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Problems > The Maine Page > 2002

Fractional Equation

Solve for x if:
 

x
1+
1
x
-
x
1-
1
x
= 1

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Infinite Series

Find the sum of the following infinite series:

1
6
+
5
36
+
19
216
+
65
1296
+
211
7776
+ ...

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Inscribed Circle

A circle C is inscribed in a square with sides of length 4 inches.

A second circle O is tangent to the square in exactly two places, and is also tangent to circle C.

What is the radius of circle O?

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Graeme's Number

This problem was submitted by Graeme MacRae.

Express the following number in simplest form. Please provide explanation with your answer!

3
108
+ 10
-
3
108
- 10

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Another Radical Expression

This problem was submitted by Sasha Joseph, this year's high scoring sophomore at the MAML state math meet.

If f(x) = 3b2x, with b a constant greater than zero, find the value of the following expression in terms of b.

f(1)-
f(2)+
f(3)-
f(4)
...

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Arithmetic and Geometric

Four distinct positive integers (A1, A2, A3, and A4) form an arithmetic sequence. A1, A2, and A4 form a geometric sequence. The sum of the four terms is a perfect cube. What is the smallest possible value of A1?
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X, Y, and Z

Find the ordered triple (x,y,z) if x, y, and z are natural numbers which satisfy the following:

xyz=256
x + y + z = 41
x
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A Sequence of Circles

I have an infinite number of circles. The first one has radius r. The second has radius r/2. The third has radius r/4, and so on.

I also have an infinite number of squares, each of which has area equal to the area of the corresponding circle.

What is the sum of the perimeters of all my squares?
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Fractional Infinite Series

Find the following sum:

Σ
1
(x2+x)
x=1

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Parabola Intersections

Write the equation of the line containing the points of intersection of:
y = 2x2 - x -3
y = x2 + x + 5
Express your answer in y = mx + b form.

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X and Y

If XY = 10 and X2Y + XY2 + X + Y = 99, find X2 + Y2
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Triangle On A Plane

Find the area of the triangle whose vertices are located at

A: (4,6)
B: (0,9)
C: (-5,-6)
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Reciprocal Roots

Let K be a constant, and the following a polynomial equation in x:
x4 + Kx3 + 11x2 + 7x -12 = 0
What is the sum of the reciprocals of the roots of this equation?
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Three-Digit Number

X is a three digit number. The product of its first two digits is between 16 and 24, inclusive. The product of its last two digits is between 35 and 45, inclusive. If the product of all its digits is equal to 72, list all possible values of X.

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Math And Music

When I was in high school, my math teacher always spoke with pride of his 'math and music' students; students who excelled in both fields. He insisted that it made sense, because music is very mathematical.

A music scale is made up of different frequencies which have been assigned names:

A A# B C C# D D# E F F# G G#

The scale then starts over at A. This is called an 'octave'. The next 'A' is said to be one octave above the first A

The ratio between the frequencies of any two successive notes is equal to the twelfth-root of 2.

In the USA, 'concert' A has a frequency of 440 Hz. However, in France, 'concert' A is considered to be 435 Hz.

What is the frequency difference (to the nearest Hz) Between an A two octaves above concert A (in the USA), and a G#, directly below concert A (in France)?
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1948

If x and y are positive integers such that
x3 + x - y3 - y - 1948 = 0
Find the ordered pair (x, y)
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Rational Trig Functions

Demonstrate that if the tangent of half of an angle is rational, any trigonometric function of the whole angle will (if defined) be rational.

Submitted by Sasha
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The Power of e

By definition, Euler's number (e) is the limit, as n increases without bound, of

(1+
1
n
)n, approximately 2.71828182846. What is the limit, as n increases without bound, of

(1+
2
n
)n?

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Money For Lunch?

In my town there are four restaurants. The price for lunch at the four restaurants is $5.00, $10.00, $15.00, and $20.00.

In my wallet I have four bills. They might be ones, fives, tens, or any combination of those (with each type of bill being equally likely) but I don't know what they are.

