PROBLEMS. Problems. x 1
|
|
- Kristopher Gray
- 6 years ago
- Views:
Transcription
1 Irish Math. Soc. Bulletin Number 76, Winter 05, 9 96 ISSN PROBLEMS IAN SHORT We begin with three integrals. Problems Problem 76.. a 0 sinx dx b 0 x log x dx c cos x e /x + dx I learnt the next problem from a popular-mathematics lecture given by Vicy Neale of the University of Oxford. Problem 76.. For each point z on the unit circle, let l z denote the closed line segment from z to z. Consider the collection of those points in the closed unit disc that each lie at the intersection of two distinct line segments l z and l w. What shape is the complement in the unit disc of this collection of points? We finish with a problem proposed by Wenchang Chu of Universitá del Salento, Italy. Problem Evaluate n=0 x tan. n n Solutions Here are solutions to the problems from Bulletin Number 74. The first problem was solved by the North Kildare Mathematics Problem Club, and it is their solution that we present. The solution was also nown to Grahame Ersine of the Open University, who suggested the problem. Received on c 05 Irish Mathematical Society
2 9 I. SHORT Problem 74.. Given a positive integer A, let B be the number obtained by reversing the digits in the base n expansion of A. The integer A is called a reverse divisor in base n if it is a divisor of B that is not equal to B. For example, using decimal expansions, if we reverse the digits of the integer 5, then we obtain 5. Since 5 is not a divisor of 5, the integer 5 is not a reverse divisor in base 0. For which of the positive integers n between and 6, inclusive, is there a two-digit reverse divisor in base n? You may also wish to attempt the more difficult problem of classifying those positive integers n for which there is a two-digit reverse divisor in base n. Solution 74.. Given a positive integer n and two integers a and b with a, b < n, let [ab] n = an + b. The number [ab] n is defined to be a two-digit reverse divisor in base n if [ba] n is divisible by, but not equal to, [ab] n. By inspection, there are no two-digit reverse divisors in bases, or 3. For n > 3, we claim that there is a two-digit reverse divisor in base n if and only if n + is composite. To see this, suppose first that n + is composite. Choose positive integers a and with n + = a + +, and define b = n a. Then bn + a = an + b, so we have a two-digit reverse divisor, unless =. In the case =, we define a = and b = n to give b n+a = aa n+b. This time we certainly have a two-digit reverse divisor, because a = n / >. Conversely, suppose that n + is equal to a prime p, and suppose that [ab] n is a two-digit reverse divisor in base n. Let = bn + a/an + b. One calculates that pb a = b a +. Thus one of b a or + is divisible by p. But b a < p and < + < b + n + = p, so we have a contradiction. Hence, contrary to our earlier assumption, there does not exist a two-digit reverse divisor in base n. The second problem was solved by Henry Ricardo New Yor Math Circle, New Yor, USA, the North Kildare Mathematics Problem Club, and the proposer, Ángel Plaza Universidad de Las Palmas de Gran Canaria, Spain. All solutions were similar. The solution
3 PROBLEMS 93 we present is that of Henry Ricardo. Nesbitt s inequality features in the solution, which says that for positive real numbers a, b, and c, we have a b + c + b c + a + c a + b 3. This inequality can be proven by using two applications of the rearrangement inequality with the triples a, b, c and /a+b, /b+ c, /c + a. Problem 74.. Let f : R + R + be an increasing, convex function with f =, and let x, y, and z be positive real numbers. Prove that for any positive integer n, n n n x y z f + f + f 3. y + z z + x x + y Solution 74.. By Nesbitt s inequality, x y + z + y z + x + z x + y 3. Then, as the function gx = fx n is convex and increasing with g =, we can use Jensen s inequality to write x x y z y+z g + g + g 3g + y z+x + z x+y y + z z + x x + y 3 as required. 