11th Vietnamese Mathematical Olympiad 1972 Problems

11th Vietnamese Mathematical Olympiad 1972 Problems

A1.  Let x = cos α, y = cos nα, where n is a positive integer. Show that for each x in the range [-1, 1], there is only one corresponding y. So consider y as a function of x and put y = T_n(x). Find T₁(x) and T₂(x) and show that T_n+1(x) = 2x T_n(x) - T_n-1(x). Show that T_n(x) is a polynomial of degree n with n roots in [-1, 1].

A2.  For any positive integer n, let f(n) = ∑ (-1)^(d-1)/2 where the sum is taken over all odd d dividing n. Show that:

f(2n) = 1

f(p) = 2 for p a prime congruent to 1 mod 4

f(p) = 0 for p a prime congruent to 3 mod 4

f(pn) = n+1 for p a prime congruent to 1 mod 4

f(pn) = 1 for p a prime congruent to 3 mod 4, and n even

f(pn) = 0 for p a prime congruent to 3 mod 4, and n odd

Show that f(mn) = f(m) f(n) for m and n relatively prime. Find f(5⁴11²⁸17¹⁹) and f(1980). Show how to calculate f(n).
B1.  ABC is a triangle. U is a point on the line BC. I is the midpoint of BC. The line through C parallel to AI meets the line AU at E. The line through E parallel to BC meets the line AB at F. The line through E parallel to AB meets the line BC at H. The line through H parallel to AU meets the line AB at K. The lines HK and FG meet at T. V is the point on the line AU such that A is the midpoint of UV. Show that V, T and I are collinear. [Next part unclear.]
B2.  ABCD is a regular tetrahedron with side a. Take E, E' on the edge AB such that AE = a/6, AE' = 5a/6. Take F, F' on the edge AC such that AF = a/4, AF' = 3a/4. Take G, G' on the edge AD such that AG = a/3, AG' = 2a/3. Find the intersection of the planes BCD, EFG and E'F'G' and its position in the triangle BCD. Calculate the volume of EFGE'F'G' and find the angles between the lines AB, AC, AD and the plane EFG.