The Maine Page Archive: 2002
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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.
view solution |
Dec 16, 2002 | 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!
view solution |
Dec 9, 2002 | Complex Calculations
Find all fourth roots of the complex number -7+24i. Express your answers in the form a+bi.
view solution |
Dec 2, 2002 | 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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Dec 2, 2002 | 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 31/3% less interior diagonals.
What is the value of N?
view solution |
Nov 25, 2002 | Exponents and Radicals
Solve for x:
4(8x)-21(4x)+21(2x)=4
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Nov 18, 2002 | 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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Nov 12, 2002 | Ratio and Proportion
(2a+3b)/(a+c)=2
(6a+c)/(a+b)=3
Find the numeric value of the ratio a:b:c.
view solution |
Nov 4, 2002 | The Third Degree
The zeros of the polynomial
x3 - 33x2 + 354x + k
are in arithmetic progression.
What is the value of k?
view solution |
Oct 28, 2002 | 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.
view solution |
Oct 21, 2002 | 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.
view solution |
Oct 14, 2002 | Arithmetic Sequences
Consider all increasing arithmetic sequences with common difference 1, {t1,t2,t3...}. If the absolute difference of
and
is 1100, find the sum of all possible values of
view solution |
Oct 7, 2002 | 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.
view solution |
Sep 30, 2002 | 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'.
view solution |
Sep 23, 2002 | 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
view solution |
Sep 16, 2002 | 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?
view solution |
Sep 3, 2002 | 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?
view solution |
Sep 2, 2002 | 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
view solution |
Aug 20, 2002 | 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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Aug 5, 2002 | 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)?
view solution |
Jul 22, 2002 | 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.
view solution |
Jul 8, 2002 | 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?
view solution |
Jun 24, 2002 | Triangle On A Plane
Find the area of the triangle whose vertices are located at
A: (4,6)
B: (0,9)
C: (-5,-6)
view solution |
Jun 9, 2002 | X and Y
If XY = 10 and X2Y + XY2 + X + Y = 99, find X2 + Y2
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Jun 2, 2002 | 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.
view solution |
May 19, 2002 | Fractional Infinite Series
Find the following sum:
view solution |
May 14, 2002 | 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?
view solution |
May 5, 2002 | 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 < y < z
view solution |
Apr 29, 2002 | 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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Apr 21, 2002 | 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.
view solution |
Apr 14, 2002 | Graeme's Number
This problem was submitted by Graeme MacRae.
Express the following number in simplest form. Please provide explanation with your answer!
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Mar 31, 2002 | 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?
view solution |
Mar 24, 2002 | Infinite Series
Find the sum of the following infinite series:
view solution |
Mar 17, 2002 | Fractional Equation
Solve for x if:
view solution |
Mar 10, 2002 |
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