一道习题的四种解法及其启示

我在高三数学培优训练中出了这样一道题 :在△ＡＢＣ中证明ｃｏｓＡｃｏｓＢｃｏｓＣ≤ 18. ( 1)学生的解法使我既惊喜又欣慰 .有些解法是我开始没有预料到的 ,有些解法又暴露出同学们对某些数学思想、方法理解中的问题 ,现呈献给大家供参考 .分析 1:积化和差会出现ｃｏｓ (Ｂ Ｃ )而在△ＡＢＣ中 ,ｃｏｓ(Ｂ Ｃ) =-ｃｏｓＡ ,从而产生下列解法 :解法 1　ｃｏｓＡｃｏｓＢｃｏｓＣ= 12 ｃｏｓＡ[ｃｏｓ(Ｂ Ｃ) ｃｏｓ(Ｂ -Ｃ) ]= - 12 ｃｏｓ2 Ａ 12 ｃｏｓ(Ｂ -Ｃ)ｃｏｓＡ= - 12 [ｃｏｓＡ - 12 ｃｏｓ(Ｂ -Ｃ) ] 2 18ｃｏｓ2 (Ｂ…     

构造法简证一类三角不等式

１９９６年,周永良先生在全国第三届初等数学研究学术交流会论文集中提出如下三角不等式在锐角三角形ＡＢＣ中,有ｃｏｓ（Ｂ－Ｃ）ｃｏｓＡ＋ｃｏｓ（Ｃ－Ａ）ｃｏｓＢ＋ｃｏｓ（Ａ－Ｂ）ｃｏｓＣ≥６（１）ｃｏｓＡｃｏｓ（Ｂ－Ｃ）＋ｃｏｓＢｃｏｓ（Ｃ－Ａ）＋ｃｏｓ...     

对《一个不等式的另几种证法与推广》的商榷与补充

本刊 2 0 0 0年第 2期刊出了《一个不等式的另几种证法与推广》[1] ,读后受益匪浅 .1　一点商榷意见及修正意见文 [1 ]在证明ａ3 ｂ3 ｃ3≥ 3ａｂｃ (ａ ,ｂ ,ｃ∈Ｒ )时 ,似有“循环论证”之嫌 .现恭抄其证明过程如下 :　　　　ａ3 ｂ3≥ａ2 ｂ ａｂ2 ≥ 2ａ3ｂ3,　　　　ｂ3 ｃ3≥ｂ2 ｃ ｂｃ2 ≥ 2ｂ3ｃ3,　　　　ｃ3 ａ3≥ｃ2 ａ ｃａ2 ≥ 2ｃ3ａ3,三式相加得ａ3 ｂ3 ｃ3≥ａ3ｂ3 ｂ3ｃ3 ｃ3ａ3　　　　　　　≥ 33 ａ6 ｂ6 ｃ6 =3ａｂｃ.在证明过程的最后 ,作者绕了一个大圈 ,最终还是利用ａ3 ｂ3 ｃ3≥ 3ａｂｃ来证明ａ3 ｂ3 ｃ3≥ …     

一个三角不等式猜想的证明及推广

刘健先生在文［１］中提出如下猜想：在任意△ＡＢＣ中,有ｃｏｓＢｃｏｓＣｓｉｎＡ２＋ｃｏｓＣｃｏｓＡｓｉｎＢ２＋ｃｏｓＡｃｏｓＢｓｉｎＣ２＜１①笔者通过研究,发现了这个不等式的一个指数形式：定理在△ＡＢＣ中,有ｃｏｓＢｃｏｓＣｓｉｎｋＡ２＋ｃｏｓＣｃｏ...     

解三角形不容忽视的两个推论

解答三角形中的问题 ,除正确运用正弦定理、余弦定理外 ,还应重视它们的两个推论 .利用推论解题 ,方法简捷 ,过程明了 .1　两个推论推论 1　在△ＡＢＣ中 ,有ａ =ｂｃｏｓＣ +ｃｃｏｓＢ ,ｂ =ｃｃｏｓＣ +ａｃｏｓＣ ,ｃ =ａｃｏｓＢ +ｂｃｏｓＡ .推论 2　在△ＡＢＣ中 ,有ｂｃｏｓＢ +ｃｃｏｓＣ=ａｃｏｓ(Ｂ -Ｃ)≤ａ (1 )ｃｃｏｓＣ +ａｃｏｓＡ =ｂｃｏｓ(Ｃ -Ａ)≤ｂ (2 )ａｃｏｓＡ +ｂｃｏｓＢ =ｃｃｏｓ(Ａ -Ｂ)≤ｃ (3 )当且仅当 :Ｂ =Ｃ时 ,(1 )中等号成立 ;Ｃ=Ａ时 ,(2 )中等号成立 ;Ａ =Ｂ时 ,(3 )中等号成立 .证　推论 1 :由…     

Ricci曲率平行的一类Riemann流形的C~∞紧性(英文)

