challenging mathematical problems with solution
challenging mathematical problems with elementary solutions
... continuum of possible outcomes - 30 CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY SOLurIONS Volume I Combinatorial Analysis and Probability Theory PROBLEMS The problems in this volume are related ... The first volume contains 100 problems and detailed solutions to them Although the problems differ greatly in formulation and method of solution, they all deal with a single branch of mathematics: ... excellent book, Mathematical Snapshots (New York: Oxford University Press, 1960), and presented in the form of problems together with detailed solutions If the book is used in this way, the solution...
- 240
- 377
- 0
MEASURE and INTEGRATION Problems with Solutions
... sequence (En )n∈N in A such that lim En = ∅ with n→∞ lim µ(En ) = n→∞ Solution Since X is a infinite set, we can find an countably infinite set {x1 , x2 , } ⊂ X with xi = xj if i = j Let En = {xn , xn+1 ... measurable (ii) For every ε > 0, there exists an open set O ⊃ E with µ∗ (O \ E) ≤ ε L (iii) There exists a Gδ -set G ⊃ E with µ∗ (G \ E) = L Solution • (i) ⇒ (ii) Suppose that E is measurable Then ∗ ... of an increasing sequence {An : n ∈ N} in A with µ(An ) = for every n ∈ N, but µ(X) = (d) Show that when X is uncountably, the µ is countably additive Solution (a) Suppose A, B ∈ A and A ∩ B =...
- 158
- 1,163
- 0
Problems with Solutions potx
... bisector of CE and therefore coincides with Q Comment In this solution we proved more general fact: SC = SP if and only if M K = M L Solution As in the previous solution, we assume that S lies on ... intersect on Γ Solution Let X be the second point of intersection of line EI with Γ, and L be the foot of the bisector of angle BAC Let G′ and T be the points of intersection of segment DX with lines ... integers ℓ and N , with ℓ ≤ s and such that an = aℓ +an−ℓ for all n ≥ N Solution First, from the problem conditions we have that each an (n > s) can be expressed as an = aj1 + aj2 with j1 , j2
- 15
- 305
- 0
100 Nice Polynomial Problems With Solutions - Amir Hossein Parvardi.pdf
... + p(ca, b2 + 1) = Solutions http://www.artofproblemsolving.com/Forum/viewtopic.php?t=382979 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=385331 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=337211 ... two polynomials with integer coefficients 78 Find all polynomials P (x), Q(x), R(x) with real coefficients such that P (x) − Q(x) = R(x) ∀x ∈ R √ 79 Let k = 3 Find a polynomial p(x) with rational coefficients ... polynomials P (x) with integer coefficients such that the polynomial Q(x) = (x2 + 6x + 10) · P (x) − is the square of a polynomial with integer coefficients 49 Find all polynomials p with real coefficients...
- 13
- 692
- 3
Hanoi Open Mathematical Olympiad - Problems and solutions pdf
... Hanoi Open Mathematical Olympiad 1.1 Hanoi Open Mathematical Olympiad 2006 1.1.1 Junior Section, Sunday, April 2006 1.1.2 Senior Section, Sunday, April 2006 1.2 Hanoi Open Mathematical ... April 2007 1.3 Hanoi Open Mathematical Olympiad 2008 1.3.1 Junior Section, Sunday, 30 March 2008 1.3.2 Senior Section, Sunday, 30 March 2008 1.4 Hanoi Open Mathematical Olympiad 2009 ... 3 5 10 10 11 12 12 14 Questions of Hanoi Open Mathematical Olympiad 1.1 Hanoi Open Mathematical Olympiad 2006 1.1.1 Junior Section, Sunday, April 2006 Q1 What is...
- 16
- 3,087
- 28
Báo cáo toán học: " Existence of positive solutions to discrete secondorder boundary value problems with indefinite weight" pptx
... positive solution and a negative solution of (1.1) and (1.2) Definition 1.1 A positive solution of problem (1.1) and (1.2) refers to a pair (r, u), where r ≠ 0, u is a solution of (1.1) with u ... boundary value problems with indefinite weight Nonlinear Anal: TMA 71, 2119–2125 (2009) doi:10.1016/j.na.2009.01.046 Agarwal, RP, Henderson, J: Positive solutions and nonlinear eigenvalue problems ... discrete Dirichlet problems with lower and upper solutions Nonlinear Anal: TMA 67(4), 1236–1245 (2007) doi:10.1016/j.na.2006.07.010 10 Rodriguez, J: Nonlinear discrete Sturm-Liouville problems J Math...
- 10
- 503
- 0
Báo cáo hóa học: " A Caccioppoli-type estimate for very weak solutions to obstacle problems with weight" potx
... regularity theory for very weak solutions of the A-harmonic equations with w(x) ≡ have been considered [4], and the regularity theory for very solutions of obstacle problems with w(x) ≡ have been explored ... gives a Caccioppoli-type estimate for solutions to obstacle problems with weight, which is closely related to the local regularity theory for very weak solutions of the A-harmonic equation (1.2) ... obstacle problems with w(x) ≡ was first considered by Li and Martio [2] in 1994, using the so-called reverse Hölder inequality Gao and Tian [3] gave a local regularity result for weak solutions...
