Problems with solutions

I have pdf solutions for three good maths problems. I'll post the problems now, and the pdfs some days later. The source of these problems might be brilliant, but I'm not sure of this. They are a bit beyond JEE, but they're doable with a day of effort :).

I'll post more hints as time goes by.

1. If F is a degree 8 function such that F(i) = 2^i for i in {0,1,2,...,8}, then find F(9).
• Given n+1 points, there is a unique function of degree n that passes through them.
• Can you construct F? Try to think of F as a sum of 9 degree 8 functions.
2. How can a circle cut the graph of ln(x) at 4 points? You don't need to find a circle which does the job, just say something interesting.
3. Show that if F is a bijective function from natural numbers to natural numbers, then there exist a and d such that F(a) < F(a+d) < F(a+2d).
• Can you prove that there exist a and d such that F(a) < F(a+d)?
• Can you construct a bijection, F, where F(a) < F(a+d) < F(a+2d) < F(a+3d) is impossible for all pairs of a and d?

Increasing Sequences and Carnegie Mellon

Andrew has 3 trees of height 3 meters, 4 trees of height 4 meters and 6 trees of height 6 meters.

He also owns 7 pavements in 7 different cities which run South-North. He will decorate his pavements with trees. Each pavement must contain one or more of his trees. Andrew is also a Feng-Shui fan, so he believes that no tree on a pavement should be taller than or even equally as tall as a tree to its south.

In how many ways can Andrew decorate his pavements?

For example, a valid decoration is:

Pavement 1: 6

Pavement 2: 3, 6

Pavement 3: 3, 4, 6

Pavement 4: 4, 6

Pavement 5: 4, 6

Pavement 6: 4,6

Pavement 7: 3

The pavements are written in a South-North manner.

Hint: this might be slightly difficult. I suggest first counting the number of decorations where each pavement can contain zero or more trees.

JEE 2013 rank list

I've been so bored that I wrote code to get all the JEE rank holders. Took me an hour to write the code and half an hour to get all the ranks.

Here is the ranklist:
http://pastebin.com/Fn1w1NUB

And here is the code:
http://pastebin.com/bp5aXfrG