3g = 3, The third problem was solved by the North Kildare Mathematics Problem Club, Ángel Plaza, and the proposer Finbarr Holland University College Cor. Finbarr has pointed out that in fact the result in the problem can quicly be derived using Bernstein polynomials. Bernstein polynomials are useful for approximating continuous functions; they provide one way of proving the Weierstrass approximation theorem. To solve the problem, simply calculate the nth Bernstein polynomial of the function fx = x j and use the value x = /. Nonetheless, we present a complete, more elementary, solution here, which does not require Bernstein polynomials. The solution is essentially the same as that of the North Kildare Mathematics Problem Club and Ángel Plaza. In this problem, we use the standard notation fn gn as n,
4 94 I. SHORT where f and g are positive functions, to mean that fn gn as n. Problem Prove that for j = 0,,,..., j n j n j as n. =0 Solution Let f 0 x = + x n, and for j =,,... let f j x = x d f j x. d x We shall prove by induction that for j = 0,,,..., f j x = n j x j + x n j + p j x, n, where p j is a polynomial in x and n, and its degree as a polynomial in n is less than j. This equation holds when j = 0 p 0 is the zero polynomial, which we assume has degree in each variable. Suppose that it holds when j = m. Then one can chec that where f m x = n m x m + x n m + qx, n, qx, n = m n m x m + x n m + x x p m x, n. The degree of q as a polynomial in n is less than m, so the inductive proof is complete. We deduce that, for x, Now, so f j x n j x j + x n j, j = 0,,,.... f 0 x = + x n = =0 x, j = f j n j n j, j = 0,,,.... =0 Finally, we return to a problem of Niall Ryan University of Limeric from issue 7. Niall had a solution to this problem, but it was quite involved. The solution we include here, provided by Finbarr Holland, deals with the odd case only.
5 Problem 7.. For each integer n 0, let K m S n = m + n + + PROBLEMS 95 m n K m m n, where [ ] m! K m =. m m! Prove that 0, n odd, S n = K n log +, n even. Solution 7.. Let = Γ a n = + n Γ Γn + n n! = K n. We aim to prove that for n = 0,,,..., S n+ = a m m + n + + m n = 0. Clearly, it is enough to prove that T n = 0 for n = 0,,,..., where T n = j S j+ = 0. j To establish this, notice that j j z + j + n!γz + n = z + j + Γz + n +, and j j z j = n n n! n z j = n n!γz n+ Γz +. Next we appeal to a well-nown result about hypergeometric series, which can be found, for example, in Section 0.3 of An introduction to the theory of functions of a complex variable by E.T. Copson
6 96 I. SHORT OUP, 970. Recall that the hypergeometric series F a, b; c; z is defined for z < by the equation Γa + mγb + m z m Γc + m m! = ΓaΓb F a, b; c; z, Γc and it is defined for z by analytic continuation. In fact, by Abel s continuity theorem, the series above can also be used to define F a, b; c;. The result we need, from page 5 of Copson s text, is that ΓcΓc a b F a, b; c; = Γc aγc b if Re[c a b] > 0. Using this observation, we see that T n a Γm + n+ m Γn + + m + nγ + m Γ + m Γ + m Γ + m + nγ + m Γ n+ + m π m!γn + + m m!γ + m π π π = 0. Γ F, ; n + ; + nγ Γ n+ F, n+ ; ; Γn + Γ πγn + Γ n nγ n+ Γn + Γ n + 3 πγn + Γ n n πγn + sin n + π Γ n + 3 The well-nown functional equation ΓzΓ z = π/ sinπz is used in the second-last line. We invite readers to submit problems and solutions. Please submissions to imsproblems@gmail.com in any format we prefer Latex. Submissions for the summer Bulletin should arrive before the end of April, and submissions for the winter Bulletin should arrive by October. The solution to a problem is published two issues after the issue in which the problem first appeared. Please include solutions to any problems you submit, if you have them. Department of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA, United Kingdom
PROBLEMS IAN SHORT. ( 1) n x n
Irish Math. Soc. Bulletin Number 8, Winter 27, 87 9 ISSN 79-5578 PROBLEMS IAN SHORT Problems The first problem this issue was contributed by Finbarr Holland of University College Cork. Problem 8.. Let
