## Easy as 2, 3, 4

sin(2x) + tan(3x) + cos(4x) = 2

## Fractional System

Please note: A complete solution

*must*demonstrate that you have all possible solutions!

## A Recursive Partitioned Function

f(n+1) = f(n) + 3 while f(n) f(n+1) = f(n) - 2 while f(n) > 100

f(1) = 50

Find the value of f(500).

## Analytic Trigonometry

Lines a_{1} and a_{2} in two-space intersect at a point, forming an acute angle A. The slope of a_{1} is √3, and cos A = 1/7. What is the slope of a_{2}? Please note: all answers must be in exact radical form.

## Conics

(

^{2}- (

^{2}= 1

45 degrees counterclockwise about its center.

Express your equation in the form Ax

^{2}+ Bxy + Cy

^{2}+ Dx + Ey + F = 0, where the coefficients A through F are relatively prime integers. You will not need to belabor your process of simplification - please, however, describe the steps you take to simplify.

## Literal Equations

ax

^{2}+ by

^{2}= c

d

xy

Find the minimum value of a + b, assuming all variables are positive.

## Pythagorean Triples

Exactly four right triangles with integer side lengths exist with a leg equal to 15. These are:

(15, 112, 113), (15, 20, 25), (15, 36, 39), and (8, 15, 17).

**How many different right triangle with integer side lengths exist with a leg equal to 42? How do you know?**

BONUS QUESTION:

42 is the product of 3 different prime numbers - 2, 3, and 7.

Resolve this problem given a leg which is the product of

*n*different primes, given that, as in this problem, one of these primes is 2.

## Theorem of Pappus

The Theorem of Pappus states that when a region **R** is rotated about a line *l*, the volume of the solid generated is equal to the product of the area of R and the distance the *centroid* of the region has traveled in one full rotation. The centroid of a region is essentially the one point on which the region should "balance." The centroid of a rectangle with vertices (0,0), (x,0), (0,y), and (x,y) is the point (x/2,y/2), for example, but finding the centroid of a non-rectangular region is a little bit trickier. Part of this week's problem will require you to come up with a unique way of locating the centroid of a semicircle.

Consider the figure below, a rectangle topped by a semicircle.

Use the Theorem of Pappus to:

1) Find the centroid of the semicircle and use it to find the volume of the solid generated when just the semicircle is rotated about *l*.

2) Find the volume of the solid generated when just the rectangle is rotated about *l*.

3) Find the distance from the centroid of the region **R** to *l*.

## Theorem of Pappus: Continued

^{2}+ 32

^{2}+ 32

Is there some height

*h*of a rectangle of length 2 such that a semi-circular "cap" will move its centroid exactly one unit to the right? If so, find it.

## An infinite color spectrum in a bottomless pit

*x*beads in a bottomless pit. Only two of them are the same color. Two beads are chosen at random. Let p(x) equal the probability that these two beads are the same color. Find

## * Operations and Matrices

a*b = a +

^{2}- ab)

"#" operates as follows:

n# = n*([n-1]*([n-2]*...*(3*(2*1))))

Given that

what are the value(s) of x and y?

## Inequalities and Geometric Similarities

Given:

AB = 4x

BC = x + 2

AC = x + 4

CD = 3x + 10

AD = 12

Find the range of values of x such that the area of triangle ABC exceeds that of triangle ACD.

## A Volume Problem

In triangle ABC, AB = 25, BC = 16, and AC = 39. If ABC is rotated about its shortest side, what is the volume of the resultant solid?

## A Recursive Sequence

u

_{0}= 4

u

_{1}= 7

u

_{n+2}= 5u

_{n+1}- 6u

_{n}for all n>=2.

Find the value of u

_{x}for all x>=2. Your answer should be a function of x.