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)

What is the integral,

òdx/(x + sqrt(1-x²))

?

Let A be the set of all possible finite sequences (n₀, n₁, ..., 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.

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).

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

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!

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

Simplify the infinite product (1+x)(1+x²)(1+x⁴)(1+x⁸)(1+x¹⁶)...,
given |x| < 1.

(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 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?