MathDB

1

Part of 2012 Vietnam Team Selection Test

Problems(2)

Sequence involving 2011 gives pefect square

Source: Vietnamese TST 2012 Problem 4

5/9/2012
Consider the sequence (xn)n≥1(x_n)_{n\ge 1} where x1=1,x2=2011x_1=1,x_2=2011 and xn+2=4022xn+1−xnx_{n+2}=4022x_{n+1}-x_n for all n∈Nn\in\mathbb{N}. Prove that x2012+12012\frac{x_{2012}+1}{2012} is a perfect square.
algebrapolynomialDiophantine equationnumber theory unsolvednumber theory
Tangents to cirumcircle meet at the fixed point T

Source: Viet Nam TST 2012 Day 1

4/17/2012
Consider a circle (O)(O) and two fixed points B,CB,C on (O)(O) such that BCBC is not the diameter of (O)(O). AA is an arbitrary point on (O)(O), distinct from B,CB,C. Let D,J,KD,J,K be the midpoints of BC,CA,ABBC,CA,AB, respectively, E,M,NE,M,N be the feet of perpendiculars from AA to BCBC, BB to DJDJ, CC to DKDK, respectively. The two tangents at M,NM,N to the circumcircle of triangle EMNEMN meet at TT. Prove that TT is a fixed point (as AA moves on (O)(O)).
geometrycircumcirclesearchgeometric transformationreflectiongeometry unsolved