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2003Find all x between 0 and 90 degrees that satisfy the equation

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

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Find the value of n if:

log_{2}(log_{3}(log_{4}2^{n})) = 2

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Solve the system of equations in positive real a,b,c:

+

+

+

+

+

= 6

+

+

= 4

Please note: A complete solution

*must* demonstrate that you have all possible solutions!

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Consider a function, f(n), defined recursively by parts:

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).

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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.

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Using the definition of a hyperbola, rotate the conic section defined by the equation

(

)

^{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.

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ax + by = c

ax^{2} + by^{2} = c

d

xy

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

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A little more fun with 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.

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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*.

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Soroban noticed that the final answer to last week's problem,

, is very close to 2, which is the length of the rectangle. (You may want to look over last week's solution.)

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.

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There are

*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

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"*" operates as follows:

a*b = a +

"#" operates as follows:

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

Given that

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

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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.

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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?

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A sequence u is defined recursively as follows:

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.

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