# Shuli s Math Problem Solving Column

Size: px
Start display at page:

Download "Shuli s Math Problem Solving Column"

Transcription

1 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 Edited d Authored by Shuli Sog Colordo Sprigs, Colordo Cotets Mth Trick: Metl Clcultio: b cd Mth Competitio Skill: Divisibility by 3 A Problem from Rel Mth Competitio 4 Aswers to All Prctice Problems i Lst Issue 5 Solutios to Cretive Thikig Problems 55 to 57 6 Clues to Cretive Thikig Problems 58 to 6 7 Cretive Thikig Problems 6 to 63 Mth Trick The Trick I the lst issue we preseted how to clculte the followig multiplictios metlly: These multiplictios re i the geerl form: b cd where, b, c, d d re digits with b close to cd I this short lesso I will preset other short cut through exmples Exmple Metl Clcultio: b cd Clculte 8 3 If we tret 8 s with multiplictio i b Step : Clculte b or b I this exmple, 8 3, the this is or 3 Step : Multiply the result i step by I this exmple, 4 Step 3: Clculte b I this exmple, 3 6 Step 3 Add them this wy: We re doe: Exmple Clculte 4 3 Tret 4 s 3 with 6 The this is multiplictio i 3 3b Step : Clculte 3 b or 3 b I this exmple, 4 6 or Step : Multiply the result i step by 3 I this exmple, Step 3: Clculte b I this exmple, 6 Step 4 Add them: We obti Exmple 3 Clculte 47 5 Tret 47 s 5 with 3 The this is multiplictio i 5 5b Step : Clculte 5 b or 5 b I this exmple, or Step : Multiply the result i step by 5 I this exmple, Step 3: Clculte b I this exmple, 3 3

2 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 Step 4 Add them: We obti Exmple 4 Clculte Tret it s multiplictio i 7 7b Step : Clculte 7 b or 7 b I this exmple, Step : Multiply the result i step by 7 I this exmple, Step 3: Clculte b I this exmple, 4 8 Step 4 Add them: or We get Prctice Problems Mth Competitio Skill Defiitios: Divisibility by Alterte Digit Differece The st, 3 rd, 5 th, digits couted from the right re clled odd plced digits The sum of ll these digits is clled the odd plced digit sum The d, 4 th, 6 th, digits re clled eve plced digits The sum of ll these digits is clled the eve plced digit sum Subtrct the eve plced digit sum from the odd plced digit sum The result is clled the lterte digit differece Exmple Wht is the lterte digit differece of 378? Aswer: The odd plced digit sum is The eve plced digit sum is 3 5 The the lterte digit differece is 5 5 Exmple Wht is the lterte digit differece of ? Aswer: 5 The odd plced digit sum is The eve plced digit sum is The the lterte digit differece is 5 5 Divisibility by We hve the followig theorem for divisibility by Theorem A umber is divisible by if d oly if the lterte digit differece of the umber is divisible by Note tht is divisible by y turl umber Exmple 3 Is 47,839 divisible? Aswer: Yes The odd plced digit sum is 9 8 4, d the eve plced digit sum is 3 7 The the lterte digit differece is, which is divisible by So 47,839 is divisible by Exmple 4 Is divisible? Aswer: No The odd plced digit sum is , d the eve plced digit sum is The the lterte digit differece is 5 5, which is ot divisible by So is ot divisible by Exmple 5 Is 789 divisible? Aswer: Yes The odd plced digit sum is 4, d the eve plced digit sum is The the lterte digit differece is 4 6 which is divisible by So 789 is divisible by Proof of the Theorem Let N be -digit umber is odd without loss of geerlity Assume Express N i the bse expsio: ' s ' s 99 Copyright 9 Shuli Sog All Rights Reserved Use with Permissio

