赛题另解

Other Solutions of Mathematics Competition Problems

题1设xn=C2n^n(n=1,2,…).证明:存在无穷多有限集合对(A,B)(A、B■Z+),使得A∩B=φ,且∏i∈Axi/∏j∈Bxj=2012.(2012,中国国家集训队测试)证明注意到,xn+1/xn=C2(n+1)^n+1=2(n+1)/n+1C2n+1^n/C2n^n=2C2n+1^n/C2n^n=2·2n+1/n+1C2n^n/C2n^n=2(2n+1)/n+1.