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

Permutation and Combination

Here is a simple problem:

How many sequences {a1, a2, a3, a4, a5} are there such that:

1. a1<=20, a2<=40, a3<=60, a4<=150, a5<=300
2. LCM(a1, a2, a3, a4, a5) = MAX(a1, a2, a3, a4, a5) = 210

MAX (a, b, c, d, e) is the largest of these 5 numbers.

Permutation and Combination

Here's a super interesting problem for you. Its been really hard to find such pretty problems.  This can be solved with JEE knowledge, so you should be able to do it :) .

We want to create a 'Divisible Sequence' of length H from a number N. In a Divisible Sequence, every term (except the starting number) is a divisor of the previous term. Examples of Divisible Sequences of length 3 starting with 10 are:
10, 10, 5
10, 2, 2
10, 10, 1
10, 1, 1

As you can see, there are many Divisible Sequences of length 3 starting with 10. Tell me the number of Divisible Sequences of length 10 starting with 264600.

N cakes are arranged in a line. Each cake is of a different flavor. You and your friend play a game to eat the cakes. When you eat a cake, your friend eats the (<=2) adjacent cakes.

For example, suppose there were 5 cakes initially named A,B,C,D and E. If you eat cake C, your friend eats  B and D, and the new configuration is A,E. Now, if you eat cake A, your friend eats cake E.

The question is simple: how many possible combinations of cakes can you finally eat? Note that the order of eating does not matter for the answer.

Hint: consider the Nth Fibonacci number, also called FN. Everyone knows that:

1. F1=1, F2=2 and Fn=Fn-1+Fn-1
2. FN is the number of ways to choose some objects from N objects (arranged in a line) such that no 2 are adjacent.
3. FN is the also number of ways to climb a flight of n stairs using one or 2 steps.

Consider f(x) as shown (the red line is the graph of f ). The domain and range of f is [0,1].

Can you plot y=f(f(f(f..........(f(x)))))....) i.e. can you tell what happens to x after an infinite number of applications of the function f?

What is the number of ways in which 20 identical toffees can be distributed among 10 students such that

1. Every student gets zero or more toffees.
2. Exactly 4 students get an odd number of toffees.

Use a calculator (an online one if the result is too large) to write only the answer.

Financial Maths

I bought a stock costing me Rs S(0). It is known that at the end of every year, the stock price will either rise by a factor of 'u' or fall by a factor of 'd'. 

Under these assumptions, there are only 4 stock prices possible after 3 years: 24, x, 32 and y.

What are x and y?

Range contd...

Continuing in the spirit of the last post, you have a ball again. But this time the the action takes place on the floor. Sadly, the collisions are not elastic (so you may assume e=1/2).

As shown in the figure, the ball bounces upto some distance and then stops after a very long time. Whats the maximum distance you can throw the ball upto?