More informationIrish Math. Soc. Bulletin Number 79, Summer 2017, ISSN PROBLEMS IAN SHORT
Irish Math Soc Bulletin Number 79, Summer 27, 95 ISSN 79-5578 PROBLEMS IAN SHORT Problems The first problem this issue was contributed by Peter Danchev of Plovdiv University, Bulgaria Problem 79 Suppose
More informationSolutions 2017 AB Exam
1. Solve for x : x 2 = 4 x. Solutions 2017 AB Exam Texas A&M High School Math Contest October 21, 2017 ANSWER: x = 3 Solution: x 2 = 4 x x 2 = 16 8x + x 2 x 2 9x + 18 = 0 (x 6)(x 3) = 0 x = 6, 3 but x
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More information1 Basic Combinatorics
1 Basic Combinatorics 1.1 Sets and sequences Sets. A set is an unordered collection of distinct objects. The objects are called elements of the set. We use braces to denote a set, for example, the set
More informationThe primitive root theorem
The primitive root theorem Mar Steinberger First recall that if R is a ring, then a R is a unit if there exists b R with ab = ba = 1. The collection of all units in R is denoted R and forms a group under
More informationSolutions to Practice Final
s to Practice Final 1. (a) What is φ(0 100 ) where φ is Euler s φ-function? (b) Find an integer x such that 140x 1 (mod 01). Hint: gcd(140, 01) = 7. (a) φ(0 100 ) = φ(4 100 5 100 ) = φ( 00 5 100 ) = (
More informationELEMENTARY PROBLEMS AND SOLUTIONS
ELEMENTARY PROBLEMS AND SOLUTIONS EDITED BY HARRIS KWONG Please submit solutions and problem proposals to Dr. Harris Kwong, Department of Mathematical Sciences, SUNY Fredonia, Fredonia, NY, 4063, or by
More informationLines, parabolas, distances and inequalities an enrichment class
Lines, parabolas, distances and inequalities an enrichment class Finbarr Holland 1. Lines in the plane A line is a particular kind of subset of the plane R 2 = R R, and can be described as the set of ordered
More informationOriginally problem 21 from Demi-finale du Concours Maxi de Mathématiques de
5/ THE CONTEST CORNER Originally problem 6 from Demi-finale du Concours Maxi de Mathématiques de Belgique 00. There were eight solution submitted for this question, all essentially the same. By the Pythagorean
More informationPUTNAM TRAINING MATHEMATICAL INDUCTION. Exercises
PUTNAM TRAINING MATHEMATICAL INDUCTION (Last updated: December 11, 017) Remark. This is a list of exercises on mathematical induction. Miguel A. Lerma 1. Prove that n! > n for all n 4. Exercises. Prove
More informationSTEP Support Programme. Pure STEP 1 Questions
STEP Support Programme Pure STEP 1 Questions 2012 S1 Q4 1 Preparation Find the equation of the tangent to the curve y = x at the point where x = 4. Recall that x means the positive square root. Solve the
More informationHow many units can a commutative ring have?
How many units can a commutative ring have? Sunil K. Chebolu and Keir Locridge Abstract. László Fuchs posed the following problem in 960, which remains open: classify the abelian groups occurring as the
More informationAll variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points.
Math 152, Problem Set 2 solutions (2018-01-24) All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. 1. Let us look at the following equation: x 5 1
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationAP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period:
WORKSHEET: Series, Taylor Series AP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period: 1 Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The
More informationMathematics 1 Lecture Notes Chapter 1 Algebra Review
Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to
More informationUniversity of Toronto Faculty of Arts and Science Solutions to Final Examination, April 2017 MAT246H1S - Concepts in Abstract Mathematics
University of Toronto Faculty of Arts and Science Solutions to Final Examination, April 2017 MAT246H1S - Concepts in Abstract Mathematics Examiners: D. Burbulla, P. Glynn-Adey, S. Homayouni Time: 7-10
More informationMathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...
