TAILIEUCHUNG - Inequality in international tests in mathematics

Inequality problem is always fear for your college exam, they are difficult because they are diverse and they must know to solve a number of skills, if the inequality experienced some difficult without knowing to a number of techniques to solve. | 1. Prove that the inequality V 2 1 - b 2 Jb2 1 - c 2 Vc2 1 - a 2 holds for arbitrary real numbers a b c. Komal Solution 1 Applying Minkowsky s Inequality to the left hand side we have y a2 1 b 2 b2 1 c 2 c2 1 a 2 y 3 a b c 2 a b c 2. Denoting a b c xwe get 3 2 g 3 a b c 2 a c 2 2 x - - and the conclusion follows. Solution 2 We have the obvious inequalities y a2 1 - b 2 y b2 1 - c 2 y c2 1 - a 2 in 3 2 and the last quantity is clearly at least . 2. If a b c E 0 1 prove that Vabc l a l 1 c 1. Junior TST 2002 Romania Solution 1 Observe that x x for x G 0 1 . Thus we have Vabc y abc and V 1 - a l - 1 - c 1 - a l - 1 - c . By the AM-GM Inequality we get r r sr r a b c V abc v abc ----- 3 and y i-a i- i-c i-a i- i-c 1 a 1 6 1 c . 1 2 Summing up we obtain Vab c y i _ a i _ 6 i _ C a b c l-a l-b l-c o as desired. Solution 2 We clearly have Vabc 4- l a l 1 c y b yfc y X b y 1 c 1 the last one being an obvious application of Cauchy-Schwarz Inequality. Solution 3 2 2 2 7F Let a sin ar b sin y c sin z where x y z G 0 J. ie ineQuality becomes sin x sin y sin z cos x cos y cos z 1 and it follows from the obvious inequalities sin x sin y sin z cos x cos y cos z sin x sin y cos x cos y cos x y 1 Dinu erbanescu 3. Let a b c be positive numbers. Prove that 1 1 1 27 b a b c b c a c a 2 a b c 2 JBMO 2002 Solution By the AM-GM Inequality 1 1 1 3 27 c b c a c a J abc a 4- 6 6 c c 4- a z 11 i i i- . f a b c On the other hand using the same inequality we inter that I --------j abc and o j i 2 a b c i a b 4- 6 4- c 4- c 4- a . z . . P----------- P----------- P -----------------1 a 6 6 c c a . o j a j Multiplying these inequalities yields __1_ 33 -33 abc a c c a 23 a b c 6 as needed. 4. 3 5. Let aq X2 tanæ2 tanxn xn be the numbers from the interval n. Prove that sin x siu. q -sin. - 7T 0 J such that 1 tanæi Bosnia and Herzegovina 2002 Solution With the substitution tan cii the problem reduces to proving that for any y 2 a- ar an n we have TT . 8 1. But this follows from .

• Toán học
• Inequality
• math exam
• international math test
• exercises inequality
• international math
• international tests
• Construct integral inequality from algebraic inequality
• Construct integral inequality
• The order preserving property of the limit
• Known algebraic inequalities
• Research and development
• Voices against socio political inequality in critical discourse analysis
• Voices against socio political inequality
• Critical discourse analysis
• Socio political inequality
• Media discourse and political discourse
• Hardy Hilbert’s inequality
• Holder’s inequality
• β function
• Inequality similar
• Hilbert type inequalities
• Linear matrix inequality approach
• Linear matrix inequality approach to robust stability
• Uncertain nonlinear discrete time systems
• Linear matrix inequality
• The effectiveness of the result
• Bài tập tiếng Anh về kinh tế
• Bài tập kinh tế
• Income inequality and poverty
• Income inequality
• Bất bình đẳng
• Bài tập về đói nghèo
• Principles of economics
• Lecture Principles of economics
• Distribution of income
• U
• Poverty rate
• Popoviciu inequality
• Taylor formula
• Green function
• Cebysev functional
• Gruss inequality
• Ostrowski inequality
• Exponential convexity
• Cauchy mean
• Abstract convexity
• Functional inequalities
• Holder inequality
• Global optimization
• Inequality theory
• Hermite Hadamard inequality
• Bullen inequality
• Simpson inequality
• Fractional integral
• Convex function
• Dynamic Shum inequalities
• Opial’s inequality
• Beesack’s inequality
• Shum’s inequality
• Time scales
• M Convex function
• Jensen inequality
• Fejer inequality
• Hermite–Hadamard inequality
• Real numbers containing the zero
• Domestic gender inequality in mekong delta
• Domestic gender inequality
• School dropouts among girls
• Transnational marriage among women
• Gender and cultural factors
• Credit market depth
• Income inequality in Vietnam
• A panel data analysis
• The impact of financial inclusion on income inequality
• Financial inclusion on income inequality in transition economies
• Two stage least squares
• The GINI coefficient
• Developing financial inclusion
• Two different financial inclusion index
• Quasi variational inequality
• Remark on a quasi variational inequality
• Maxwell type equation
• Boundary conditions
• Holes has a solution
• Status seeking hypothesis
• Household and consumption savings
• Higher income inequality
• Unmarried headed
• Women headed households
• Financial development
• An empirical analysis
• Algebraic problems on the triangle inequality
• Inequalities of areas
• quadrilateral’s diagonals
• Miscellaneous problems on the triangle inequality
• The quadrilateral
• trình bày báo cáo
• tài liệu báo cáo khoa học
• báo cáo y học
• công trình y học
• nghiên cứu y học
• policy
• options
• guangdong
• province
• strategy
• china
• shared
• growth
• reducing
• báo cáo khoa học
• báo cáo hóa học
• công trình nghiên cứu về hóa học
• tài liệu về hóa học
• cách trình bày báo cáo
• Exchange Rate
• Policy Coordination
• thesis
• internship reports
• project course
• graduate thesis