The Maine Page Archive: 2003
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| A Recursive Sequence
A sequence u is defined recursively as follows:
u0 = 4
u1 = 7
un+2 = 5un+1 - 6un for all n>=2.
Find the value of ux for all x>=2. Your answer should be a function of x.
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Oct 26, 2003 | 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?
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Oct 16, 2003 | The Binomial Theorem
It is a frequently unacknowledged yet extremely useful fact that the binomial theorem works for all powers of binomials, not just integer powers. That is, for any a,
(1+x)a = 1 + ax + a(a-1)/2!x2 + ... + a(a-1)...(a-k+1)/k!xk + ...
Knowing this, find with proof the coefficient of xn in the binomial expansion of (1+x)½ for n>=2.
Provide an answer in closed form. That is, n! is acceptable, while n(n-1)(n-2)...(3)(2)(1) is not.
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Oct 2, 2003 | 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.
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Sep 14, 2003 | * Operations and Matrices
"*" operates as follows:
a*b = a + (b2 - ab)/(a-b)
"#" 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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Sep 6, 2003 | Back to the basics
A sphere of radius X is inscribed in a regular tetrahedron of arbitrary side length, and a sphere of radius Y is circumscribed about the same tetrahedron. (A tetrahedron is a solid with four congruent equilateral triangle faces.) What is the numerical value of the ratio Y:X? Express your answer in the simplest possible form.
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May 29, 2003 | An infinite color spectrum in a bottomless pit
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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May 15, 2003 | Theorem of Pappus: Continued
Soroban noticed that the final answer to last week's problem, 9π2 + 32/3π2 + 32, 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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Apr 28, 2003 | 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.
view solution |
Apr 20, 2003 | Logarithms
Find all x such that
(logx2)(logx8) + 6 = 3logx8
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Apr 7, 2003 | Pythagorean Triples
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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Mar 17, 2003 | Literal Equations
ax + by = c
ax2 + by2 = c
d <= x + y
xy <= c
Find the minimum value of a + b, assuming all variables are positive.
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Mar 13, 2003 | Conics
Using the definition of a hyperbola, rotate the conic section defined by the equation
((x-3)/3)2 - ((y-1)/4)2 = 1
45 degrees counterclockwise about its center.
Express your equation in the form Ax2 + Bxy + Cy2 + 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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Mar 3, 2003 | Analytic Trigonometry
Lines a1 and a2 in two-space intersect at a point, forming an acute angle A. The slope of a1 is √3, and cos A = 1/7.
What is the slope of a2?
Please note: all answers must be in exact radical form.
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Feb 25, 2003 | A Recursive Partitioned Function
Consider a function, f(n), defined recursively by parts:
f(n+1) = f(n) + 3 while f(n) <= 100
f(n+1) = f(n) - 2 while f(n) > 100
f(1) = 50
Find the value of f(500).
view solution |
Feb 17, 2003 | Fractional System
Solve the system of equations in positive real a,b,c:
a/b + b/a + a/c + c/a + b/c + c/b = 6
a/bc + b/ac + c/ab = 4
Please note: A complete solution must demonstrate that you have all possible solutions!
view solution |
Feb 3, 2003 | Logs Within Logs
Find the value of n if:
log2(log3(log42n)) = 2
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Jan 27, 2003 | Easy as 2, 3, 4
Find all x between 0 and 90 degrees that satisfy the equation
sin(2x) + tan(3x) + cos(4x) = 2
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Jan 20, 2003 |
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