More informationPre-Final Round 2018
Pre-Final Round 2018 DATE OF RELEASE: 29. OCTOBER 2018 IYMC.PFR.2018 1 Important: Read all the information on this page carefully! General Information Please read all problems carefully! We recommend to
More informationUNDERSTANDING RULER AND COMPASS CONSTRUCTIONS WITH FIELD THEORY
UNDERSTANDING RULER AND COMPASS CONSTRUCTIONS WITH FIELD THEORY ISAAC M. DAVIS Abstract. By associating a subfield of R to a set of points P 0 R 2, geometric properties of ruler and compass constructions
More informationCompletion Date: Monday February 11, 2008
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #4 Completion Date: Monday February, 8 Department of Mathematical and Statistical Sciences University of Alberta Question. [Sec..9,
More informationPROBLEMS ON CONGRUENCES AND DIVISIBILITY
PROBLEMS ON CONGRUENCES AND DIVISIBILITY 1. Do there exist 1,000,000 consecutive integers each of which contains a repeated prime factor? 2. A positive integer n is powerful if for every prime p dividing
More informationELEMENTARY PROBLEMS AND SOLUTIONS
ELEMENTARY PROBLEMS AND SOLUTIONS EDITED BY HARRIS KWONG Please submit solutions and problem proposals to Dr Harris Kwong, Department of Mathematical Sciences, SUNY Fredonia, Fredonia, NY, 4063, or by
More informationTheory of Numbers Problems
Theory of Numbers Problems Antonios-Alexandros Robotis Robotis October 2018 1 First Set 1. Find values of x and y so that 71x 50y = 1. 2. Prove that if n is odd, then n 2 1 is divisible by 8. 3. Define
More informationMATH 361: NUMBER THEORY FOURTH LECTURE
MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need
More information2016 EF Exam Texas A&M High School Students Contest Solutions October 22, 2016
6 EF Exam Texas A&M High School Students Contest Solutions October, 6. Assume that p and q are real numbers such that the polynomial x + is divisible by x + px + q. Find q. p Answer Solution (without knowledge
More informationZsigmondy s Theorem. Lola Thompson. August 11, Dartmouth College. Lola Thompson (Dartmouth College) Zsigmondy s Theorem August 11, / 1
Zsigmondy s Theorem Lola Thompson Dartmouth College August 11, 2009 Lola Thompson (Dartmouth College) Zsigmondy s Theorem August 11, 2009 1 / 1 Introduction Definition o(a modp) := the multiplicative order
More informationChapter 1. Sets and Numbers
Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend
More information1 + lim. n n+1. f(x) = x + 1, x 1. and we check that f is increasing, instead. Using the quotient rule, we easily find that. 1 (x + 1) 1 x (x + 1) 2 =
Chapter 5 Sequences and series 5. Sequences Definition 5. (Sequence). A sequence is a function which is defined on the set N of natural numbers. Since such a function is uniquely determined by its values
More informationMASTERS EXAMINATION IN MATHEMATICS SOLUTIONS
MASTERS EXAMINATION IN MATHEMATICS PURE MATHEMATICS OPTION SPRING 010 SOLUTIONS Algebra A1. Let F be a finite field. Prove that F [x] contains infinitely many prime ideals. Solution: The ring F [x] of
More informationEven perfect numbers among generalized Fibonacci sequences
Even perfect numbers among generalized Fibonacci sequences Diego Marques 1, Vinicius Vieira 2 Departamento de Matemática, Universidade de Brasília, Brasília, 70910-900, Brazil Abstract A positive integer
More informationMath 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement.
