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!

Match 1 results

Name                                 Section1      Section2      Section3      Score
Simarjit                             24.5          5             50.85         250
lalitkundu1995                       24.5          1500          47.122        250
deboshish                            24.5          6             18.84         250
rishabh                              24.5          900000        24.60504      250
Pulkit                               24.500        5.000         143.325       250
himanshu                             24.5          0             0.206         250
shivam                               24.500        750.000       0.457         250
luv.this.world@gmail.com             24.500        5.000         143.325       250
rohin                                24.500        31363636.363  18.840        250
prateekchaudhry                      24.500        450000.000    18.840        250
rockon                               28.000        0             0             0
rahuljain2410                        0             0             0             0
addas1395                            29.500        0             0             0
happy                                14.500        0             0.206         0
abhishek                             14.500        0             0.206         0
hbk                                  0             0             0             0
abcd                                 0             0             0             0
lakshaybansal                        21.500        450000.000    26.92714      0
sandeep                              0             0             0             0
rajatjain                            0             0             0             0
admin                                12            3123          312           0
Mayank                               0             0             0             0
akhilkumar                           29.952        4500.000      47.100        0
sumantra                             0             0             0             0
vipulverma                           0             0             0             0
shubhamsinghspecial                  0             0             0             0
vinayak                              0             0             0             0
vipulverma10                         0             0             0             0
MayankJain                           0             0             0             0
NEXUS                                0             0             0             0

One Round Match 1: Editorial

Question 1

I thought that Question 1 would be pretty simple, yet many people got a wrong answer.
Here, we subtract the colored area in the 1st figure from the colored area in the 2nd figure.
A1=.5*(1*3+11*4)
A2=.5*(8*1+13*3+13*1+18*2)
The answer is A2-A1=24.5 units.

Question2

The basic idea was to take an elemental ring at a distance x from the apex. The common mistake in this question would be in the wrong interpretation of the "width" of the ring.

In the 1st diagram, the small black line is dx, while the small red line is the length we are most interested in. Now, we simply write the equations and use an online integrator to get the answer:

Question 3

The obvious way to start is to find the field inside the charged material. Assume that the charged material is composed of infinitely many 'large' charged plates. Use the fact that the field due to each plate is independent of its distance from the plate. :)

Answer: 750 V

If you find a mistake in the solutions, please leave a comment.

One Round Contest 1: Review

We had a lot of complications during the contest. First, the images did not appear. Next, our database for storing your answers was not working. Thankfully, these issues were resolved by 5:45 pm and contestants had a chance of re-submitting.

12 people entered the contest and 3 people submitted answers. This was very disappointing for us as a lot of effort had gone into creating the website.

We'll post the results and solutions in a few days. Till then, feel free to try the problems, and we'll check your submissions anyway. Frankly, they are not difficult at all and I would expect every JEE aspirant to be able to think of the solutions atleast to the 250 and 500 point problems.

One Round Contest 1:

We are launching our new website now. The first contest will start on 08/01/2012 at 5pm IST and end at 6pm.

To participate in the test, you should register yourself on the website and familiarize yourself with the interface. The contest will be rated i.e. you will receive a 'rating'
based on your performance.

For the rules, read this.

We hope that the contest will be fun and educating at the same time!

2012 is here, time to try something new

Disclaimer: the following may never happen. Still, I am saying it loud as suggested in "The Monk who sold his Ferrari"

We 'plan' to start a new website where we will host tests. You will have to register for a test. The test will start at a certain time and close automatically at a later time. There will be a small number of questions, of level equal to those asked in the blog.

You will be assigned a 'rating' based on your performance in the tests. Performance will be judged by the correctness of your answer and by the time taken to submit the correct answer. From the 2nd test onwards, there will be different problem sets for people in different rating ranges.

Such a test is called a Match. "A One Round Match". We hope to conduct a match by 4th Jan. Hope you all participate!