MathDB

2

Part of 1996 China Team Selection Test

Problems(2)

f(n+1) >= f(n) >= n/(n=1) * f(2n)

Source: China TST 1996, problem 2

5/17/2005
SS is the set of functions f:N→Rf:\mathbb{N} \to \mathbb{R} that satisfy the following conditions: I. f(1)=2f(1) = 2 II. f(n+1)≥f(n)≥nn+1f(2n)f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n) for n=1,2,…n = 1, 2, \ldots Find the smallest M∈NM \in \mathbb{N} such that for any f∈Sf \in S and any n∈N,f(n)<Mn \in \mathbb{N}, f(n) < M.
functionlimitinductionstrong inductionalgebra unsolvedalgebra
China TST 1996 sum inequalities

Source: China TST 1996, problem 5

5/17/2005
Let α1,α2,…,αn\alpha_1, \alpha_2, \dots, \alpha_n, and β1,β2,…,βn\beta_1, \beta_2, \ldots, \beta_n, where n≥4n \geq 4, be 2 sets of real numbers such that ∑i=1nαi2<1and∑i=1nβi2<1.\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1. Define \begin{align*} A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\ B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\ W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2. \end{align*} Find all real numbers λ\lambda such that the polynomial xn+λ(xn−1+⋯+x3+Wx2+ABx+1)=0,x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0, only has real roots.
inequalitiesalgebraSums and ProductsReal RootsChina