Math 421, Homework #6 Solutions (1) Let E R n Show that (Ē) c = (E c ) o, i.e. the complement of the closure is the interior of the complement. 1 Proof. Before giving the proof we recall characterizations
More informationMath 109 HW 9 Solutions
Math 109 HW 9 Solutions Problems IV 18. Solve the linear diophantine equation 6m + 10n + 15p = 1 Solution: Let y = 10n + 15p. Since (10, 15) is 5, we must have that y = 5x for some integer x, and (as we
More informationSection 19 Integral domains
Section 19 Integral domains Instructor: Yifan Yang Spring 2007 Observation and motivation There are rings in which ab = 0 implies a = 0 or b = 0 For examples, Z, Q, R, C, and Z[x] are all such rings There
More informationFunctions. Remark 1.2 The objective of our course Calculus is to study functions.
Functions 1.1 Functions and their Graphs Definition 1.1 A function f is a rule assigning a number to each of the numbers. The number assigned to the number x via the rule f is usually denoted by f(x).
More informationIntermediate Math Circles February 14, 2018 Contest Prep: Number Theory
Intermediate Math Circles February 14, 2018 Contest Prep: Number Theory Part 1: Prime Factorization A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. An integer
More informationPRACTICE PROBLEMS: SET 1
PRACTICE PROBLEMS: SET MATH 437/537: PROF. DRAGOS GHIOCA. Problems Problem. Let a, b N. Show that if gcd(a, b) = lcm[a, b], then a = b. Problem. Let n, k N with n. Prove that (n ) (n k ) if and only if
More informationMath 113 Homework 5. Bowei Liu, Chao Li. Fall 2013
Math 113 Homework 5 Bowei Liu, Chao Li Fall 2013 This homework is due Thursday November 7th at the start of class. Remember to write clearly, and justify your solutions. Please make sure to put your name
More informationJuly 21 Math 2254 sec 001 Summer 2015
July 21 Math 2254 sec 001 Summer 2015 Section 8.8: Power Series Theorem: Let a n (x c) n have positive radius of convergence R, and let the function f be defined by this power series f (x) = a n (x c)
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationSubtraction games. Chapter The Bachet game
Chapter 7 Subtraction games 7.1 The Bachet game Beginning with a positive integer, two players alternately subtract a positive integer < d. The player who gets down to 0 is the winner. There is a set of
More information11.10a Taylor and Maclaurin Series
11.10a 1 11.10a Taylor and Maclaurin Series Let y = f(x) be a differentiable function at x = a. In first semester calculus we saw that (1) f(x) f(a)+f (a)(x a), for all x near a The right-hand side of
More information40th International Mathematical Olympiad
40th International Mathematical Olympiad Bucharest, Romania, July 999 Determine all finite sets S of at least three points in the plane which satisfy the following condition: for any two distinct points
More informationPRIME NUMBERS YANKI LEKILI
PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume
More informationBichain graphs: geometric model and universal graphs
Bichain graphs: geometric model and universal graphs Robert Brignall a,1, Vadim V. Lozin b,, Juraj Stacho b, a Department of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA, United
More informationExercise Sheet 3 - Solutions
Algebraic Geometry D-MATH, FS 2016 Prof. Pandharipande Exercise Sheet 3 - Solutions 1. Prove the following basic facts about algebraic maps. a) For f : X Y and g : Y Z algebraic morphisms of quasi-projective
More informationHMMT November 2012 Saturday 10 November 2012
HMMT November 01 Saturday 10 November 01 1. [5] 10 total. The prime numbers under 0 are:,, 5, 7, 11, 1, 17, 19,, 9. There are 10 in. [5] 180 Albert s choice of burgers, sides, and drinks are independent
More informationUNCC 2001 Algebra II
UNCC 2001 Algebra II March 5, 2001 1. Compute the sum of the roots of x 2 5x + 6 = 0. (A) 3 (B) 7/2 (C) 4 (D) 9/2 (E) 5 (E) The sum of the roots of the quadratic ax 2 + bx + c = 0 is b/a which, for this
More informationA Few Elementary Properties of Polynomials. Adeel Khan June 21, 2006
A Few Elementary Properties of Polynomials Adeel Khan June 21, 2006 Page i CONTENTS Contents 1 Introduction 1 2 Vieta s Formulas 2 3 Tools for Finding the Roots of a Polynomial 4 4 Transforming Polynomials
More informationTest 2. Monday, November 12, 2018
Test 2 Monday, November 12, 2018 Instructions. The only aids allowed are a hand-held calculator and one cheat sheet, i.e. an 8.5 11 sheet with information written on one side in your own handwriting. No
More informationSydney University Mathematical Society Problems Competition Solutions.
