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<b>Tốn học, Olympic tốn tồn quốc - Việt </b>

<b>nam 2000</b>

Bài từ Tủ sách Khoa học VLOS.
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A1. Define a sequence of positive reals x0, x1, x2, ... by x0 = b, xn+1 = �"(c - �"(c +
xn)). Find all values of c such that for all b in the interval (0, c), such a sequence exists
and converges to a finite limit as n tends to infinity.

A2. C and C' are circles centers O and O' respectively. X and X' are points on C and C'
respectively such that the lines OX and O'X' intersect. M and M' are variable points on C
and C' respectively, such that "XOM = "X'O'M' (both measured clockwise). Find the
locus of the midpoint of MM'. Let OM and O'M' meet at Q. Show that the circumcircle of
QMM' passes through a fixed point.

A3. Let p(x) = x3 + 153x2 - 111x + 38. Show that p(n) is divisible by 32000 for at least
nine positive integers n less than 32000. For how many such n is it divisible?

B1. Given an angle ±� such that 0 < ±� < À�, show that there is a unique real monic
quadratic x2 + ax + b which is a factor of pn(x) = sin ±� xn - sin(n±�) x + sin(n±�
-±�) for all n > 2. Show that there is no linear polynomial x + c which divides pn(x) for
all n > 2.

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