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.