Sydney University Mathematical Society Problems Competition 005 Solutions 1 Suppose that we look at the set X n of strings of 0 s and 1 s of length n Given a string ɛ = (ɛ 1,, ɛ n ) X n, we are allowed
More informationRINGS ISOMORPHIC TO THEIR NONTRIVIAL SUBRINGS
RINGS ISOMORPHIC TO THEIR NONTRIVIAL SUBRINGS JACOB LOJEWSKI AND GREG OMAN Abstract. Let G be a nontrivial group, and assume that G = H for every nontrivial subgroup H of G. It is a simple matter to prove
More informationSimplifying Rational Expressions and Functions
Department of Mathematics Grossmont College October 15, 2012 Recall: The Number Types Definition The set of whole numbers, ={0, 1, 2, 3, 4,...} is the set of natural numbers unioned with zero, written
More informationMATH NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS
MATH. 4433. NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS TOMASZ PRZEBINDA. Final project, due 0:00 am, /0/208 via e-mail.. State the Fundamental Theorem of Algebra. Recall that a subset K
More informationCSC 344 Algorithms and Complexity. Proof by Mathematical Induction
CSC 344 Algorithms and Complexity Lecture #1 Review of Mathematical Induction Proof by Mathematical Induction Many results in mathematics are claimed true for every positive integer. Any of these results
More information3. Which of these numbers does not belong to the set of solutions of the inequality 4
Math Field Day Exam 08 Page. The number is equal to b) c) d) e). Consider the equation 0. The slope of this line is / b) / c) / d) / e) None listed.. Which of these numbers does not belong to the set of
More informationA Note on the 2 F 1 Hypergeometric Function
A Note on the F 1 Hypergeometric Function Armen Bagdasaryan Institution of the Russian Academy of Sciences, V.A. Trapeznikov Institute for Control Sciences 65 Profsoyuznaya, 117997, Moscow, Russia E-mail:
More informationMath 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible.
Math 320: Real Analysis MWF pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 4.3.5, 4.3.7, 4.3.8, 4.3.9,
More informationMATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017
MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A
More information41st International Mathematical Olympiad
41st International Mathematical Olympiad Taejon, Korea, July 2000 1 Two circles Γ 1 and Γ 2 intersect at M and N Let l be the common tangent to Γ 1 and Γ 2 so that M is closer to l than N is Let l touch
More information2018 LEHIGH UNIVERSITY HIGH SCHOOL MATH CONTEST
08 LEHIGH UNIVERSITY HIGH SCHOOL MATH CONTEST. A right triangle has hypotenuse 9 and one leg. What is the length of the other leg?. Don is /3 of the way through his run. After running another / mile, he
More informationMATH 2400: PRACTICE PROBLEMS FOR EXAM 1
MATH 2400: PRACTICE PROBLEMS FOR EXAM 1 PETE L. CLARK 1) Find all real numbers x such that x 3 = x. Prove your answer! Solution: If x 3 = x, then 0 = x 3 x = x(x + 1)(x 1). Earlier we showed using the
More informationAS1051: Mathematics. 0. Introduction
AS1051: Mathematics 0 Introduction The aim of this course is to review the basic mathematics which you have already learnt during A-level, and then develop it further You should find it almost entirely
More informationx n cos 2x dx. dx = nx n 1 and v = 1 2 sin(2x). Andreas Fring (City University London) AS1051 Lecture Autumn / 36
We saw in Example 5.4. that we sometimes need to apply integration by parts several times in the course of a single calculation. Example 5.4.4: For n let S n = x n cos x dx. Find an expression for S n
More informationMATH FINAL EXAM REVIEW HINTS
MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any
More informationFRESHMAN PRIZE EXAM 2017
FRESHMAN PRIZE EXAM 27 Full reasoning is expected. Please write your netid on your paper so we can let you know of your result. You have 9 minutes. Problem. In Extracurricula all of the high school students
More informationnot to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results
REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division
More informationChapter 8. P-adic numbers. 8.1 Absolute values
Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.