3 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 m ' s 9' Note tht d 999 m s 9 where m is eve re lwys divisible by Therefore, N is divisible by if d oly if 3, which is the lterte digit differece 3 3, is divisible by Remider upo Divisio by For umber, clculte the lterte digit differece If the result is more th or equl to, clculte the lterte digit differece gi Or subtrct from it If the result is still more th or equl to, subtrct gi util the result is less th If the lterte digit differece is less th, dd to it If the result is still less th, dd gi util the result is lrger th or equl to The fil result is the remider of the umber upo divisio by Exmple 6 Wht is the remider of upo divisio by? Aswer: 6 From exmple, the lterte digit differece is The 6 is the remider Exmple 7 Wht is the remider of upo divisio by? Aswer: The odd plced digit sum is , d the eve plced digit sum is 3 6 The lterte digit differece is The is the swer Exmple 8 Six-digit umber Aswer: 8 Exmples of Problem Solvig 7m 3 is divisible by Fid m The odd plced digit sum is 3 5, d the eve plced digit sum is m 7 8 m The lterte digit differece is 5 8 m 3 m The oly vlue of digit m is 8 such tht 3 m is divisible by Exmple 9 _ Five-digit umber m497 is divisible by where m d re digits with m How my differet pirs of vlues for m d re there? Aswer: 8 The odd plced digit sum is m 6, d the eve plced digit sum is 4 The lterte digit differece is m The m is divisible by So m m could be from to 8, d from to 9 respectively Therefore, there re 8 differet pirs of vlues for m d Exmple! b where d b re digits Fid d b Aswer: d b 4! is divisible by 9 The the digit sum b 48 is divisible by 9 So b 6 or b 5! is divisible by The odd plced digit sum is 5, d the eve plced digit sum is 3 b The lterte digit differece is b The b is divisible by So b or b 9 Solvig for d b we hve d b 4 Exmple Prove tht y six-digit umber bc, bc is divisible by where, b, d c re digits with Proof: The odd plced digit sum is c b, d the eve plced digit sum is b c The lterte digit differece is, which is divisible by Therefore, bc,bc is divisible by Prctice Problems Circle the umbers divisible by : 34 3, Fid the remider for ech upo divisio by : Is divisible by? 4 Whe is divided by, wht is the remider? 5 Eight-digit umber Fid m m 37 is divisible by 6 9! b where d b re digits Fid d b _ 7 Five-digit umber m48 is divisible by where m d re digits How my differet pirs of vlues for m d re there? _ 8 Prove tht y six-digit umber bccb is divisible by where, b, d c re digits with Copyright 9 Shuli Sog All Rights Reserved Use with Permissio 3

4 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 A Problem from Rel Mth Competitio Tody s problem comes from MthCouts The problem or similr problem ppered i MthCouts d other mth competitios occsiolly (MthCouts 995 Ntiol Sprit Problem 7) A chord of the lrger of two cocetric circles is tget to the smller circle d mesure 8 iches Fid the umber of squre iches i the re of the shded regio 8 Prctice Problems (MthCout 7 Ntiol Trget Problem 7) Two cocetric circles with rdii of 9 d 9 uits boud shded regio A third circle will be drw with re equl to tht the shded re Wht must the rdius of the third circle be? Express your swer i simplest rdicl form 9 9 Aswer: 8 Theorem I the figure below, C d C re two cocetric circles Chord AB of circle C is tget to circle C Let l be the legth of AB The the re of the rig betwee the two circles re solely determied by l, idepedet of the sizes of the two circles, provided tht l is fixed B l A C (5th AMCB 4 Problem d 55 th AMC B 4 Problem ) A ulus is the regio betwee two cocetric circles The cocetric circles i the figure hve rdii b d c Let OX be rdius of the lrger circle, let XZ be tget to the smller circle t Z, d let OY be the rdius of the lrger circle tht cotis Z Let XZ, y YZ, d e XY Wht is the re of the ulus? Y d Z O c e b X C Proof of the Theorem: Let T be the tget poit, d O be the ceter of the two circles Let R d r be the rdii of circles C d C respectively Drw OA d OT The OT AB d T is the midpoit l of AB Obviously, AT, OT r d OA R I right OTA B l r 4 R So R l l The the re of the rig is R r 4 determied by l solely l C r T O r 4, which is With l 8 the swer to the problem is R A C A) B) b C) c D) 3 (th AHSME 969 Problem 6) d E) e The re of the rig betwee two cocetric circles is squre iches The legth of chord of the lrger circle tget to the smller circle, i iches, is A) 5 B) 5 C) 5 D) E) 4 (6th AMCA 9 Problem 9) Adre iscribed circle iside regulr petgo, circumscribed circle roud the petgo, d clculted the re of the regio betwee the two circles Bethy did the sme with regulr heptgo (7 sides) The res of the two regios were A d B, respectively Ech polygo hd side legth of Which of the followig is true? 5 5 A) A B B) A B C) A B D) A B E) A B 5 5 Copyright 9 Shuli Sog All Rights Reserved Use with Permissio 4

