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)

## Tuesday, June 26, 2007

## Friday, June 15, 2007

### pretty good

Prove that among any 10 consecutive integers at least one is relatively prime to each of the others.

hint: picture has nothing to do with the problem

hint: picture has nothing to do with the problem

Labels:
number theory

## Tuesday, June 12, 2007

### A generalization

Prove that any positive integer x lying between a

x = a

^{k}and a^{k+1}can be represented uniquely as followsx = a

^{k}+ b_{k-1}a^{k-1 }+...+b_{k-r}a^{k-r}+..+b_{0}a^{0where a is a positive integer >1 and each bi (non negative integer) is less than a.I generalized this based on the problems found in Niven's number theory book( pg 19 pro, 44 and 45 , fifth edition)}
Labels:
number theory

## Thursday, June 07, 2007

### 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]

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]

Labels:
general

## Tuesday, June 05, 2007

### A functional equation

Prove that this equation does not admit any real continuous function R ->R ( try R

f(x).f(x+1) + f(x) + 1 = 0

My progress so far: turn this into a recursive equation

f(x+1)= -1/ [ f(x) +1]

suppose if f(x

then , f(x

Let us select x and x+$ , two real numbers, then

epsilon=f(x)-f(x+$) = $/ (1+x)

//neglecting $ as it does not matter in the analysis that follows//

if x is negative and greater than -1, epsilon> $.

will this help? am not able to proceed from here .

^{n}->R^{m})f(x).f(x+1) + f(x) + 1 = 0

My progress so far: turn this into a recursive equation

f(x+1)= -1/ [ f(x) +1]

suppose if f(x

_{o})=kthen , f(x

_{o}+4)=k for arbitrary x_{o}and k belonging to R.Let us select x and x+$ , two real numbers, then

epsilon=f(x)-f(x+$) = $/ (1+x)

^{2}//neglecting $ as it does not matter in the analysis that follows//

if x is negative and greater than -1

will this help? am not able to proceed from here .

Labels:
analysis

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