27th Vietnamese Mathematical Olympiad 1989 Problems

27th Vietnamese Mathematical Olympiad 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, 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_n² + 1956a_n + 1960 is divisible by 1989. Does there exist a term such that 1985a_n² + 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 m² + 19n² = 198·10¹⁹⁸⁹? 

B2.  Define the sequence of polynomials p₀(x), p₁(x), p₂(x), ... by p₀(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.