- 7
- 376
- 0
báo cáo hóa học: " Sign-changing solutions for some nonlinear problems with strong resonance" pot
... theory with applications to differential equations CBMS Reg Cof Ser Math 65 (1986) doi:10.1186/1687-2770-2011-18 Cite this article as: Qian: Sign-changing solutions for some nonlinear problems with ... sign-changing solutions (Mj denotes the eigenspace corresponding to lj) Remark 1.2 In the article [1], they only show the existence of solutions to problem (1.3), while we obtain its sign-changing solutions ... condition In problems which not have resonance at infinity, the (PS) condition is easy to verify On the other hand, a weaker condition than the condition (PS) is needed to study problems with strong...
- 9
- 277
- 0
Báo cáo hóa học: " Existence of positive solutions to periodic boundary value problems with sign-changing Green’s function" ppt
... article as: Zhong and An: Existence of positive solutions to periodic boundary value problems with sign-changing Green’s function Boundary Value Problems 2011 2011:8 ... has a fixed point in KR \Kr, which is the positive solution of (2) (9) Zhong and An Boundary Value Problems 2011, 2011:8 http://www.boundaryvalueproblems.com/content/2011/1/8 Page of Remark Theorem ... periodic boundary value problem with vanishing Green’s function Appl Math Lett 21, 176–180 (2008) doi:10.1016/j.aml.2007.02.019 Webb, TRL: Boundary value problems with vanishing Green’s function...
- 6
- 366
- 0
Báo cáo hóa học: " Research Article Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application" ppt
... equilibrium problems, ” Journal of Global Optimization, vol 36, no 3, pp 339– 349, 2006 19 N J Huang, J Li, and H B Thompson, “Implicit vector equilibrium problems with applications,” Mathematical ... quasiequilibrium problems and optimization problems, ” Journal of Global Optimization, vol 37, pp 195–213, 2007 25 J Li, N J Huang, and J K Kim, “On implicit vector equilibrium problems, ” Journal of Mathematical ... coincides with the solution set of DSVEP F, K Remark 3.8 Theorem 3.7 establishes the necessary and sufficient conditions for the vector equilibrium problem to have nonempty and bounded solution...
- 15
- 387
- 0
báo cáo hóa học:" Research Article Global Structure of Nodal Solutions for Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities" pot
... structure of positive solutions of 1.1 , 1.2 with f0 ∞ However, to the best of our knowledge, there is no paper to discuss the global structure of nodal solutions of 1.1 , 1.2 with f0 ∞ In this paper, ... value problems, ” Boundary Value Problems, vol 2008, Article ID 254593, 10 pages, 2008 R Ma and Y An, “Global structure of positive solutions for superlinear second order m-point boundary value problems, ” ... Boundary Value Problems Theorem 1.6 Let (A1)–(A4) hold a If f∞ ν 0, then 1.1 , 1.2 has at least one solution in Tk for any λ ∈ 0, ∞ ν b If f∞ ∈ 0, ∞ , then 1.1 , 1.2 has at least one solution in...
- 12
- 309
- 0
báo cáo hóa học:" Existence of positive solutions to discrete secondorder boundary value problems with indefinite weight" ppt
... positive solution and a negative solution of (1.1) and (1.2) Definition 1.1 A positive solution of problem (1.1) and (1.2) refers to a pair (r, u), where r ≠ 0, u is a solution of (1.1) with u ... boundary value problems with indefinite weight Nonlinear Anal: TMA 71, 2119–2125 (2009) doi:10.1016/j.na.2009.01.046 Agarwal, RP, Henderson, J: Positive solutions and nonlinear eigenvalue problems ... discrete Dirichlet problems with lower and upper solutions Nonlinear Anal: TMA 67(4), 1236–1245 (2007) doi:10.1016/j.na.2006.07.010 10 Rodriguez, J: Nonlinear discrete Sturm-Liouville problems J Math...
- 10
- 331
- 0
báo cáo hóa học:" Existence of positive solutions for fourth-order semipositone multi-point boundary value problems with a sign-changing nonlinear term" docx
... Existence of positive solutions for fourth-order semipositone multi-point boundary value problems with a sign-changing nonlinear term Yan Sun Department of ... operators together with approximating technique They guarantee the existence of at least one positive solution for nonlinear fourth-order semipositone multi-point boundary value problems The interesting ... positive solutions; multi-point boundary value problems 2000 Mathematics Subject Classification: 34B10; 34B18; 47N20 Introduction In this article, we consider the existence of positive solutions...
- 25
- 221
- 0
báo cáo hóa học:" Research Article Positive and Dead-Core Solutions of Two-Point Singular Boundary Value Problems with φ-Laplacian" pot
... 1.2 with n replaced by kn In Section see Theorem 3.1 we show that any sequential solution of problem 1.1 , 1.2 is either a positive solution or a pseudo-dead-core solution or a dead-core solution ... problem 1.1 , 1.2 has a sequential solution ii Any sequential solution of problem 1.1 , 1.2 is either a positive solution, a pseudo-deadcore solution, or a dead-core solution Proof i Fix λ > By Lemma ... problem 3.17 , 1.2 has a sequential solution for each λ > 0, and any sequential solution is either a positive solution or a pseudo-dead-core solution or a dead-core solution If the values of λ are...
