4th Annual Harvard-MIT November Tournament Team Round Problems

4th Annual Harvard-MIT November Tournament
Saturday 12 November 2011
Team Round
p-Polynomials
1. [2] Find the number of positive integers x less than 100 for which
3x + 5x + 7x + 11x + 13x + 17x + 19x
is prime.
2. [4] Determine the set of all real numbers p for which the polynomial Q(x) = x3 + px2
− px − 1 has
three distinct real roots.
3. [6] Find the sum of the coefficients of the polynomial P(x) = x4
− 29x3 + ax2 + bx + c, given that
P(5) = 11, P(11) = 17, and P(17) = 23.
4. [7] Determine the number of quadratic polynomials P(x) = p1x2 + p2x − p3, where p1, p2, p3 are not
necessarily distinct (positive) prime numbers less than 50, whose roots are distinct rational numbers.
C Coloring
5. [3] Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be
colored either red or blue. A quadrant operation on the grid consists of choosing one of the four
two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the
adjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changing
the orientation of the C. Given that two colorings are the considered same if and only if one can be
obtained from the other by a series of quadrant operations, determine the number of distinct colorings
of the Cs.
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
6. [5] Ten Cs are written in a row. Some Cs are upper-case and some are lower-case, and each is written
in one of two colors, green and yellow. It is given that there is at least one lower-case C, at least one
green C, and at least one C that is both upper-case and yellow. Furthermore, no lower-case C can be
followed by an upper-case C, and no yellow C can be followed by a green C. In how many ways can
the Cs be written?
7. [7] Julia is learning how to write the letter C. She has 6 differently-colored crayons, and wants to write
Cc Cc Cc Cc Cc. In how many ways can she write the ten Cs, in such a way that each upper case C
is a different color, each lower case C is a different color, and in each pair the upper case C and lower
case C are different colors?
[GG]eometry
8. [4] Let G,A1,A2,A3,A4,B1,B2,B3,B4,B5 be ten points on a circle such that GA1A2A3A4 is a regular
pentagon and GB1B2B3B4B5 is a regular hexagon, and B1 lies on minor arc GA1. Let B5B3 intersect
B1A2 at G1, and let B5A3 intersect GB3 at G2. Determine the degree measure of \GG2G1.
9. [4] Let ABC be a triangle with AB = 9, BC = 10, and CA = 17. Let B0 be the reflection of the point
B over the line CA. Let G be the centroid of triangle ABC, and let G0 be the centroid of triangle
AB0C. Determine the length of segment GG0.
10. [8] Let G1G2G3 be a triangle with G1G2 = 7, G2G3 = 13, and G3G1 = 15. Let G4 be a point
outside triangle G1G2G3 so that ray
−−−!
G1G4 cuts through the interior of the triangle, G3G4 = G4G2,
and \G3G1G4 = 30±. Let G3G4 and G1G2 meet at G5. Determine the length of segment G2G5.