An intriguing one -II

This is the continuation of the post 'An intriguing one'. Here I make an attempt to look at a possible approach.

Consider two concentric circles with radii and . It is clear that there cannot exist two points within the b-circle.

I go on placing points on the circumference of a-circle with gap of at least 'b' of course, I do not know what to do with . As of now we shall distribute evenly. This was to give an idea. You must have figured out that we can pack still more, it is not difficult to see that we can place
points. So the picture now looks like fig1.


I have written only two circles with centers and respectively, to show available region for further points. After doing this, pick points like and in the next iteration. But analysis becomes non trivial from this step itself.

Of course the problem is solved of if points like lie within b-circle.

Miscellany: Seemingly related but a easy one i found at Colorado mathematical Olympiad

(a) We need to protect from the rain a cake that is in the shape of an equilateral triangle of side 2.1. All we have are identical tiles in the shape of an equilateral triangle of side 1. Find the smallest number of tiles needed.
(b) Suppose the cake is in the shape of an equilateral triangle of side 3.1. Will 11 tiles be enough to protect it from the rain?

I found that we require 6 and 11 .

Two not so easy ones

Take this first one from USAMO 2004.

1.Suppose are integers whose greatest common divisor is 1. Let be a set of integers with the following properties:

(a) For , .
(b) For (not necessarily distinct), .
(c) For any integers , if , then .

Prove that must be equal to the set of all integers.
*****************************************************

If you hit the above one, try this, as of now I haven't got it yet.

2. A friend of mine is organizing a board game tournament with 5 rounds. There are 12 competing players. One of the games is a 6-player game. The other 4 games are different 4-player games. My friend has enough copies of every game, so each round will be played with multiple parallel game tables.

The question is: can he assign the players to the tables in such a way that
every player plays every other player exactly 1 or 2 times during the tournament? "Playing" means: sitting at the same table during any round.

So, the assignment should look like this:

round1: XXXXXX XXXXXX (2 tables of 6 people each)

round2: XXXX XXXX XXXX (3 tables of 4 people each)

round3: XXXX XXXX XXXX

round4: XXXX XXXX XXXX

round5: XXXX XXXX XXXX

I hope someone finds a solution for my friend!!
(courtesy :Big fury monster)

A simple one [S]

This objective type question is said to have appeared in one of the old ISI selection papers.
A club with 'x' members is organized into 4 committees such that
[a] each member is in exactly two committees
[b] any two committees have exactly one member in common

Then,
1. exactly two values both between 4 and 8.
2. exactly one value lying between 4 and 8.
3. exactly two values between 8 and 16.
4. exactly one value between 8 and 16. [S]

An apparent paradox

You have infinite number of balls numbered 1,2,... at your disposal.At time(t)=0, you put balls numbered 1 to 10 into a box.At t=1/2, you put balls numbered 11 to 20 and remove ball-1 from it.Similarly at t=(1/2 + 1/4 ), you put balls numbered 21 to 30 into the box and remove ball-2 from the box.
t follows 1/2 +1/4 +1/8 +...
how many balls are present at t=1 ?

Surprisingly, number of balls in the box at t=1 is 0. For any ball numbered 'k' we can find t(=1/2 + 1/4 +...+ 1/2^k)when it is removed.
For a finite number of operations we can argue that 9 balls are added per each operation but this argument does not work when n->inf.In the language of limits,

Lt (x+9) - Lt (x) = 0
(x->inf) (x->inf)

and mind you LHS(not=)x+9-x=9

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]

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 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]

Divsion problem[S]

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

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]

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]