5 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 Aswers to All Prctice Problems i Lst Issue Metl Clcultio Divisibility by 9 The umbers divisible by 9 re: 34 33, Yes A Problem from Rel Mth Competitio 6 divisio by 8 So it is i the leftmost colum Therefore, cot be the sum If is the sum of the ie umbers i mtrix, the cetrl umber must be Sice mod 8, the cetrl umber is t the third colum from the left Therefore, c be the sum of the ie umbers i mtrix 56 Coverig with Tetromios You my hve got o s the swer if you hve tried It is correct But re you ble to expli why you cot? I gve the problem to child oce He hd tried little while, d the he sid I kow the swer is o, d I feel tht the T-shped is strge Yes, the T-shped is the odd m Why is it? Color the bord with the stdrd chessbord colorig: Solutios to Cretive Thikig Problems 55 to x 3 Mtrix Look t the mtrix If we pir the umbers s show, we kow tht the sum is 9 times the cetrl umber The y oe of the followig four tetromioes covers two blck d two white squres: Therefore, the sum must be multiple of 9 Two umbers i ech pir hs sum of 58 = x 9 95 is ot multiple of 9 So the sum cot be 95 However, if umber is multiple of 9, it is ot ecessrily the sum For exmple, 44 is multiple of 9, but it cot be the sum of the ie umbers i mtrix The reso follows If 44 is the sum of the ie umbers i mtrix, the 44 cetrl umber must be 6 However, 6 is i the 9 rightmost colum If umber is i the leftmost colum, it cot be the cetrl umber of mtrix either Both d re multiples of 9 If is the sum of the ie umbers i mtrix, the cetrl umber must be Recll the shortcut for the divisibility by 8 i Issue 6, Volume yields remider of upo There re blck d white squres Wherever you plce these four shpes o the 4 5 rectgle, two ( 4 ) blck d two white squres will be left ucovered These four ucovered squres (two blck d two white) cot be covered by becuse the T-shped tetromio covers three blck d oe white squres, or oe blck d three white squres oly 57 Weighig Met II Agi, let us study from smll umbers We must hve weight of poud for piece of met of poud As we tlked i the Weigh Met I problem (Issue 5, Volume ), we do t wt to mke other weight of poud to weigh piece of met of pouds Isted we would like to mke hevier weight A weight of pouds works Copyright 9 Shuli Sog All Rights Reserved Use with Permissio 5

