1: Find, with proof, the largest number which is the product of positive integers whose sum is 2013.
2: What arrangement of the digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 into the product of five two-digit numbers yields the largest product? For example, 10 × 23 × 45 × 67 × 89 = 61, 717, 050.
3: Without calculators, determine which is bigger:
(a) √19 + √99, or √20 + √98?
(b) 2000! or 10002000?
(c) If x1, ..., xn are real numbers, (i=1∑n xi)2 or n(i=1∑n xi2)?
4: (a) Given an isosceles right triangle with legs of length 1, suppose a bug wants to travel from the midpoint of one leg to the vertex at the right angle, but must touch the hypotenuse somewhere in between. What is the shortest distance the bug can travel?
(b) Given a point (a, b) with 0 < b < a, determine the minimum perimeter of a triangle with one vertex at (a, b), one on the x-axis, and one on the line y = x. You may assume that a triangle of minimum perimeter exists.
5: Center of a Ladder. A ladder lies vertically against a wall with its foot on the floor. The ladder slides down the wall so that its top is always against the wall and its bottom is always touches the floor. The ladder stops when it lies completely on the floor. What path does the center of the ladder take during this process?