## Infinite Series

## 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?

## Graeme's Number

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

## Another Radical Expression

If f(x) = 3b

^{2}

^{x}, with b a constant greater than zero, find the value of the following expression in terms of b.

## Arithmetic and Geometric

_{1}, A

_{2}, A

_{3}, and A

_{4}) form an arithmetic sequence. A

_{1}, A

_{2}, and A

_{4}form a geometric sequence. The sum of the four terms is a perfect cube. What is the smallest possible value of A

_{1}?

## X, Y, and Z

xyz=256

x + y + z = 41

x

## A Sequence of Circles

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?

## Parabola Intersections

Write the equation of the line containing the points of intersection of:

y = 2x^{2} - x -3

y = x^{2} + x + 5

Express your answer in y = mx + b form.

## Triangle On A Plane

A: (4,6)

B: (0,9)

C: (-5,-6)

## Reciprocal Roots

x

^{4}+ Kx

^{3}+ 11x

^{2}+ 7x -12 = 0

What is the sum of the reciprocals of the roots of this equation?

## 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.

## Math And Music

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)?

## 1948

x

^{3}+ x - y

^{3}- y - 1948 = 0

Find the ordered pair (x, y)

## Rational Trig Functions

**Submitted by Sasha**

## The Power of e

*e*) is the limit, as n increases without bound, of

(1+

^{n}, approximately 2.71828182846. What is the limit, as n increases without bound, of

(1+

^{n}?

## Money For Lunch?

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?

## Stars And Pounds

a * b =

a # b =

Find the value X such that:

(22 * X) # (X * 33) = 27

## 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'.

## What is this number?

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.

## Arithmetic Sequences

_{1},t

_{2},t

_{3}...}. If the absolute difference of

_{x}

and

_{2x}

is 1100, find the sum of all possible values of

_{4x}

## Expansion

^{-12}+x

^{-13})

^{n}, where n is a positive integer, contains a term in the form c*x

^{-176}, where c is a constant, for three different values of n. Find the value(s) of c.

## Circumscriptive Geometry

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*.

## The Third Degree

x

^{3}- 33x

^{2}+ 354x + k

are in arithmetic progression.

What is the value of k?

## Ratio and Proportion

Find the numeric value of the ratio a:b:c.

## A Row of Marbles

## Toothpick Polygon

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

*interior*diagonals.

What is the value of N?

## Complex Calculations

*i*. Express your answers in the form a+b

*i*.

## Complex Proof

*i*is the square of x + y

*i*, 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.

## Thanks, Soroban

Let x, y be integers.

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

^{3}then

a

^{2}+ b

^{2}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!

## Euler's Number

*e*) is the limit as n increases without bound of

(1+

^{n}, or the limit as v approaches 0 from the positive direction of (1+v)

^{1v}, approximately 2.71828182846.

What is the limit, as n increases without bound, of (1+

^{n}? Derive your solution without the tools of Calculus.