6 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 However, we my plce piece of met d weights o oe p If we plce piece of met of pouds with the existig weight of poud together, we would like to mke weight of 3 pouds to blce them With the weight of 3 pouds, we c weigh piece of met of 3 pouds We c weigh piece of met of 4 pouds by combiig the two existig weights To weigh piece of met of 5 pouds, we eed ew weight Sice we my plce the met of 5 pouds with the two existig weights ( poud d 3 pouds) together o oe p, we would like to mke weight of 9 pouds to blce them The we c use the 9-poud weight to blce piece of met of 6 pouds d the 3-poud weight Combie the 9-poud weight d the -poud weight to blce piece of met of 7 pouds d the 3-poud weight Use the 9-poud weight to blce piece of met of 8 pouds d the -poud weight Usig the 9-poud weight we c weigh piece of met of 9 pouds Combiig the 9-poud weight d the -poud weight we c weigh piece of met of pouds Combie the 9-poud weight d the 3-poud weight to blce piece of met of pouds d the -poud weight Combiig the 9-poud weight d the 3-poud weight we c weigh piece of met of pouds Usig ll three existig weights we c weigh piece of met of 3 pouds To weigh piece of met of 4 pouds, we eed ew weight Similrly we would mke weight of 7 pouds to blce the met of 4 pouds d ll three existig weights (-poud, 3-poud, d 9-poud) Now we hve the ptter: the weights re powers of 3 So the fifth weight is 8 pouds Five weights re eough, which re poud, 3 pouds, 9 pouds, 7 pouds, d 8 pouds respectively I fct, with these five weights we c weigh piece of met of up to pouds To weigh piece of met of up to pouds, you my hve differet set of five weights Oe set my be five weights of poud, 3 pouds, 9 pouds, 7 pouds, d 6 pouds respectively Clues to Cretive Thikig Problems 58 to 6 58 A Divisio Fctorize Movig Checkers Let me tell you the first move: 6 A Checkerbord Gme For gme strtegy, workig bckwrds very ofte helps Cretive Thikig Problems 6 to 63 6 Four Tromios to Oe Arrge the four L-shped tromios together to mke lrger shpe similr to them 6 Mgic Circles Fill to ito the smll circles such tht the five umbers o ech of the three lrge circles hve sum of 4 Which umber must replce the questio mrk? 63 Two Smrt Studets of Dr Mth Dr Mth hs two smrt studets Al d Bob Dr Mth picks up two itegers m d from to 9 with m Dr Mth clcultes the sum of m d d tells Al the sum, d clcultes the product of m d d tells Bob the product The Al d Bob hve the followig coverstios: Al: I do t kow wht m d re, but I m sure you do t kow either Bob: Now I kow wht m d re Wht does Dr Mth tell Al? First move (Clues d solutios will be give i the ext issues)? Copyright 9 Shuli Sog All Rights Reserved Use with Permissio 6

### AQA Level 2 Further mathematics Further algebra. Section 3: Inequalities and indices

AQA Level Further mthemtics Further lgebr Sectio : Iequlities d idices Notes d Emples These otes coti subsectios o Iequlities Lier iequlities Qudrtic iequlities Multiplyig epressios The rules of idices

### Logarithms APPENDIX IV. 265 Appendix

APPENDIX IV Logarithms Sometimes, a umerical expressio may ivolve multiplicatio, divisio or ratioal powers of large umbers. For such calculatios, logarithms are very useful. They help us i makig difficult

### First Round Solutions Grades 4, 5, and 6

First Round Solutions Grdes 4, 5, nd 1) There re four bsic rectngles not mde up of smller ones There re three more rectngles mde up of two smller ones ech, two rectngles mde up of three smller ones ech,

### Unit 5: Estimating with Confidence

Uit 5: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Uit 5 Estimatig with Cofidece 8.1 8.2 8.3 Cofidece Itervals: The Basics Estimatig a Populatio

### Domination and Independence on Square Chessboard

Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 A.A. Omrn Deprtment of Mthemtics, College of Eduction for Pure Science, University of bylon, bylon, Irq pure.hmed.omrn@uobby lon.edu.iq Domintion

### EQ: What are the similarities and differences between matrices and real numbers?

Unit 4 Lesson 1 Essentil Question Stndrds Objectives Vocbulry Mtrices Mtrix Opertions Wht re the similrities nd differences between mtrices nd rel numbers? M.ALGII.2.4 Unit 4: Lesson 1 Describe how you

### Example: Modulo 11: Since Z p is cyclic, there is a generator. Let g be a generator of Z p.

Qudrtic Residues Defiitio: The umbers 0, 1,,, ( mod, re clled udrtic residues modulo Numbers which re ot udrtic residues modulo re clled udrtic o-residues modulo Exmle: Modulo 11: Itroductio to Number

