27th Vietnam 1989 problems

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 a1 = 1, a2 = 1, an+2 = an+1 + an be the Fibonacci sequence. Show that there are infinitely many terms of the sequence such that 1985an2 + 1956an + 1960 is divisible by 1989. Does there exist a term such that 1985an2 + 1956an + 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 m2 + 19n2 = 198·101989?
B2.  Define the sequence of polynomials p0(x), p1(x), p2(x), ... by p0(x) = 0, pn+1(x) = pn(x) + (x - pn(x)2)/2. Show that for any 0 ≤ x ≤ 1, 0 < √x - pn(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.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
23 July 2002