本文证明,在Ｇｒｏｍｏｖ－Ｈａｕｓｄｏｒｆｆ拓扑下,Ｒｉｃｃｉ曲率平行,截面曲率和单一半径有下界,体积有上界的Ｒｉｅｍａｎｎ流形的集合是ｃ∞紧的．作为应用,我们证明一个ｐｉｎｃｈｉｎｇ结果,即在某些条件下,Ｒｕｃｃｉ平坦的流形必定平坦．     

注意三角形中的一个隐含条件

在三角形中 ,隐含着一个非常重要的条件 ,而同学们在解题时常常忽略该条件 ,从而造成解题失误 .这个条件就是 :三角形中 ,任意两内角的余弦之和为正 ,即△ＡＢＣ中 ,ｃｏｓＡ ｃｏｓＢ >0 ,ｃｏｓＢ ｃｏｓＣ >0 ,ｃｏｓＣ ｃｏｓＡ >0 .证明　∵△ＡＢＣ中 ,0 <Ａ Ｂ <π ,∴ 0 <Ａ <π -Ｂ <π ,又余弦函数在 (0 ,π)上是减函数 ,∴ｃｏｓＡ >ｃｏｓ(π -Ｂ) ,即ｃｏｓＡ >-ｃｏｓＢ ,故ｃｏｓＡ ｃｏｓＢ >0 .同理可证 :ｃｏｓＢ ｃｏｓＣ >0 ,ｃｏｓＣ ｃｏｓＡ >0 .下面举例说明这一条件的应用 .例 1　已知△ＡＢＣ中 ,ｃｏｓＡ…     

三面角的棱面角的计算公式

定理　在三面角Ｓ—Ａ1 Ｂ1 Ｃ1 中 ,三个面角∠Ｃ1 ＳＢ1 =α ,∠Ａ1 ＳＣ1 =β ,∠Ａ1 ＳＢ1 =γ ,且棱ＳＡ1 和平面Ｃ1 ＳＢ1 所成棱面角为θ1 ,棱ＳＢ1 和平面Ａ1 ＳＣ1 所成棱面角为θ2 ,棱ＳＣ1 和平面Ａ1 ＳＢ1 所成棱面角为θ3,则ｃｏｓθ1 =ｃｏｓ2 β ｃｏｓ2 γ- 2ｃｏｓαｃｏｓβｃｏｓγｓｉｎαｃｏｓθ2 =ｃｏｓ2 γ ｃｏｓ2 α- 2ｃｏｓαｃｏｓβｃｏｓγｓｉｎβｃｏｓθ3=ｃｏｓ2 α ｃｏｓ2 β- 2ｃｏｓαｃｏｓβｃｏｓγｓｉｎγ(三面角的棱面角余弦公式 )证明　如图 .在ＳＡ1 上取一点Ｐ ,作ＰＱ⊥平面Ｂ1 ＳＣ1 …     

数学问题解答

20 0 0年 9月号问题解答(解答由问题提供人给出 )1 2 71　已知ａ,ｂ,ｃ∈Ｒ ,求证 :当ａｂｃ≤ 1时 ,ａｂ ｂｃ ｃａ ≥ａ ｂ ｃ.当ａｂｃ≥ 1时 ,ａｂ ｂｃ ｃａ >1ａ 1ｂ 1ｃ证明　当ａｂｃ≤ 1时 ,(1 )当ａ ｂ ｃ≥ａｂ ｂｃ ｃａ时 ,∵　 ａｂ ｂｃ ｃａ ａｂ ｂｃ ｃａ≥ 2 (ａ ｂ ｃ)∴　 ａｂ ｂｃ ｃａ≥ 2 (ａ ｂ ｃ) - (ａｂ ｂｃ ｃａ)≥ａ ｂ ｃ(2 )当ａｂ ｂｃ ｃａ ≥ａ ｂ ｃ时 ,∵　ａ2 ｃ ｂ2 ａ ｃ2 ｂ ｃ ａ ｂ≥ 2 (ａｃ ｂａ ｃｂ)∴　ａ2 ｃ ｂ2 ａ ｃ2 ｂ≥ 2 (ａｃ ｂａ ｃｂ) - (ｃ ａ…     

一个不等式的证法与推广

若ａ、ｂ、ｃ∈Ｒ＋,且ａ＋ｂ＋ｃ＝１,则ｂｃａ＋ａｃｂ＋ａｂｃ＋ａｂｃ＋ｂａｃ＋ｃａｂ≥１０．证明ｂｃａ＋ａｃｂ＋ａｂｃ＋ａｂｃ＋ｂａｃ＋ｃａｂ＝１２［ａ（ｃｂ＋ｂｃ）＋ｂ（ｃａ＋ａｃ）＋ｃ（ｂａ＋ａｂ）＋１ａ（ｃｂ＋ｂｃ）＋１ｂ（ｃａ＋ａｃ）＋１ｃ...     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.