Math 289 – Winter 2006 – Problem Set 4
Due February 8, 2006
1.(10) Find all the integer solutions of the equation:
qx√5−qy√5 = q6√5−10 .
2.(10) In the round robin of an Ultimate Frisbee tournament there were four different
teams in a certain group and each team played all the other teams exactly once. One
of the smart ultimate players pointed out an interesting fact, the number of “foul calls”
was different in every game but the number of “foul calls” in each game divided the total
number of “foul calls” in all the games. What is the smallest possible total number of “foul
calls” (in this particular group)?
3.(10) Find all the pairs of integers (a, b) for which a+bis a root of the quadratic equation
x2+ax +b= 0 .
4.(15) Given is the sequence of real numbers (an)∞
n=1:
a11 = 4, a22 = 2, a33 = 1,
an+3 −an+2
an−an+1
=an+3 +an+2
an+an+1
for n≥1.
Prove that
ak
1+ak
2+. . . +ak
100
is a square of an integer for each k,k= 1,2, . . ..
5.(15) Solve the following system of equations for the unknown real numbers xand y:
sin2x+ cos2y=y2,
sin2y+ cos2x=x2.
6.(15) Initially, we have three piles with a, b, and cchips, respectively. In one step, you
may transfer one chip from any pile with xchips onto any other pile with ychips. Let
d=y−x+ 1. If d > 0, the bank pays you ddollars. If d < 0, you pay the bank |d|dollars.
Repeating this step several times you observe that the original distributions of chips has
been restored. What maximum amount can you have gained at this stage?
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