07/06/07 21:52:14
>51 上
[A.422]
Let x_1,x_2,…,x_n,x_(n+1) be positive real numbers with x_1+x_2+…+x_n = x_(n+1).
Prove that
Σ[i=1,n] √{x_i[x_(n+1) - x_i]} ≦ √{ Σ[i=1,n] x_(n+1)[x_(n+1) - x_i]},
(略解)
(左辺) ≦ (1/n){Σ[i=1,n] √x_i)}{Σ[j=1,n] √(x_{n+1} - x_j)} (← 逆順序積 ≦ 乱順序積)
≦ √{Σ[i=1,n] x_i}･√{Σ[j=1,n] (x_{n+1} - x_j)} (← コーシー)
= √(x_{n+1})･√{Σ[j=1,n] (x_{n+1} - x_j)}
= √{Σ[j=1,n] x_{n+1}[x_{n+1} - x_j]}
= (右辺).
[B.3989]
a,b,c are positive real numbers, such that a^2 +b^2 +c^2 +abc = 4.
Prove that a+b+c ≦ 3.
(略解)
1-a,1-b,1-c のうち2つは同符号、よって (1-b)(1-c) ≧0 としてもよい。
3(3-a-b-c) + (a^2 +b^2 +c^2 +abc -4)
= (1/2)(2-a-b)^2 + (1/2)(2-b-c)^2 + (1/2)(2-c-a)^2 -(1-a)(1-b)(1-c)
= (1/4)(3-2c-a)^2 + (1/4)(3-2b-a)^2 + (1/2)(1-a)^2 + a(1-b)(1-c)
≧ a(1-b)(1-c).
[C.892]
Prove that if x,y,z are positive real numbers and xyz=1, the values of the expressions
1/(1+x+xy), y/(1+y+yz), xz/(1+z+xz)
cannot all be greater than 1/3.
(略解)
1/(1+x+xy) = yz/(1+y+yz) = z/(1+z+xz),
xy/(1+x+xy) = y/(1+y+yz) = 1/(1+z+xz),
x/(1+x+xy) = 1/(1+y+yz) = xz/(1+z+xz),
辺々たす.