MathDB

3

Part of 2005 Vietnam National Olympiad

Problems(2)

Find the smallest n

Source: VietnamMO2005,problem3

3/10/2005
Let A1A2A3A4A5A6A7A8A_1A_2A_3A_4A_5A_6A_7A_8 be convex 8-gon (no three diagonals concruent). The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of A1A2A3A4A5A6A7A8A_1A_2A_3A_4A_5A_6A_7A_8 and such convex quadrilaterals will be called "sub quadrilaterals".Find the smallest nn satisfying: We can color n "button" such that for all i,k∈{1,2,3,4,5,6,7,8},i≠k,s(i,k)i,k \in\{1,2,3,4,5,6,7,8\},i\neq k,s(i,k) are the same where s(i,k)s(i,k) denote the number of the "sub quadrilaterals" has Ai,AkA_i,A_k be the vertices and the intersection of two its diagonals is "button".
combinatorics proposedcombinatorics
x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n.Find the limit

Source: VietNamMO 2005,problem 6

3/11/2005
Let {xn}\{x_n\} be a real sequence defined by: x1=a,xn+1=3xn3−7xn2+5xnx_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n For all n=1,2,3...n=1,2,3... and a is a real number. Find all aa such that {xn}\{x_n\} has finite limit when n→+∞n\to +\infty and find the finite limit in that cases.
limitinductionalgebra proposedalgebra