## An Increasing Function for the New Year

Let f be a function from Z+ to Z+ where Z+ is the set of positive integers, such that f satisfies these two conditions:

(1) f(n+1) > f(n); that is, f is strictly increasing

And

(2) f(n+f(m)) = f(n)+m+1

Find all values of f(2003)

## A Geometrical Diversion

The diagonals of a square meet at O.

The bisector of angle OAB meets

BO and BC at N and P respectively.

The length of NO is 24.

How long is PC?

## Interesting Integer Sequences

Let A be the set of all possible finite sequences (n_{0}, n_{1}, ..., n_{k}) of integers such that,

for each i = 0, 1, ..., k

i appears in the sequence n_{i} times.

Here are some sequences in set A:

1,2,1,0

2,0,2,0

2,1,2,0,0

3,2,1,1,0,0,0

4,2,1,0,1,0,0,0

k-3,2,1,0,0,...,1,0,0,0

Are there other sequences in set A? If so, what are they?

Now prove it.

## Another Increasing Function

Maybe you remember that back in January 2003, I offered you an increasing function that met two criteria (click the problem archives to see it). This month, I will challenge you with another increasing function that meets two criteria...

Let f map positive integers to positive integers with the conditions:

i) f(n+1) > f(n)

ii) f(f(n)) = 3n

Find f(955).

## Strange Sum

Let A_{1776} be the set { 1, ^{1}/_{2}, ^{1}/_{3}, ..., ^{1}/_{1776} }

Remove any two elements, say a and b, from A_{1776}, and replace them with the single number ab+a+b to form set A_{1775}.

Continue in this manner, until you have performed 1775 such operations, to form set A_{1}, which contains a single element.

What is this element?

Prove it!

## A Little Number Theory to Begin the School Year

Prove that if p and p²+8 are prime then so is p³+4.

## It's Cotton Candy for the Mind

Simplify the infinite product (1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})...,

given |x| < 1.

## Party Hardy

(Part A): There are 100 people in a ballroom. Every person knows at least 67 other people (and if I know you, then you know me). Prove that there is a set of four people in the room such that every two from the four know each other. (We will call such a set a "clique.")

(Part B): Two people in the room are Joe and Grace, who know each other. Is there a clique of four people which includes Joe and Grace?

(Part C): Oops, I miscounted. There are actually 101 people in the room, But it's still true that each knows at least 67 others. That can't make a difference, can it?

## A Strong Will

A father in his will left all his money to his children in the following manner:

$1000 to the first born and 1/10 of what then remains, then

$2000 to the second born and 1/10 of what then remains, then

$3000 to the third born and 1/10 of what then remains, and so on.

When this was done each child had the same amount. How many children were there?

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