22nd Vietnamese Mathematical Olympiad 1984 Problems

 22nd Vietnamese Mathematical Olympiad 1984 Problems

A1.  (1) Find a polynomial with integral coefficients which has the real number 2^1/2 + 3^1/3 as a root and the smallest possible degree.

(2) Find all real solutions to 1 + √(1 + x²) (√(1 + x)³ - √(1 - x)³ ) = 2 + √(1 - x²).

A2.  The sequence a₁, a₂, a₃, ... is defined by a₁ = 1, a₂ = 2, a_n+2 = 3a_n+1 - a_n. Find cot^-1a₁ + cot^-1a₂ + cot^-1a₃ + ... .

A3.  A cube side 2a has ABCD as one face. S is the other vertex (apart from B and D) adjacent to A. M, N are variable points on the lines BC, CD respectively.

(1) Find the positions of M and N such that the planes SMA and SMN are perpendicular, BM + DN ≥ 3a/2, and BM·DN has the smallest value possible.

(2) Find the positions of M and N such that angle NAM = 45 deg, and the volume of SAMN is (a) a maximum, (b) a minimum, and find the maximum and minimum.

(3) Q is a variable point (in space) such that ∠AQB = ∠AQD = 90^o. p is the plane ABS. Q' is the intersection of DQ and p. Find the locus K of Q'. Let CQ meet K again at R. Let R' be the intersection of DR and p. Show that sin²Q'DB + sin²R'DB is constant.

B1.  (1) m, n are integers not both zero. Find the minimum value of | 5m² + 11mn - 5n² |.

(2) Find all positive reals x such that 9x/10 = [x]/( x - [x] ). 

B2.  a, b are unequal reals. Find all polynomials p(x) which satisfy x p(x - a) = (x - b) p(x) for all x.

B3.  (1) ABCD is a tetrahedron. ∠CAD = z, ∠BAC = y, ∠BAD = x, the angle between the planes ACB and ACD is X, the angle between the planes ABC and ABD is Z, the angle between the planes ADB and ADC is Y. Show that sin x/sin X = sin y/sin Y = sin z/sin Z and that x + y = 180^o iff X + Y = 180^o.

(2) ABCD is a tetrahedron with ∠BAC = ∠CAD =∠DAB = 90^o. Points A and B are fixed. C and D are variable. Show that ∠CBD + ∠ABD + ∠ABC is constant. Find the locus of the center of the insphere of ABCD.