The Maine Page: Math Problems For MAML Competitors

Polynomial equations

f(x) = x³ + (2-a³)x² + (8a-4)x - a³

For which real value(s) of a does this function have only one real zero?

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A Counting Problem

Draw six cards from a standard deck of 52 playing cards without replacement. How many distinct ways can you choose these six cards so that:

1) The first card drawn is a spade,
2) the second card drawn is also a spade,
3) the third card drawn is a club,
4) the fourth card drawn is a diamond,
5) the fifth card drawn is a red card (either a diamond or a heart), and
6) the sixth and final card drawn is an ace?

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A Matrix Problem

Welcome back to the Maine Page! This year, all of our problems are going to involve topics covered in the five annual Maine Association of Math Leagues meets. New problems will be posted every week after the results of the first meet are posted on www.maml.net. Until then, here's a little problem to hold you over:

Suppose that A, B, C, and D are real numbers such that the following matrix equation holds true:

Determine the value of the following determinant:

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A Recursive Sequence

A sequence u is defined recursively as follows:

u₀ = 4
u₁ = 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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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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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!x^k + ...

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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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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* 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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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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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

∞
∑	p(x)
X=3

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