- 17
- 272
- 0
Báo cáo hóa học: " Research Article The Existence of Positive Solutions for Third-Order p-Laplacian m-Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales" ppt
... positive solutions of semilinear differential equations with singularities,” Journal of the London Mathematical Society, vol 63, no 3, pp 690–704, 2001 21 D J Guo and V Lakshmikantham, Nonlinear Problems ... that positive solution is concave down If the nonlinearity is negative somewhere, then the solution needs no longer to be concave down As a result, it is difficult to find positive solutions of the ... particular, little work has been done on the existence of positive solutions for higher order p-Laplacian m-point boundary value problems with nonlinearity f being nonnegative on time scales Therefore,...
- 14
- 282
- 0
Báo cáo hóa học: "Research Article Optimality Conditions of Globally Efficient Solution for Vector Equilibrium Problems with Generalized Convexity" pptx
... necessary and sufficient conditions for weakly efficient solution, Henig efficient solution, and superefficient solution for the vector equilibrium problems with constraints under the condition of cone-convexity ... existence of solutions and the properties of the solutions, there are few papers which deal with the optimality conditions Giannessi et al 16 turned the vector variational inequalities with constraints ... and Applications Vector equilibrium problems, which contain vector optimization problems, vector variational inequality problems, and vector complementarity problems as special case, have been...
- 13
- 279
- 0
Báo cáo hóa học: "Research Article Existence and Multiple Solutions for Nonlinear Second-Order Discrete Problems with Minimum and Maximum" ppt
... least two difference solutions u1 k , u2 k Since u k is a solution of 3.30 , if and only if u k A is a solution of 1.5 , we see that ui k ui k A, i 1, 3.32 are two different solutions of 1.5 and ... equations with general boundary conditions,” Bulletin of the American Mathematical Society, vol 48, pp 692–704, 1942 R Ma and N Castaneda, “Existence of solutions of boundary value problems for ... multiple solutions for some discrete Sturm-Liouville problems, ” Computers & Mathematics with Applications, vol 54, no 4, pp 535–543, 2007 10 R Ma, H Luo, and C Gao, “On nonresonance problems...
- 16
- 233
- 0
THERE ONCE WAS A CLASSICAL THEORY: Introductory Classical Mechanics, with Problems and Solutions pdf
... your mail-order physics degree! ♣ One last note: the problems with included solutions are called Problems. ” The problems without included solutions are called “Exercises.” There is no fundamental ... Students invariably want extra practice problems, with solutions, to work on, and (2) I find them rather fun The problems are marked with a number of asterisks Harder problems earn more asterisks, on ... textbook, consisting of text, along with exercises suitable for homework assignments The other half takes the form of a problem book, with all sorts of problems (with solutions) of varying degrees...
- 623
- 464
- 0
Introductory Classical Mechanics, with Problems and Solutions doc
... your mail-order physics degree! ♣ One last note: the problems with included solutions are called Problems. ” The problems without included solutions are called “Exercises.” There is no fundamental ... Students invariably want extra practice problems, with solutions, to work on, and (2) I find them rather fun The problems are marked with a number of asterisks Harder problems earn more asterisks, on ... textbook, consisting of text, along with exercises suitable for homework assignments The other half takes the form of a problem book, with all sorts of problems (with solutions) of varying degrees...
- 623
- 568
- 1
Tìm thêm: xác định các mục tiêu của chương trình khảo sát chương trình đào tạo của các đơn vị đào tạo tại nhật bản khảo sát chương trình đào tạo gắn với các giáo trình cụ thể tiến hành xây dựng chương trình đào tạo dành cho đối tượng không chuyên ngữ tại việt nam điều tra với đối tượng sinh viên học tiếng nhật không chuyên ngữ1 khảo sát thực tế giảng dạy tiếng nhật không chuyên ngữ tại việt nam nội dung cụ thể cho từng kĩ năng ở từng cấp độ xác định mức độ đáp ứng về văn hoá và chuyên môn trong ct phát huy những thành tựu công nghệ mới nhất được áp dụng vào công tác dạy và học ngoại ngữ mở máy động cơ lồng sóc mở máy động cơ rôto dây quấn các đặc tính của động cơ điện không đồng bộ đặc tuyến hiệu suất h fi p2 đặc tuyến tốc độ rôto n fi p2 đặc tuyến dòng điện stato i1 fi p2 sự cần thiết phải đầu tư xây dựng nhà máy thông tin liên lạc và các dịch vụ phần 3 giới thiệu nguyên liệu từ bảng 3 1 ta thấy ngoài hai thành phần chủ yếu và chiếm tỷ lệ cao nhất là tinh bột và cacbonhydrat trong hạt gạo tẻ còn chứa đường cellulose hemicellulose chỉ tiêu chất lượng theo chất lượng phẩm chất sản phẩm khô từ gạo của bộ y tế năm 2008