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?

This used to be a physics blog!

You're standing in the middle of a 5 meter wide street with very high walls, as shown. You have a ball in you hand, and its obviously smooth and elastic.
The fun thing to do would be to throw the ball as hard as possible on the wall and watch it bounce against the wall, as shown. Have you wondered the how many bounces you could possibly achieve? The figure shows 3 bounces.

Make any assumption(s) you want (possibly using your vast knowledge of cricket). Please don't post any hints, just write down the integer answer.
Hint: If you watch cricket very closely, you would know that you can't throw at > 140 kmph (if you could, you wouldn't be solving physics problems)!

Google Rude

Today I was trying a website which generated rude comments everytime it was refreshed. Some of the rude comments were:
"Ya silly ninny!"
"Ya mouth breather!"
"Ya COBOL expert!"

Looking closely, I realized that the website chose a random comment whenever it was refreshed. It chose the comment randomly from a set of 100 rude comments.

After a while, I started seeing comments I had seen before, and the website became boring for me.
Can you tell me the probability that I have already seen the comment that I see on my 10th refresh?

2-SAT & Who will read my blog

How many positive integral solutions can you find to this system of equations?

x1+x2=8
x2-x3=-3
x3+x4=14
x4-x5=-5
x5+x6=18
x6-x7=4
x7-x8=-3

Note:
1)xi>=1.
2)Neither I nor Shivanker could find a simple enough solution for this problem. Can you come up with the awesome and simple solution?

The JEE - A Final Word

To all JEE-2012 aspirants,

All the very best for tomorrow. :)

I hope you are content with your preparation. Because that feeling is what's the
most important. Even if you're not, tell your heart that you are. Ah, yes, aal iz well. :)
Believe me, it helps.

Now this is yet another simple exam. You can't change anything
at this point even if you treat it like something special. Its just another Test Series Examination!
(No I seriously don't mean the all-devastating AITS! Just another 6 hour exam.)

You can do it. But still, keep overconfidence at bay. Yes, you can do it.
Just be open. Open your eyes, mind, and conscience. Open to all the good things around you. :)

All the best once again.
(Nah.. I don't say best of luck.. Why luck? Believe in yourself.)

Cheers!
Anuj, Shivanker and Sambhav

Some Thermochemistry

Find ΔS and ΔG for the conversion of 10.0g of supercooled water at -10° C and 1 atm to ice at -10° C and 1 atm. Average c_p values for ice and supercooled water in the range 0° C and -10° C are 0.50 and 1.01 cal/(°C g) respectively. Given that the latent heat of fusion of ice is 79.7 cal/g at the normal boiling point.

GCD-LCM problems are always fun. The solution is simple, yet it takes a long time to get there.

Here is one for you. Do not Google it because you'll obviously find the solution there.

How many unordered pairs of numbers (a,b) are there such that their GCD is 100 and their LCM is 137062800?

Noobs may try this also:
How many unordered pairs of numbers (a,b) are there such that their LCM is 137062800?

Quiz:

Here are some moderately difficult problems for your revision:

1)
Consider dD(t)/dt=k1*L(t)-k2*D(t). Also, L(t)=Lo*e^(-k1*t). Find an expression for the critical time Tc: the time when D(t) is maximum. Take D(0)=Do. Note that k1,k2, Lo and Do are constants and k2>k1.
--3 marks
What is this time (Tc) when k1*Lo is less than Do*(k2-k1)?
--3 marks

2)What is the magnetic moment of a disc sector (angle=pi/3) of radius R and charge Q (uniformly spread) rotated with angular velocity w about its natural axis??
--2 marks

3)A point charge is kept at the center of a cylinder of length L and radius R. What is the ratio of electric flux thru the curved surface to the flux thru the end caps?
--2 marks

4)The volume charge density varies with the distance from origin 'r' as rho=a*r+b*r*r.
What is the variation of electric field with r?
--2 marks

5)A 3:1 (by moles) mixture of oxygen and nitrogen effuses through a tiny orifice. What is the molecular mass of the mixture effusing out? (Do NOT consult your class notes!).
--1 mark

Note: Part 2 of Q1 is my favorite. Please try it! It looks (and is) difficult but it is solvable by you!

Note: You're through with JEE if you can solve 4 problems!

Expectations

 

A random variable is a variable whose value results from a measurement on some type of random process. Expectation E(X) of a random variable X is given by :



Suppose that n balls are tossed into n bins. Each toss is independent and each ball is equally likely to end up in any bin.
Using (or maybe not) the fact that E(X+Y) = E(X) + E(Y) even if X&Y are dependent variables, give the expected value of:

• Number of balls in a bin?
• Number of tosses till a given bin contains a ball?
• Number of tosses till every bin has a ball?
• Number of tosses till a given bin contains two balls?
• Number of tosses till at least a bin contains two balls?
• Number of empty bins?
• Number of bins with exactly one ball?

P.S.: Take n to be very very large.

PlayLists

I have 10 songs with me. On my way home, I want to listen to 30 songs (some songs will be repeated). However, I don't want to listen the same song again too soon. Precisely, there should be at-least an interval of 5 songs before I hear the same song again.

Example: Say, I have songs A, B, C, D, E,..
Then, the playlist A,B,C,D,E,F,A,B,C,D,E,F...is valid but the playlist A,B,C,A,... is not valid because there are only 2 songs between 2 occurrences of song A.

How many playlists can I have?

Sneak Peek

I was going to throw in these problems in the next One Round Match, but it turns out 6th semester at IIT is pretty heavy and I don't get enough time to set up the contest website. So, here is a simple question for you. Choose ONE word/phrase which best explains the phenomenon:

Q1)Washing machine detergents are different from hand-wash detergents.
Q2)Life is all about replication (and duplication). So, life is carbon based and not silicon based.
Q3)Life is also about making and breaking bonds with oxygen. So, life is carbon based and not silicon based.

Note: 'Life' here refers to biological life (cells, DNA etc). And finally we have a chemistry question on the blog!