Math Problems

Here are a bunch of math problems that I have discovered, uniquely or independently. Work on them and have fun with them! How many can you solve?

Parabolic Rectangle
Let ABCD be a rectangle such that points A and B are located on the x-axis and points C and D touch the curve y = 4 - x2. What is the relationship between sides AB and BC? When the rectangle is a square, what is the side length?
Circles and Squares
A square is inscribed in a circle, which is inscribed in a square. If the length of the larger side is 1, how much area is inside the circle but not inside the smaller square?
Weird Operations
Suppose we define two operations: a (+) b = 1/(1/a + 1/b), and a (-) b = 1/(1/a - 1/b). What is the relationship between a and b if (1) a (-) b = b/a; (2) a (+) b = 2a (-) b; (3) a (-) b = b?
Find a value of c such that the equation 2x = x + c has only one solution.
Dissimilar Triangles
Triangle ABC and triangle DEC touch at point C, and points B, C, and E are collinear. Suppose that AB = 3, ED = 2, and BD = 6. If angles ACB and DCE are the same, what are their measures? If angles ACB and DCE add up to 90 degrees, what are their measures?
Triangle Inside
Find the radius of a circle that can have a triangle with sides of lengths 3, 4, and 5 inscribed in it. What is the radius of a circle that can have a triangle with sides of length 3, 4, and 4 inscribed in it? What is the radius of a circle that can have a triangle with sides of length a, b, and c inscribed in it?
Two-Dimensional Walking
A man starts at point (0,0). Each minute, he walks one unit either in the positive x direction or the positive y direction randomly, with a probability of 0.5 for each. What is the probability that he will, at some time, be at the point (3,3)?
Magic Squares
Can you construct a 4x4 magic square such that the product of all rows, columns, and major diagonals is the same, and where all of the cells have different numbers?
Circle Segments
A digon AB is constructed such that one path from A to B is a straight line of length 9 and the other is an arc of a circle. If the straight line is considered to be the bottom, the maximum height of the digon is 3/2 * Sqrt(3). What is the radius of the circle of which the arc belongs?
Tetrahedral Skeleton
Take a tetrahedron of which all the sides are equilateral triangles with side length 1. What is the distance from one of the corners to the center of the tetrahedron? What is the angle between two of these corner-center lines?
Square Dance
Take a square ABCD. Let M be the midpoint of AB, N be the midpoint of BC, O be the midpoint of CD, and P be the midpoint of DA. Draw the lines AN, BO, CP, and DM. What is the area of the central, smaller square formed by these lines? Draw the lines AO, BP, CM, and DN. What is the area of each of the regions formed?
If you've finished, check out my solution to this problem.
A standard deck has 4 suits of 13 numbers each, and you are dealt five cards. Is it more likely to be dealt a flush (all of the same suit) or a straight (five numbers in a row (order doesn't matter), but no wrapping around - the highest card and the lowest card are not considered next to each other)? In a generalized deck of m suits and n numbers each, where you are dealt k cards, is it more likely to be dealt a flush or a generalized straight?