不等式（1）通常称为柯西（Cauchy）不等式。有关这个不等式,已有不少中学数学杂志论及,本文在这里主要通过实例来说明它在解数学竞赛题时所起的重要作用,以及如何利用柯西不等式来解题,从而为中学数学课外活动提供一点辅导资料。 例1 ...不等式（1）通常称为柯西（Cauchy）不等式。有关这个不等式,已有不少中学数学杂志论及,本文在这里主要通过实例来说明它在解数学竞赛题时所起的重要作用,以及如何利用柯西不等式来解题,从而为中学数学课外活动提供一点辅导资料。 例1 已知a1,a2,…,ak,…为两两各不相同的正整数,求证对任意正整数n,都有 sum from k=1 to n（ak/k2）≥sum from k=1 to n（1/k） （3） （第20届国际数学竞赛题,1978年）展开更多
第四届（1989年）全国中学生数学冬令营试题的第二题是: 设x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>都是正数（n≥2）,且sum from i=1 to n x<sub>i</sub>=1,求证: 二/X。 sum f...第四届（1989年）全国中学生数学冬令营试题的第二题是: 设x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>都是正数（n≥2）,且sum from i=1 to n x<sub>i</sub>=1,求证: 二/X。 sum from i=1 to n x<sub>i</sub>/1-x<sub>i</sub><sup>1/2</sup>≥sum from i=1 to n x<sub>i</sub><sup>1/2</sup>/n-1<sup>1/2</sup>.（1） 本文对这道试题作出如下推广: 设x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>都是正数（n≥2）,且sum from i=1 to n x<sub>i</sub>=A】0,若α≥1,β】0,0【γ【1,展开更多
Let S=A_oA_1…A. be a simplex in n-dimensional Euclidean Space E^n with centroid G. The straight line A,G intersects the surface A_0A_1…Ai_l A_i+l…A. of S at the point G, and the circumsphere F of Sat the point Ai(i...Let S=A_oA_1…A. be a simplex in n-dimensional Euclidean Space E^n with centroid G. The straight line A,G intersects the surface A_0A_1…Ai_l A_i+l…A. of S at the point G, and the circumsphere F of Sat the point Ai(i=0,1,…,n). Let the edge-lengths of S be a_ij=A_iA_j(i,j=0,1,…,n, i≠j) and the me dians be m,=A_iG_i(i=O,1,…,n). Following theorem will be proved. Theorem. For the simplex in E^n, wo have sum from i=0 to x A_iA_i≥2n/n+1t=1 sum from i=0 to x m_i; (1) sum from i=0 to x A_i A_i≥2^(2/1)~n(N+1) sum from 0≤i<f≤x a_if; (2) sum from i=v to x (A_i A_i)~2≥4n+1 sum from 0≤i<f≤x ai; (3) The equalities in (1) and (3) hold if and only if the centroid G and the center O of the circumsphere of S are concurrent. The equality in (2) holds if and only if S is regular simplex.展开更多
Let A={A_1, A_2,…, A_(n+1)} be a simplex in E^n which its center O of circumscribed sphere is in inside of A. If R and R_i are radiuses of A_i respectively (A_i={A_1, A_2,…, A_(i-1), O, A_(i+1),…,A_(n+1)} ,i=1,2,…...Let A={A_1, A_2,…, A_(n+1)} be a simplex in E^n which its center O of circumscribed sphere is in inside of A. If R and R_i are radiuses of A_i respectively (A_i={A_1, A_2,…, A_(i-1), O, A_(i+1),…,A_(n+1)} ,i=1,2,…,n+1),then we have The equality holds if and only if A is a regular simplex.展开更多
文摘不等式（1）通常称为柯西（Cauchy）不等式。有关这个不等式,已有不少中学数学杂志论及,本文在这里主要通过实例来说明它在解数学竞赛题时所起的重要作用,以及如何利用柯西不等式来解题,从而为中学数学课外活动提供一点辅导资料。 例1 已知a1,a2,…,ak,…为两两各不相同的正整数,求证对任意正整数n,都有 sum from k=1 to n（ak/k2）≥sum from k=1 to n（1/k） （3） （第20届国际数学竞赛题,1978年）
文摘第四届（1989年）全国中学生数学冬令营试题的第二题是: 设x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>都是正数（n≥2）,且sum from i=1 to n x<sub>i</sub>=1,求证: 二/X。 sum from i=1 to n x<sub>i</sub>/1-x<sub>i</sub><sup>1/2</sup>≥sum from i=1 to n x<sub>i</sub><sup>1/2</sup>/n-1<sup>1/2</sup>.（1） 本文对这道试题作出如下推广: 设x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>都是正数（n≥2）,且sum from i=1 to n x<sub>i</sub>=A】0,若α≥1,β】0,0【γ【1,
文摘Let S=A_oA_1…A. be a simplex in n-dimensional Euclidean Space E^n with centroid G. The straight line A,G intersects the surface A_0A_1…Ai_l A_i+l…A. of S at the point G, and the circumsphere F of Sat the point Ai(i=0,1,…,n). Let the edge-lengths of S be a_ij=A_iA_j(i,j=0,1,…,n, i≠j) and the me dians be m,=A_iG_i(i=O,1,…,n). Following theorem will be proved. Theorem. For the simplex in E^n, wo have sum from i=0 to x A_iA_i≥2n/n+1t=1 sum from i=0 to x m_i; (1) sum from i=0 to x A_i A_i≥2^(2/1)~n(N+1) sum from 0≤i<f≤x a_if; (2) sum from i=v to x (A_i A_i)~2≥4n+1 sum from 0≤i<f≤x ai; (3) The equalities in (1) and (3) hold if and only if the centroid G and the center O of the circumsphere of S are concurrent. The equality in (2) holds if and only if S is regular simplex.
文摘Let A={A_1, A_2,…, A_(n+1)} be a simplex in E^n which its center O of circumscribed sphere is in inside of A. If R and R_i are radiuses of A_i respectively (A_i={A_1, A_2,…, A_(i-1), O, A_(i+1),…,A_(n+1)} ,i=1,2,…,n+1),then we have The equality holds if and only if A is a regular simplex.