If I randomly select a restaurant, what is the probability I'll have enough money for lunch?
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Stars And Pounds

Let the operations '#' and '*' be defined as follows:

a * b =
1
a
+
1
b


a # b =
(a + b)
(a - b)


Find the value X such that:

(22 * X) # (X * 33) = 27
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Rectangle in a Triangle

A rectangle is placed inside an isoceles right triangle in such a way that the two vertices of the rectangle lie on the hypotenuse, and the other two vertices lie on the legs.

The area of the triangle is 2 square units, and the area of the rectangle is one quarter of that.

In the diagram thus created, find the area of the triangle which does not touch the hypotenuse of the larger, isoceles triangle.
 



Note: For clarity, this problem has been reworded to say 'hypotenuse of the larger, isoceles triangle', rather than 'base of the larger, isoceles triangle'.

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What is this number?

n is a positive three-digit integer in base ten with the following properties:

1) If the digits of n are reversed, the new three- digit number is 51 more than a third of n.

2) There are between 128 and 130 zeros, inclusive, following the final non-zero digit in n!.

Find n.
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Arithmetic Sequences

Consider all increasing arithmetic sequences with common difference 1, {t1,t2,t3...}. If the absolute difference of

100
Σtx
x=1

and

50
Σt2x
x=1

is 1100, find the sum of all possible values of

25
Σt4x
x=1
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Expansion

The binomial expansion of (x
-1
2
+x
-1
3
)n, where n is a positive integer, contains a term in the form c*x
-17
6
, where c is a constant, for three different values of n. Find the value(s) of c.
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Circumscriptive Geometry

About a unit circle, a regular hexagon is circumscribed, about which another circle is circumscribed, about which an equilateral triangle is circumscribed, about which a third circle is circumscribed. Find, in simplified and exact form, the ratio of the area of the smallest circle to that of the largest.

Note: within a regular polygon, the distance from its center to any given vertex is called its radius. The length of a perpendicular from its center to any given side is called its apothem.
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The Third Degree

The zeros of the polynomial

x3 - 33x2 + 354x + k

are in arithmetic progression.

What is the value of k?
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Ratio and Proportion

(2a+3b)
(a+c)
=2
(6a+c)
(a+b)
=3

Find the numeric value of the ratio a:b:c.
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A Row of Marbles

A boy has six yellow marbles and four blue marbles, each distinct from the others. He arranges his ten marbles in a row. What is the probability that no blue marble lies next to another?
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Exponents and Radicals

Solve for x:

4(8x)-21(4x)+21(2x)=4
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Toothpick Polygon

A regular polygon is created by placing N toothpicks of equal size end to end.

The polygon is then altered by moving exactly two of the toothpicks. The resulting figure is still a polygon, but it has 3
1
3
% less interior diagonals.

What is the value of N?

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Complex Calculations

Find all fourth roots of the complex number -7+24i. Express your answers in the form a+bi.
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Complex Proof

Let x and y be real integers such that x > y > 0. If a + bi is the square of x + yi, prove that a and b are the legs in a pythagorean triple -- that is, if a and b are whole number lengths of the legs in a right triangle, the hypotenuse will also be a whole number. (i is the imaginary unit.)

For 2 bonus points, specify less restrictive conditions on the integers x and y for which the above statement is still true.
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Thanks, Soroban

After last week's problem, Soroban emailed me a more 'generalized' statement, from which this week's problem is derived.


Let x, y be integers.

Prove: If a + bi = (x + yi)3 then

a2 + b2 is a perfect cube.


For those of you who solved last week's problem, this one should be easy; if you had trouble with last week's, check the solution, and that should help you solve this one!

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Euler's Number

By definition, Euler's number (e) is the limit as n increases without bound of
(1+
1
n
)n, or the limit as v approaches 0 from the positive direction of (1+v)
1
v
, approximately 2.71828182846.



What is the limit, as n increases without bound, of (1+
2
n
)n? Derive your solution without the tools of Calculus.
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