### Permutation Enumeration

RMT 2012 Power Roud Rubric February 18, 2012 Permutatio Eumeratio 1 (a List all permutatios of {1, 2, 3} (b Give a expressio for the umber of permutatios of {1, 2, 3,, } i terms of Compute the umber for

### The Solution of the More General Traveling Salesman Problem

AMSE JOURNALS 04-Series: Adces A; Vol. ; N ; pp -40 Submitted No. 0; Reised Dec., 0; Accepted July 0, 04 The Solutio of the More Geerl Trelig Slesm Problem C. Feg, J. Lig,.Deprtmet of Bsic Scieces d Applied

### Polar Coordinates. July 30, 2014

Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xy-plne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes

### 9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations

54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show

### Extra Practice 1. Name Date. Lesson 1.1: Patterns in Division

Master 1.22 Extra Practice 1 Lesso 1.1: Patters i Divisio 1. Which umbers are divisible by 4? By 5? How do you kow? a) 90 b) 134 c) 395 d) 1724 e) 30 f) 560 g) 3015 h) 74 i) 748 2. Write a 5-digit umber

### PERMUTATIONS AND COMBINATIONS

www.sakshieducatio.com PERMUTATIONS AND COMBINATIONS OBJECTIVE PROBLEMS. There are parcels ad 5 post-offices. I how may differet ways the registratio of parcel ca be made 5 (a) 0 (b) 5 (c) 5 (d) 5. I how

### Student Book SERIES. Patterns and Algebra. Name

E Student Book 3 + 7 5 + 5 Nme Contents Series E Topic Ptterns nd functions (pp. ) identifying nd creting ptterns skip counting completing nd descriing ptterns predicting repeting ptterns predicting growing

### 10.4 AREAS AND LENGTHS IN POLAR COORDINATES

65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the

### TCOM370. Solutions to Homework 99-1

COM7 Solutios to Homework 99- Problem hops (poit-to-poit liks) betwee two termils odes; rsmissio rte 96 bps o ll liks; overhed bits [Heder + riler] for ech pcket; ms per-hop sigl propgtio dely. sec. Cll

### 1 tray of toffee 1 bar of toffee. 10 In the decimal number, 0 7, the 7 refers to 7 tenths or

Chpter 3 Deciml Numers Do you know wht DECIMAL is? In chpter, we delt with units, s, 0 s nd 00 s. When you tke single unit nd divide it into (or 0 or 00) its, wht we then hve re deciml frctions of whole

### Grade 6 Math Review Unit 3(Chapter 1) Answer Key

Grade 6 Math Review Uit (Chapter 1) Aswer Key 1. A) A pottery makig class charges a registratio fee of \$25.00. For each item of pottery you make you pay a additioal \$5.00. Write a expressio to represet

### x y z HD(x, y) + HD(y, z) HD(x, z)

Massachusetts Istitute of Techology Departmet of Electrical Egieerig ad Computer Sciece 6.02 Solutios to Chapter 5 Updated: February 16, 2012 Please sed iformatio about errors or omissios to hari; questios

### Addition Mission: Facts Practice

Addition Mission: Fcts Prctice ++ Tble of Contents Addition Mission: Fcts Prctice Mth Counting Apple Mth Hippity Hop Crunchy Celery Addition Superstr Cherry Mth At the Librry Mth Sld -0 Sunny Summer Mth

### COMBINATORICS 2. Recall, in the previous lesson, we looked at Taxicabs machines, which always took the shortest path home

COMBINATORICS BEGINNER CIRCLE 1/0/013 1. ADVANCE TAXICABS Recall, i the previous lesso, we looked at Taxicabs machies, which always took the shortest path home taxipath We couted the umber of ways that

### THE LUCAS TRIANGLE RECOUNTED. Arthur T. Benjamin Dept. of Mathematics, Harvey Mudd College, Claremont, CA Introduction

THE LUCAS TRIANLE RECOUNTED Arthur T Bejami Dept of Mathematics, Harvey Mudd College, Claremot, CA 91711 bejami@hmcedu 1 Itroductio I 2], Neville Robbis explores may properties of the Lucas triagle, a

### Example. Check that the Jacobian of the transformation to spherical coordinates is

lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.

