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2 nd Midterm Exam Complex Variable Basic. 1. Be f
analytic in a region
A
, is
w point A analytic in a region
A
, is
, and let D (
) a disk of positive radius R A content. Then, for each z disk it holds that f (z) is the sum k from zero to infinity f (k) (w) · (zw) k / k! where f (K)
(w) represents the k-th derivative of f w
evaluated. 2. Articulate and demonstrate the Liouville theorem. 3. Be f analytic in a region A and suppose there positive R such that
(
z - a
) / (1 - bz
) b where is the conjugate to , prove that r
25 comply pq + r = 2004 and pqr + 1 is a perfect square. 2. What is the highest number of positive integers that can be found so that any two of them a, b (with to than b) meet
CA in N
. Be L point on CA such that NL = AB
(and L the same side of N that A ). The line cuts ML AB in <1.
K
. Shows that KA = NC
. 
4. At the end of a soccer tournament in which each pair of teams played each other exactly once and where there were no draws, it was observed that for all three teams A, B, C
if
A
B beat and
B C defeated
, then B beat and
B C defeated
A beat C . Each team calculated the difference (positive) between the number of games won and lost many games. The sum of all these differences proved to be 5000. How many teams participated in the tournament? Find all possible answers. 5. Sean X, Y two circles such that the center of
O And is about X . Sean C, D the two points of intersection of the circles. Take a point in
A X and a point B in
And
such that AC is tangent to And in C and BC is tangent to A at the same point C . The segment AB cuts back to And in E , and the same segment cuts back to X in F . The line CE cut back on X G, and CF line intersects the line GD in H . Proves that the intersection of GO and EH is the center of the circumcircle of triangle DEF
.
6. What is the maximum number of direction changes in a course on the lines of a 2004x2004 grid boxes, if the route does not passes twice through the same place? feeed footer
And
such that AC is tangent to And in C and BC is tangent to A at the same point C . The segment AB cuts back to And in E , and the same segment cuts back to X in F . The line CE cut back on X G, and CF line intersects the line GD in H . Proves that the intersection of GO and EH is the center of the circumcircle of triangle DEF
.
6. What is the maximum number of direction changes in a course on the lines of a 2004x2004 grid boxes, if the route does not passes twice through the same place? feeed footer
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