1) There is a round table divided into 4 equal quadrants, with one cup in each quadrant. The quadrants are labeled with letters (A, B, C, D) that do not move. Initially, each cup is randomly face-up or face-down. You are blindfolded and put in front of the table. On each turn of the game, you...
The problem:
Given N distinct points in the unit square [0,1]x[0,1], including the origin (0,0) as one of the N points. Can you construct N rectangles, contained in the unit square, with sides parallel to the coordinate axes, pairwise non-intersecting, such that each of our N given points is...
These we're the problems that I've tried to do. Solutions with steps would greatly be appreciated.
1. Suppose that a, b, c, are distinct non-zero digits.
(A) Find a formula, depending on a,b, c, for the sum of all six digit integers whose only digits are a,b,c.
(B) What is the sum of all...
Find a nonzero polynomial f(w, x, y, z) in the four indeterminates w, x, y, and z of minimum degree such that switching any two indeterminates in the polynomial gives the same polynomial except that its sign is reversed. For example, f(z, x, y,w) = -f(w, x, y, z). Prove that the degree of the...