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Problems > High School Math > 2003

## 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)

## Evaluate This Indefinite Integral

What is the integral,

òdx/(x + sqrt(1-x²))

?

## 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 (n0, n1, ..., nk) of integers such that,

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

i appears in the sequence ni 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).

## Oh, You Can't Be Serious!

If 4x + 4-x = 7, then what is 8x + 8-x?

## Strange Sum

Let A1776 be the set { 1, 1/2, 1/3, ..., 1/1776 }

Remove any two elements, say a and b, from A1776, and replace them with the single number ab+a+b to form set A1775.
Continue in this manner, until you have performed 1775 such operations, to form set A1, 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+x2)(1+x4)(1+x8)(1+x16)...,
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?    Like us on Facebook to get updates about new resources