A1. Show that the absolute value of sin(kx) /N + sin(kx + x) /(N+1) + sin(kx + 2x) /(N+2) + ... + sin(kx + nx) /(N+n) does not exceed the smaller of (n+1) |x| and 1/(N sin(x/2) ), where N is a positive integer and k is real and satisfies 0 ≤ k ≤ N.
A2. Let a_{1} = 1, a_{2} = 1, a_{n+2} = a_{n+1} + a_{n} be the Fibonacci sequence. Show that there are infinitely many terms of the sequence such that 1985a_{n}^{2} + 1956a_{n} + 1960 is divisible by 1989. Does there exist a term such that 1985a_{n}^{2} + 1956a_{n} + 1960 + 2 is divisible by 1989?
A3. ABCD is a square side 2. The segment AB is moved continuously until it coincides with CD (note that A is brought into coincidence with the opposite corner). Show that this can be done in such a way that the region passed over by AB during the motion has area < 5π/6.
B1. Do there exist integers m, n not both divisible by 5 such that m^{2} + 19n^{2} = 198·10^{1989} ?
B2. Define the sequence of polynomials p_{0}(x), p_{1}(x), p_{2}(x), ... by p_{0}(x) = 0, p_{n+1}(x) = p_{n}(x) + (x - p_{n}(x)^{2})/2. Show that for any 0 ≤ x ≤ 1, 0 < √x - p_{n}(x) ≤ 2/(n+1).
B3. ABCDA'B'C'D' is a parallelepiped (with edges AB, BC, CD, DA, AA', BB', CC', DD', A'B', B'C', C'D', D'A'). Show that if a line intersects three of the lines AB', BC', CA', AD', then it also intersects the fourth. |