### Combinatorics. Chapter Permutations. Reading questions. Counting Problems. Counting Technique: The Product Rule

Chapter 3 Combiatorics 3.1 Permutatios Readig questios 1. Defie what a permutatio is i your ow words. 2. What is a fixed poit i a permutatio? 3. What do we assume about mutual disjoitedess whe creatig

### Ch 9 Sequences, Series, and Probability

Ch 9 Sequeces, Series, ad Probability Have you ever bee to a casio ad played blackjack? It is the oly game i the casio that you ca wi based o the Law of large umbers. I the early 1990s a group of math

### PERMUTATION AND COMBINATION

MPC 1 PERMUTATION AND COMBINATION Syllabus : Fudametal priciples of coutig; Permutatio as a arragemet ad combiatio as selectio, Meaig of P(, r) ad C(, r). Simple applicatios. Permutatios are arragemets

### Lecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is

### Student Book SERIES. Fractions. Name

D Student Book Nme Series D Contents Topic Introducing frctions (pp. ) modelling frctions frctions of collection compring nd ordering frctions frction ingo pply Dte completed / / / / / / / / Topic Types

### Section 16.3 Double Integrals over General Regions

Section 6.3 Double Integrls over Generl egions Not ever region is rectngle In the lst two sections we considered the problem of integrting function of two vribles over rectngle. This sitution however is

### Polar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a

Polr coordintes 5C r cos Are cos d (cos + ) sin + () + 8 cos cos r cos is circle centre, nd rdius. The re of the semicircle is. 8 Person Eduction Ltd 8. Copying permitted for purchsing institution only.

### CS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates

Bsic Logic Gtes : Computer Architecture I Boolen Algebr Instructor: Prof. Bhgi Nrhri Dept. of Computer Science Course URL: www.ses.gwu.edu/~bhgiweb/cs35/ Digitl Logic Circuits We sw how we cn build the

### CHAPTER 2 LITERATURE STUDY

CHAPTER LITERATURE STUDY. Introduction Multipliction involves two bsic opertions: the genertion of the prtil products nd their ccumultion. Therefore, there re two possible wys to speed up the multipliction:

### Introduction to Wireless Communication Systems ECE 476/ECE 501C/CS 513 Winter 2003

troductio to Wireless Commuicatio ystems ECE 476/ECE 501C/C 513 Witer 2003 eview for Exam #1 March 4, 2003 Exam Details Must follow seatig chart - Posted 30 miutes before exam. Cheatig will be treated

### Exercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION

Exercise 1-1 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You

### CP 405/EC 422 MODEL TEST PAPER - 1 PULSE & DIGITAL CIRCUITS. Time: Three Hours Maximum Marks: 100

PULSE & DIGITAL CIRCUITS Time: Three Hours Maximum Marks: 0 Aswer five questios, takig ANY TWO from Group A, ay two from Group B ad all from Group C. All parts of a questio (a, b, etc. ) should be aswered

### Math Circles Finite Automata Question Sheet 3 (Solutions)

Mth Circles Finite Automt Question Sheet 3 (Solutions) Nickols Rollick nrollick@uwterloo.c Novemer 2, 28 Note: These solutions my give you the nswers to ll the prolems, ut they usully won t tell you how

### Make Your Math Super Powered

Mke Your Mth Super Powered: Use Gmes, Chllenges, nd Puzzles Where s the fun? Lern Mth Workshop model by prticipting in one nd explore fun nocost/low-cost gmes nd puzzles tht you cn esily bring into your

### BOUNDS FOR OUT DEGREE EQUITABLE DOMINATION NUMBERS IN GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN 2303-4874 (p), ISSN (o) 2303-4955 www.imvibl.org/bulletin Vol. 3(2013), 149-154 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

### Stability A Simple Example

Stbility A Simple Exmple We wt the m to ty t x, but wid gve ome iitil peed (f(t) ). Wht will hppe? f (t) x( ) F( ) x(t) f (t) x(t) x( ) F( ) B f (t) x( ) F( ) x(t) B B f (t) x( ) F( ) x(t) B How to chrcterize

### 1. How many possible ways are there to form five-letter words using only the letters A H? How many such words consist of five distinct letters?

