MathDB

1

Part of 1991 China Team Selection Test

Problems(2)

Polynomial inequality

Source: China TST 1991, problem 1

6/27/2005
Let real coefficient polynomial f(x)=xn+a1⋅xn−1+…+anf(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n has real roots b1,b2,…,bnb_1, b_2, \ldots, b_n, n≥2,n \geq 2, prove that ∀x≥max{b1,b2,…,bn}\forall x \geq max\{b_1, b_2, \ldots, b_n\}, we have f(x+1)≥2⋅n21x−b1+1x−b2+…+1x−bn.f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.
algebrapolynomialinequalitiesfunctionalgebra unsolved
circumference of radius 1

Source: China TST 1991, problem 4

6/27/2005
We choose 5 points A1,A2,…,A5A_1, A_2, \ldots, A_5 on a circumference of radius 1 and centre O.O. PP is a point inside the circle. Denote QiQ_i as the intersection of AiAi+2A_iA_{i+2} and Ai+1PA_{i+1}P, where A7=A2A_7 = A_2 and A6=A1.A_6 = A_1. Let OQi=di,i=1,2,…,5.OQ_i = d_i, i = 1,2, \ldots, 5. Find the product ∏i=15AiQi\prod^5_{i=1} A_iQ_i in terms of di.d_i.
geometry unsolvedgeometry