11th Vietnamese College Students 2003

11th Vietnamese Mathematics Olympiad for College Students 2003


R is the set of real numbers. [a, b] represents the closed interval, and (a, b) represents the open interval.

A1. A is the 4 x 4 matrix a₁₁ = a₂₂ = a₃₃ = a₄₄ = a, a₁₂ = a₂₁ = a₃₄ = a₄₃ = b, a₂₃ = a₃₂ = -1, other entries 0, where a, b are real with a > \|b\|. Show that the eigenvalues of A are positive reals.
A2. B is the 3 x 3 matrix with b₁₁ = a, b₂₂ = d, b₃₃ = q, b₁₂ = b^α/β, b₁₃ = c^α/γ, b₂₁ = b^β/α, b₂₃ = p^β/γ, b₃₁ = c^γ/α, b₃₂ = p^γ/β, where a, b, c, d, p, q are reals and α, β, γ are non-zero reals. Show that B has real eigenvalues.
A3. D_k is the k x k matrix with 0s down the main diagonal, 1s for all other entries in the first row and first column, and x for all other entries. Find det D₂ + det D₃ + ... + det D_n.
A4. I_n denotes the n x n unit matrix (so I₁₁ = I₂₂ = ... = I_nn = 1, other entries 0). P and Q are n x n matrices such that PQ = QP and P^r = Q^s = 0 for some positive integers r, s. Show that I_n + (P + Q) and I_n - (P + Q) are inverses.
A5. A is a square matrix such that A²⁰⁰³ = 0. Show that rank A = rank(A + A² + ... + Aⁿ) for all n.
A6. A is the 4 x 4 matrix with a₁₁ = 1 + x₁, a₂₂ = 1 + x₂, a₃₃ = 1 + x₃, a₄₄ = 1 + x₄, and all other entries 1, where x_i are the roots of x⁴ - x + 1. Find det A.
A7. p(x) is a polynomial of order n > 1 with real coefficients and m real roots. Show that (x² + 1)p(x) + p'(x) has at least m real roots.
B1. Find all continuous functions f : R→R such that f(x + 2002) ( f(x) + √2003) = -2004 for all x.
B2. Find all continuous functions f : [0, 1]→R which are differentiable on the open interval (0, 1) and satisfy f(0) = f(1) = 1, and 2003 f ' (x) + 2004 f(x) ≥ 2004 for all x in (0, 1).
B3. Given a < b, and a differentiable real-valued function f on [a, b] such that f(a) = - (b - a)/2, f(b) = (b - a)/2 and f( (a+b)/2 ) is non-zero, prove that there are three distinct numbers c₁, c₂, c₃ in (a, b) such that the product f '(c₁) f '(c₂) f '(c₃) = 1.
B4. Let x_k = 1/2! + 2/3! + 3/4! + ... + k/(k+1)!. Find lim (x₁ⁿ + x₂ⁿ + ... + x₂₀₀₃ⁿ )^1/n as n→∞.
B5. f : [0, π/2]→R is continuous, and f(0) > 0, ∫₀^π/2 f(t) dt ≤ 1. Prove that for some z in (0, π/2) we have f(z) = sin z.
B6. Given a < b, we are given any continuous functions f, g : [a, b]→[a, b] such that f( g(x) ) = g( f(x) ) for all x, and f is monotonic. Show that f(z) = g(z) = z for some z in [a, b].

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