COMBINATORICS EXERCISES Stepha Wager 1. How may possible ways are there to form five-letter words usig oly the letters A H? How may such words cosist of five distict letters? 2. How may differet umber

### Alignment in linear space

Sequece Aligmet: Liear Space Aligmet i liear space Chapter 7 of Joes ad Pevzer Q. Ca we avoid usig quadratic space? Easy. Optimal value i O(m + ) space ad O(m) time. Compute OPT(i, ) from OPT(i-1, ). No

### Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 12

EECS 70 Discrete Mathematics ad Probability Theory Sprig 204 Aat Sahai Note 2 Probability Examples Based o Coutig We will ow look at examples of radom experimets ad their correspodig sample spaces, alog

### ECONOMIC LOT SCHEDULING

ECONOMIC LOT SCHEDULING JS, FFS ad ELS Job Shop (JS) - Each ob ca be differet from others - Make to order, low volume - Each ob has its ow sequece Fleible Flow Shop (FFS) - Limited umber of product types

### General Model :Algorithms in the Real World. Applications. Block Codes

Geeral Model 5-853:Algorithms i the Real World Error Correctig Codes I Overview Hammig Codes Liear Codes 5-853 Page message (m) coder codeword (c) oisy chael decoder codeword (c ) message or error Errors

### MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNEL-SHAPED NODES

MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNEL-SHAPED NODES Romn V. Tyshchuk Informtion Systems Deprtment, AMI corportion, Donetsk, Ukrine E-mil: rt_science@hotmil.com 1 INTRODUCTION During the considertion

### Kirchhoff s Rules. Kirchhoff s Laws. Kirchhoff s Rules. Kirchhoff s Laws. Practice. Understanding SPH4UW. Kirchhoff s Voltage Rule (KVR):

SPH4UW Kirchhoff s ules Kirchhoff s oltge ule (K): Sum of voltge drops round loop is zero. Kirchhoff s Lws Kirchhoff s Current ule (KC): Current going in equls current coming out. Kirchhoff s ules etween

### SECOND EDITION STUDENT BOOK GRADE

SECOND EDITION STUDENT BOOK GRADE 5 Bridges in Mthemtics Second Edition Grde 5 Student Book Volumes 1 & 2 The Bridges in Mthemtics Grde 5 pckge consists of: Bridges in Mthemtics Grde 5 Techers Guide Units

### EECE 301 Signals & Systems Prof. Mark Fowler

EECE 3 Sigals & Systems Prof. Mark Fowler Note Set #6 D-T Systems: DTFT Aalysis of DT Systems Readig Assigmet: Sectios 5.5 & 5.6 of Kame ad Heck / Course Flow Diagram The arrows here show coceptual flow

### Density Slicing Reference Manual

Desity Slicig Referece Maual Improvisio, Viscout Cetre II, Uiversity of Warwick Sciece Park, Millbur Hill Road, Covetry. CV4 7HS Tel: 0044 (0) 24 7669 2229 Fax: 0044 (0) 24 7669 0091 e-mail: admi@improvisio.com

### }, how many different strings of length n 1 exist? }, how many different strings of length n 2 exist that contain at least one a 1

1. [5] Give sets A ad B, each of cardiality 1, how may fuctios map A i a oe-tooe fashio oto B? 2. [5] a. Give the set of r symbols { a 1, a 2,..., a r }, how may differet strigs of legth 1 exist? [5]b.