More informationStudent: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
More informationJim Lambers MAT 169 Fall Semester Lecture 6 Notes. a n. n=1. S = lim s k = lim. n=1. n=1
Jim Lambers MAT 69 Fall Semester 2009-0 Lecture 6 Notes These notes correspond to Section 8.3 in the text. The Integral Test Previously, we have defined the sum of a convergent infinite series to be the
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)
Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
More informationAnalytic Number Theory Solutions
Analytic Number Theory Solutions Sean Li Cornell University sxl6@cornell.edu Jan. 03 Introduction This document is a work-in-progress solution manual for Tom Apostol s Introduction to Analytic Number Theory.
More informationMath 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
More informationCLASS XI Maths : Sample Paper-1
CLASS XI Maths : Sample Paper-1 Allotted Time : 3 Hrs Max. Marks : 100 Instructions 1. This questionnaire consists of 30 questions divided in four sections. Please verify before attempt. 2. Section I consists
More informationSection X.55. Cyclotomic Extensions
X.55 Cyclotomic Extensions 1 Section X.55. Cyclotomic Extensions Note. In this section we return to a consideration of roots of unity and consider again the cyclic group of roots of unity as encountered
More informationWriting proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases
Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases September 22, 2018 Recall from last week that the purpose of a proof
More informationq-series Michael Gri for the partition function, he developed the basic idea of the q-exponential. From
q-series Michael Gri th History and q-integers The idea of q-series has existed since at least Euler. In constructing the generating function for the partition function, he developed the basic idea of
More informationChapter 1 The Real Numbers
Chapter 1 The Real Numbers In a beginning course in calculus, the emphasis is on introducing the techniques of the subject;i.e., differentiation and integration and their applications. An advanced calculus
More informationAbout This Document. MTH299 - Examples Weeks 1-6; updated on January 5, 2018
About This Document This is the examples document for MTH 299. Basically it is a loosely organized universe of questions (examples) that we think are interesting, helpful, useful for practice, and serve
More information1. Consider the conditional E = p q r. Use de Morgan s laws to write simplified versions of the following : The negation of E : 5 points
Introduction to Discrete Mathematics 3450:208 Test 1 1. Consider the conditional E = p q r. Use de Morgan s laws to write simplified versions of the following : The negation of E : The inverse of E : The
More informationMath1a Set 1 Solutions
Math1a Set 1 Solutions October 15, 2018 Problem 1. (a) For all x, y, z Z we have (i) x x since x x = 0 is a multiple of 7. (ii) If x y then there is a k Z such that x y = 7k. So, y x = (x y) = 7k is also
More informationPreliminary algebra. Polynomial equations. and three real roots altogether. Continue an investigation of its properties as follows.
978-0-51-67973- - Student Solutions Manual for Mathematical Methods for Physics and Engineering: 1 Preliminary algebra Polynomial equations 1.1 It can be shown that the polynomial g(x) =4x 3 +3x 6x 1 has
More informationSupport for UCL Mathematics offer holders with the Sixth Term Examination Paper
1 Support for UCL Mathematics offer holders with the Sixth Term Examination Paper The Sixth Term Examination Paper (STEP) examination tests advanced mathematical thinking and problem solving. The examination
More informationn f(k) k=1 means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other words: n f(k) = f(1) + f(2) f(n). 1 = 2n 2.
