Challenge Problem 1
Is square_root((5/2)+square_root(6))-square_root(3/2) rational or irrational?
Challenge Problem 2
Describe all real polynomials p(x) for which it is true
that p(2x) = 2p(x).
Challenge Problem 3
Give five different complex numbers
which have the property that
conjugate(z) = 1/z.
Challenge Problem 4
Put the complex number square_root(1+2i) into complex normal form.
(Hint: You should use the quadratic formula.)
Challenge Problem 5
Show that every real-valued function y = f(x) which is
defined for all real numbers is uniquely expressible
as f(x) = E(x) + O(x) where E(x) is an even function and
O(x) is an odd function.
Challenge Problem 6
There are properties which,
if a graph fails to have the property, then it cannot
be the graph of a rational function.
List five such properties which allow you to
recognize at a glance that a graph is not the
graph of a rational function.
Challenge Problem 7
The hyperbolic sine function is defined to be
sinh(x) = (e^x - e^(-x))/2.
The function y = sinh(x) is one-to-one, so it has an inverse. What is the inverse?
Challenge Problem 8
The function f(x) = A^(A^x) is one-to-one for A>1,
so it has an inverse. Let's write LOG_A(x) (capitalized,
since this is not the usual logarithm function) for the
inverse of y = f(x). Find a change of base formula
expressing LOG_B(x) in terms of LOG_A-values.
Challenge Problem 9
Recall that the Fibonacci numbers are defined by
F_1 = F_2 = 1 and F_n = F_{n-1} + F_{n-2}.
(The Fibonacci sequence starts out <1,1,2,3,5,8,13,21,34,55,...>.)
Question: Is the number N = (F_1/100) + (F_2/(100)^2) + (F_3/(100)^3) + (F_4/(100)^4) + ... a rational number? (This number is N = .01010203050813213455... Hint: try `playing around' with this number using a calculator and see if you can guess an answer. Then try to show that your guess is correct.)