MathDB

1

Part of 2004 China National Olympiad

Problems(2)

Exactly one sequence x_i

Source: Chinese MO 2004

9/25/2011
For a given real number aa and a positive integer nn, prove that: i) there exists exactly one sequence of real numbers x0,x1,…,xn,xn+1x_0,x_1,\ldots,x_n,x_{n+1} such that {x0=xn+1=0,12(xi+xi+1)=xi+xi3−a3, i=1,2,…,n.\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases} ii) the sequence x0,x1,…,xn,xn+1x_0,x_1,\ldots,x_n,x_{n+1} in i) satisfies ∣xi∣≤∣a∣|x_i|\le |a| where i=0,1,…,n+1i=0,1,\ldots,n+1.
Liang Yengde
functioninductionalgebra proposedalgebra
Find expression for F_1C/CG_1 in terms of lambda

Source: Chinese MO 2004

9/25/2011
Let EFGH,ABCDEFGH,ABCD and E1F1G1H1E_1F_1G_1H_1 be three convex quadrilaterals satisfying:
i) The points E,F,GE,F,G and HH lie on the sides AB,BC,CDAB,BC,CD and DADA respectively, and AEEB⋅BFFC⋅CGGD⋅DHHA=1\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1; ii) The points A,B,CA,B,C and DD lie on sides H1E1,E1F1,F1,G1H_1E_1,E_1F_1,F_1,G_1 and G1H1G_1H_1 respectively, and E1F1∣∣EF,F1G1∣∣FG,G1H1∣∣GH,H1E1∣∣HEE_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE.
Suppose that E1AAH1=λ\frac{E_1A}{AH_1}=\lambda. Find an expression for F1CCG1\frac{F_1C}{CG_1} in terms of λ\lambda.
Xiong Bin
geometry proposedgeometry