1: Let s be any arc of the unit circle lying entirely in the first quadrant. Let A be the area of the region lying below s and above the x-axis and let B be the area of the region lying to the right of the y-axis and to the left of s. Prove that A+B depends only on the arc length, and not on the position, of s.
2: Find the least number A such that for any two squares of combined area 1, a rectangle of area A exists such that the two squares can be packed in the rectangle (without interior overlap). You may assume that the sides of the squares are parallel to the sides of the rectangle.
3: Let T be an acute triangle. Inscribe a rectangle R in T with one side along a side of T. Then inscribe a rectangle S in the triangle formed by the side of R opposite the side on the boundary of T, and the other two sides of T, with one side along the side of R. For any polygon X, let A(X) denote the area of X. Find the maximum value, or show that no maximum exists, of (A(R)+A(S)/A(T)), where T ranges over all triangles and R, S over all rectangles as above.
4: Find the positive value of m such that the area in the first quadrant enclosed by the ellipse (x2/9) +y2 = 1, the x-axis, and the line y = 2x/3 is equal to the area in the first quadrant enclosed by the ellipse (x2/9) + y2 = 1, the y-axis, and the line y = mx.
5: (a) An explorer lives on the planet Qubit which is in the shape of a perfect cube with a side of length q. She wishes to walk from the North Pole (which is a vertex of the cube) to the South Pole (which is the opposite vertex). What is the minimum distance she needs to walk?
(b) The moon of Qubit is a rectangular solid with sides s, 2s and 3s. What is the minimum distance one needs to walk to get from one vertex of the moon to the vertex opposite it?