MathDB

3

Part of 1992 Vietnam National Olympiad

Problems(2)

tend to infty

Source: 30-th Vietnamese Mathematical Olympiad 1992

2/17/2007
Let a,b,ca,b,c be positive reals and sequences {an},{bn},{cn}\{a_{n}\},\{b_{n}\},\{c_{n}\} defined by ak+1=ak+2bk+ck,bk+1=bk+2ck+ak,ck+1=ck+2ak+bka_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}} for all k=0,1,2,...k=0,1,2,.... Prove that lim⁡k→+∞ak=lim⁡k→+∞bk=lim⁡k→+∞ck=+∞\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty.
limitinequalitiescalculuscalculus computations
array of squares

Source: 30-th Vietnamese Mathematical Olympiad 1992

2/17/2007
Label the squares of a 1991×19921991 \times 1992 rectangle (m,n)(m, n) with 1≤m≤19911 \leq m \leq 1991 and 1≤n≤19921 \leq n \leq 1992. We wish to color all the squares red. The first move is to color red the squares (m,n),(m+1,n+1),(m+2,n+1)(m, n), (m+1, n+1), (m+2, n+1)for some m<1990,n<1992m < 1990, n < 1992. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?
geometryrectanglecombinatorics proposedcombinatorics