Handout on induction and written assignment 1. MA113 Calculus I Spring 2007 Why study mathematical induction? For many students, mathematical induction is an unfamiliar topic. Nonetheless, this is an important
More informationDiscrete Math I Exam II (2/9/12) Page 1
Discrete Math I Exam II (/9/1) Page 1 Name: Instructions: Provide all steps necessary to solve the problem. Simplify your answer as much as possible. Additionally, clearly indicate the value or expression
More informationNotes on Systems of Linear Congruences
MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the
More information2008 Euclid Contest. Solutions. Canadian Mathematics Competition. Tuesday, April 15, c 2008 Centre for Education in Mathematics and Computing
Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario 008 Euclid Contest Tuesday, April 5, 008 Solutions c 008
More informationA Simple Proof that ζ(n 2) is Irrational
A Simple Proof that ζ(n 2) is Irrational Timothy W. Jones February 3, 208 Abstract We prove that partial sums of ζ(n) = z n are not given by any single decimal in a number base given by a denominator of
More information2018 Best Student Exam Solutions Texas A&M High School Students Contest October 20, 2018
08 Best Student Exam Solutions Texas A&M High School Students Contest October 0, 08. You purchase a stock and later sell it for $44 per share. When you do, you notice that the percent increase was the
More informationThis class will demonstrate the use of bijections to solve certain combinatorial problems simply and effectively.
. Induction This class will demonstrate the fundamental problem solving technique of mathematical induction. Example Problem: Prove that for every positive integer n there exists an n-digit number divisible
More informationSome Basic Logic. Henry Liu, 25 October 2010
Some Basic Logic Henry Liu, 25 October 2010 In the solution to almost every olympiad style mathematical problem, a very important part is existence of accurate proofs. Therefore, the student should be
More informationThe theory of numbers
1 AXIOMS FOR THE INTEGERS 1 The theory of numbers UCU Foundations of Mathematics course 2017 Author: F. Beukers 1 Axioms for the integers Roughly speaking, number theory is the mathematics of the integers.
More informationD-MATH Algebra I HS18 Prof. Rahul Pandharipande. Solution 1. Arithmetic, Zorn s Lemma.
D-MATH Algebra I HS18 Prof. Rahul Pandharipande Solution 1 Arithmetic, Zorn s Lemma. 1. (a) Using the Euclidean division, determine gcd(160, 399). (b) Find m 0, n 0 Z such that gcd(160, 399) = 160m 0 +
More informationBARUCH COLLEGE MATH 1030 Practice Final Part 1, NO CALCULATORS. (E) All real numbers. (C) y = 1 2 x 5 2
BARUCH COLLEGE MATH 1030 Practice Final Part 1, NO CALCULATORS 1. Find the domain of f(x) = x + x x 4x. 1. (A) (, 0) (0, 4) (4, ) (B) (, 0) (4, ) (C) (, 4) (4, ) (D) (, ) (, 0) (0, ) (E) All real numbers.
More informationNewton, Fermat, and Exactly Realizable Sequences
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.2 Newton, Fermat, and Exactly Realizable Sequences Bau-Sen Du Institute of Mathematics Academia Sinica Taipei 115 TAIWAN mabsdu@sinica.edu.tw
More informationSequences of height 1 primes in Z[X]
Sequences of height 1 primes in Z[X] Stephen McAdam Department of Mathematics University of Texas Austin TX 78712 mcadam@math.utexas.edu Abstract: For each partition J K of {1, 2,, n} (n 2) with J 2, let
More informationHigh School Math Contest
High School Math Contest University of South Carolina February th, 017 Problem 1. If (x y) = 11 and (x + y) = 169, what is xy? (a) 11 (b) 1 (c) 1 (d) (e) 8 Solution: Note that xy = (x + y) (x y) = 169
More information