MathDB

1

Part of 2001 China Team Selection Test

Problems(2)

China TST 2001 interior points of convex quadrilateral

Source: China TST 2001, problem 1

5/22/2005
EE and FF are interior points of convex quadrilateral ABCDABCD such that AE=BEAE = BE, CE=DECE = DE, ∠AEB=∠CED\angle AEB = \angle CED, AF=DFAF = DF, BF=CFBF = CF, ∠AFD=∠BFC\angle AFD = \angle BFC. Prove that ∠AFD+∠AEB=π\angle AFD + \angle AEB = \pi.
geometryChina
Minimum value of fraction sum product

Source: China TST 2001, problem 4

5/22/2005
For a given natural number n>3n > 3, the real numbers x1,x2,…,xn,xn+1,xn+2x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2} satisfy the conditions 0<x1<x2<⋯<xn<xn+1<xn+20 < x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}. Find the minimum possible value of (∑i=1nxi+1xi)(∑j=1nxj+2xj+1)(∑k=1nxk+1xk+2xk+12+xkxk+2)(∑l=1nxl+12+xlxl+2xlxl+1)\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j + 1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l x_{l + 1}})} and find all (n+2)(n + 2)-tuplets of real numbers (x1,x2,…,xn,xn+1,xn+2)(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}) which gives this value.
algebra unsolvedalgebra