宁海中学高一段组织了围棋比赛,共有10名选手进入了决赛,决赛阶段实行单循环赛(即每两名参赛选手都要赛一局,且每局比赛都决出胜负),若一号选手胜a 1 局,输b 1 局;二号选手胜a 2 局,输b 2 局,…,十号选手胜a 10 局,输b 10 局.试比较a 1 2 +a 2 2 +…+a 10 2 与b 1 2 +b 2 2 +…+b 10 2 的大小,并叙述理由.
依题意可知,a 1 +b 1 =9,a 2 +b 2 =9,a 3 +b 3 =9…, 且a 1 +a 2 +…+a 10 =b 1 +b 2 +…+b 10 =45, ∴(a 1 2 +a 2 2 +…+a 10 2 )-(b 1 2 +b 2 2 +…b 10 2 )=(a 1 2 -b 1 2 )+(a 2 2 -b 2 2 )+…+(a 10 2 -b 10 2 ) =(a 1 +b 1 )(a 1 -b 1 )+(a 2 +b 2 )(a 2 -b 2 )+…+(a 10 +b 10 )(a 10 -b 10 ) =9[(a 1 +a 2 +…+a 10 )-(b 1 +b 2 +…+b 10 )] =0, ∴a 1 2 +a 2 2 +…+a 10 2 =b 1 2 +b 2 2 +…b 10 2 . |