absird

True of False: If f is an entire function satisfying f(0) = 1 and f'(n) = f(n) for all integers n>=1, then f(z) = e^z for all z in C.

If a complex-valued function fails to satisfy the Cauchy-Riemann equations, does this imply that it is not differentiable anywhere or that it merely fails to be analytic (meaning there might still exist points at which it is differentiable)?

I'm having trouble evaluating the line integral around the curve y(t) = e^it, 0 <= t <= 2pi (unit circle) of Re(f) where f(z) = z. I substituted y(t) in for f which gave me

Re(y(t)) = Re(e^it) = Re(cos(t) + isin(t)) = cos(t)

and y'(t) = ie^it

so our integral is equivalent to the integral from 0 to 2pi of their product, namely cos(t)ie^it. Is this the correct procedure? Because at this point I tried to evaluate this integral by parts and got 0 but the answer should be i(pi).

Given a sequence a_1 = 2, a_2 = 8, ... , a_n = 4a_(n-1) - a_(n-2), n >= 3, show that

π/12 = sum (n=1 to ∞) arctan((a_n)^2).

Prove sum (from k = 0 to n-1) C(n, k)C(n+2-k, 3) = (n/3)(n + 2)(n + 7)2^(n-4)

Let p be a prime number with p > 3 and let n = (2^(2p) - 1)/3. Prove n|(2^n - 2).

Prove that there is no integer n>1 for which n|(2^n - 1).

Does there exist a power of 3 (number of the form 3^n) with a string of 2008 consecutive zeros in its decimal expansion?

There are three schools each with n students. Every student is friends with at least n+1 students from the other schools. Prove that there is some trio of students (one from each school) that are mutual friends.

There are 2n people at a hotel where n is a postive integer. Each person knows exactly n other people. Prove that they can be paired up and assigned to rooms so that everyone knows his/her roommate.

Note: The knowing relation is mutual so A knows B implies B knows A

There is a game involving 5 boxes numbered 1, 2, 3, 4, and 5 containing 0, 20, 30, 40, and 50 coins respectively. The game has two players (A and B) who take turns alternately. It begins with A taking some coins from some box n and moving them to box n-1. (For example A could take 23 coins from box 4 and move them to box 3 so that the boxes have 0, 20, 53, 17, 50 coins respectively. Then B might take 2 coins from box 2 and move them to box 1 so that the boxes have 2, 18, 53, 17, 50 coins respectively.) You cannot move any coins that are in box 1. When all the coins have been moved to box 1 the game is over. The player who makes the last move wins. Which player has a winning strategy and what is it?

If n is a natural number such that 2n+1 and 3n+1 are perfect squares, prove that 5n+3 can't be a prime number.

Let a and b be positive real numbers with a >= b. Let p be the maximum value of a/b for which the system of equations

a² + y² = b² + x² = (a - x)² + (b - y)²

has a solution in (x, y) satisfying 0 <= x < a and 0 <= y < b. Find p.

Let f(n) = n + [sqrt(n)]. Prove that for every positive integer m, the sequence

m, f(m), f(f(m)), f(f(f(m))), ...

contains the square of an integer.

Note: [x] represents the greatest integer less than or equal to x.

Let n be a natural number. Prove that

[(n+1) / 2] + [(n+2) / 4] + [(n+4) / 8] + [(n+8) / 16] + ... = n.

For a cleaner version of the problem see the link below:

http://www.artofproblemsolving.com/Classroom/cbe6/...

Two nonoverlapping arcs of a circle are chosen. Eight distinct points are then chosen on each arc. All 64 segments connecting a chosen point on one arc to a chosen point on the other arc are drawn. How many triangles are formed that have at least one of the 16 chosen points as a vertex?

How many 10-digit numbers have all distinct digits and are multiples of 11111?

Find the least positive integer n such that

[1 / (sin(45)sin(46))] + [1 / (sin(47)sin(48))] + ... + [1 / (sin(133)sin(134))] = 1 / sin(n).

Let a, b, and c be the sides of a triangle and S its area. Prove

a^2 + b^2 + c^2 >= (4S)sqrt(3).

Where does equality hold?

Compute

[(10^4 + 324)(22^4 + 324)(34^4 + 324)(46^4 + 324)(58^4 + 324)] / [(4^4 + 324)(16^4 + 324)(28^4 + 324)(40^4 + 324)(52^4 + 324)].