4th Annual Harvard-MIT November Tournament General Test Problems

1. [3] Find all ordered pairs of real numbers (x, y) such that x2y = 3 and x + xy = 4.
2. [3] Let ABC be a triangle, and let D, E, and F be the midpoints of sides BC, CA, and AB, respectively.
Let the angle bisectors of \FDE and \FBD meet at P. Given that \BAC = 37± and \CBA = 85±,
determine the degree measure of \BPD.
3. [4] Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultane-
ously listening to exactly two songs, and each song is being listened to by exactly two people. In how
many ways can this occur?
4. [4] Determine the remainder when
2
1·2
2 + 2
2·3
2 + · · · + 2
2011·2012
2
is divided by 7.
5. [5] Find all real values of x for which
1
px + px − 2
+
1
px + 2 + px
=
1
4
.
6. [5] Five people of heights 65, 66, 67, 68, and 69 inches stand facing forwards in a line. How many
orders are there for them to line up, if no person can stand immediately before or after someone who
is exactly 1 inch taller or exactly 1 inch shorter than himself?
7. [5] Determine the number of angles µ between 0 and 2¼, other than integer multiples of ¼/2, such that
the quantities sin µ, cos µ, and tan µ form a geometric sequence in some order.
8. [6] Find the number of integers x such that the following three conditions all hold:
• x is a multiple of 5
• 121 < x < 1331
• When x is written as an integer in base 11 with no leading 0s (i.e. no 0s at the very left), its
rightmost digit is strictly greater than its leftmost digit.
9. [7] Let P and Q be points on line l with PQ = 12. Two circles, ! and ­, are both tangent to l at P
and are externally tangent to each other. A line through Q intersects ! at A and B, with A closer to
Q than B, such that AB = 10. Similarly, another line through Q intersects ­ at C and D, with C
closer to Q than D, such that CD = 7. Find the ratio AD/BC.
10. [8] Let r1, r2, . . . , r7 be the distinct complex roots of the polynomial P(x) = x7 − 7. Let
K = Y
1·i<j·7
(ri + rj),
that is, the product of all numbers of the form ri+rj , where i and j are integers for which 1 · i < j · 7.
Determine the value of K2.