Vietnam 98/A1 solution

36th VMO 1998

Problem A1

Define the sequence x₁, x₂, x₃, ... by x₁ = a ≥ 1, x_n+1 = 1 + ln(x_n(x_n²+3)/(1 + 3x_n²) ). Show that the sequence converges and find the limit.

Solution

(x-1)³ ≥ 0, so x(x²+3)/(1+3x²) ≥ 1. So the ln term is well-defined and non-negative and 1 + ln(x(x²+3)/(1+3x²)) ≥ 1. So by a trivial induction all x_n ≥ 1.

Also x² ≥ 1 implies (x²+3)/(1+3x²) ≤ 1, so x(x²+3)/(1+3x²) ≤ x and hence 1 + ln(x(x²+3)/(1+3x²)) ≤ 1 + ln x ≤ x (*). So the sequence is monotonically decreasing and bounded below by 1. So it must converge. Suppose the limit is L. Then L = 1 + ln(L(L²+3)/(1+Lx²)). But we only have equality in (*) at 1. Hence L = 1.

36th VMO 1998