let NN be the number of good functions that are not excellent. Find the remainder when NN is divided by 10001000.
Proposed by Nathan Ramesh