A probability problem [S]

Let there be a biased coin with probability 'p' of getting head and probability 'q' of getting tail.We keep tossing the coins till we get first tail(head) after a series of non zero heads(tails).

(i)Calculate the average number of tossing required. [S]
(ii)Does this function take minima when p=q=1/2. [S]

A beautiful problem [S]

I found this problem in Bay area mathematical olympiad 2006.The solution hardly needs any mathematics(the way I solved), classic example of pure thought.
Problem:All the chairs in a classroom are arranged in a square n×n array (in other words, n columns and n rows),
and every chair is occupied by a student. The teacher decides to rearrange the students according to the
following two rules:
(a) Every student must move to a new chair.
(b) A student can only move to an adjacent chair in the same row or to an adjacent chair in the same
column. In other words, each student can move only one chair horizontally or vertically.
(Note that the rules above allow two students in adjacent chairs to exchange places.)
Show that this procedure can be done if n is even, and cannot be done if n is odd.[S]

Special case is difficult than the general case [S]

Consider the traditional combinatorial problem

How many non-negative solutions does this equation have ?

X_1 + X_2 +....+X_n = N

For example, X_1 + X_2 = 10
Let me write a string of ten zeroes.
0 0 0 0 0 0 0 0 0 0

Let me insert 1 in any of the 11 available positions adjacent to any of the zeroes.This 1 acts as a separator, thus giving the value of X_1 and X_2.Hence there are (10+1)C 1 ways to arrange ie,. 11 solutions.

generalizing the same
original equation has (N+n-1)C(n-1) solutions.


How many solutions are there if X_i < style="color: rgb(255, 0, 0);">[S]

Circle n arc [S]

Say, we have a circle of radius 'a'. Now, draw an arc of a circle of radius 'b' such that its centre lies on the circumference of the first circle, and the end points of the arc are on the circumference as well, as shown below.Here comes the question: What is the ratio of b/a, such that, the area of the blue region is equal to that of the red region.


I wrote a python program and found out that the ratio is constant (=1.159). But, i could not figure out a proper theoretical proof. Im hoping that someone will do it in the comments box.


P.S: A special thanks for Sushama for asking me this cute problem, which fortunately or unfortunately, she doesnt have an answer to.[S]

A false conjecture(shelved)

Observe that sum of the digits of multiples of 9 is always less than 18 atleast till surprisingly large orders of 10^10.where does this break ?( any other solution than writing a program ?)(shelved)

The train Problem[R]

You and your friend are in two adjacent boggies of a train with infinite number of boggies.Both of you donot know in which boggie the other person is.Both of you move towards front or back boggie with the intention of meeting the other.A move consists of a person moving from one boggie to other and assume that both make moves simultaneously.What is the average(expected) number of moves after which both of you meet ?[reviewing the solution...]

The Google problem[S]

This problem is said to have appeared in Google's entrance exam.
Problem:
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1

what is the next line ? [S]

Colouring Problem

Problem : Six sides of a die are painted with six different colours.In how many ways can this be done ?
See comment for clue

Divsion problem[S]

Problem : find the maximum value of 'n' such that 18^n divides 181!
See comment for clue[S]

Bug split

A bug either splits into two perfect copies of itself or dies. If the probability of splitting is p (and is independent of the bug's ancestry), what is the probability that a bug's descendants die out?

Weighing problem[S]

There are 101 coins among which one is counterfeit. we do not know whether the counterfeit coin is lighter or heavier ( than the rest of the coins ) which has to be determined within three weighings.how do you do it ?[S]

A problem from cryptography[S]

This field deals with converting plain text into cipher text ( un understandable ) using a mathematical construct to keep the information secure.A monoalplabetic substitution of +3 applied on the plain text "ram" gives cipher text "udp".Here 'mono alphabetic substitution' is the algorithm and '+3' is the key.
by knowing the algorithm but not the key can we retrieve plain text from cipher text ? [S]

A simple inequality[S]

For real nums a,b,c prove that
(i) (a+b)(1/a +1/b)>=4

(ii)(a+b+c)(1/a +1/b +1/c) >=9

(iii) well, if this seems trivial prove in general for reals nums a_1,a_2,...a_n prove that
(a_1 +a_2 +...+a_n) (1/ a_1 +1/ a_2 +...+1/ a_n)>=n^2

(iv) well you may still go further proving (1+1/2 +1/3 +....) is not convergent.[S]

child and knives

There are ten knives kept on the table with their sharp edges pointing in one direction.

i i i i i i i i i i

a child looks at this and reverses the orientation but maintaining the neighbouring knives in opposite direction.one way child could do this is
i ! i i i i ! i i !
find the total num of ways in which child could do ?

A research problem

There are 'n' homes on the circumference of a circle.In each home there reside husband and wife.People of each home know only their neighbors (on both sides) and no one else.once,all the people of ten homes meet at a party.men dance with women.what is the probability that people do not know whom they danced with?
try out for n=5.

triplet problem[S]

(a,b,c) form a triplet if all a,b,c are natural numbers and a^2=b^2+c^2
prove that (1)one among the triplet is even.(2)one among the triplet is a multiple of 3.
[S]

treasure problem[S]

you are in 2d space and you may call you original location as (0,0).I shall tell you at what dist you are from the treasure, but not the direction.you have to guess where the treasure is and you will be transfered to that point.if you find the treasure there well and good.else I shall tell you the dist again.this constitutes one move.write an algorithm to find out the treasure.prove that in worst case you can do it in 3 moves.[S]