MAΘ Problem Set 1 January 5, 2004 The Mississippi School for Mathematics and Science 1. Minimize x + 1 x over the positive real numbers x. 2. The graph of a certain fourth degree polynomial of leading coefficient 1 is shown to the right. Find the coefficient of the x3 term for this polynomial. 4 3.5 3 2.5 2 1.5 1 0.5 1 2 3 4 -0.5 -1 -1.5 3. For how many of the first 100 counting numbers a is it true that there exists no natural number b for which ab + 1 is divisible by 6? 4. If it takes two hens three weeks to lay four dozen eggs, how many days should it take three hens to lay 60 eggs? 5. A square ABCD is inscribed in a circle O of radius r. A point in the interior of circle O is selected to be the center of a circle O 0 of radius r/10. What is the probability that circle O 0 does not intersect any of the sides of ABCD. 6. Let f (x) be defined by f (x) = x bxc is even 1/x bxc is odd Find the exact value x such that f (x) − f (x − 21 ) = 21.