### GENERATE AND MEASURE STANDING SOUND WAVES IN KUNDT S TUBE.

Acoustics Wavelegth ad speed of soud Speed of Soud i Air GENERATE AND MEASURE STANDING SOUND WAVES IN KUNDT S TUBE. Geerate stadig waves i Kudt s tube with both eds closed off. Measure the fudametal frequecy

### Module 9. DC Machines. Version 2 EE IIT, Kharagpur

Module 9 DC Mchines Version EE IIT, Khrgpur esson 40 osses, Efficiency nd Testing of D.C. Mchines Version EE IIT, Khrgpur Contents 40 osses, efficiency nd testing of D.C. mchines (esson-40) 4 40.1 Gols

### NONCLASSICAL CONSTRUCTIONS II

NONLSSIL ONSTRUTIONS II hristopher Ohrt UL Mthcircle - Nov. 22, 2015 Now we will try ourselves on oncelet-steiner constructions. You cn only use n (unmrked) stright-edge but you cn ssume tht somewhere

### Triangles and parallelograms of equal area in an ellipse

1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper

### VIII. Shell-Voicings

VIII. Shell-Voicigs A. The Cocept The 5th (ad ofte the root as well) ca be omitted from most 7th-chords. Ratioale: Most chords have perfect 5ths. The P5th is also preset as the rd partial i the overtoe

### Skills Practice Skills Practice for Lesson 4.1

Skills Prctice Skills Prctice for Lesson.1 Nme Dte Tiling Bthroom Wll Simplifying Squre Root Expressions Vocbulry Mtch ech definition to its corresponding term. 1. n expression tht involves root. rdicnd

### PRACTICAL FILTER DESIGN & IMPLEMENTATION LAB

1 of 7 PRACTICAL FILTER DESIGN & IMPLEMENTATION LAB BEFORE YOU BEGIN PREREQUISITE LABS Itroductio to Oscilloscope Itroductio to Arbitrary/Fuctio Geerator EXPECTED KNOWLEDGE Uderstadig of LTI systems. Laplace

### Department of Electrical and Computer Engineering, Cornell University. ECE 3150: Microelectronics. Spring Due on April 26, 2018 at 7:00 PM

Departmet of Electrical ad omputer Egieerig, orell Uiersity EE 350: Microelectroics Sprig 08 Homework 0 Due o April 6, 08 at 7:00 PM Suggested Readigs: a) Lecture otes Importat Notes: ) MAKE SURE THAT

### Lecture 4: Frequency Reuse Concepts

EE 499: Wireless & Mobile Commuicatios (8) Lecture 4: Frequecy euse Cocepts Distace betwee Co-Chael Cell Ceters Kowig the relatio betwee,, ad, we ca easily fid distace betwee the ceter poits of two co

### Translate and Classify Conic Sections

TEKS 9.6 A.5.A, A.5.B, A.5.D, A.5.E Trnslte nd Clssif Conic Sections Before You grphed nd wrote equtions of conic sections. Now You will trnslte conic sections. Wh? So ou cn model motion, s in E. 49. Ke

### Abacaba-Dabacaba! by Michael Naylor Western Washington University

Abcb-Dbcb! by Michel Nylor Western Wshington University The Abcb structure shows up in n mzing vriety of plces. This rticle explores 10 surprising ides which ll shre this pttern, pth tht will tke us through

### Wavelet Transform. CSEP 590 Data Compression Autumn Wavelet Transformed Barbara (Enhanced) Wavelet Transformed Barbara (Actual)

Wavelet Trasform CSEP 59 Data Compressio Autum 7 Wavelet Trasform Codig PACW Wavelet Trasform A family of atios that filters the data ito low resolutio data plus detail data high pass filter low pass filter

### ECE 274 Digital Logic. Digital Design. Datapath Components Shifters, Comparators, Counters, Multipliers Digital Design

ECE 27 Digitl Logic Shifters, Comprtors, Counters, Multipliers Digitl Design..7 Digitl Design Chpter : Slides to ccompny the textbook Digitl Design, First Edition, by Frnk Vhid, John Wiley nd Sons Publishers,

### Roberto s Notes on Infinite Series Chapter 1: Series Section 2. Infinite series

Roberto s Notes o Ifiite Series Chapter : Series Sectio Ifiite series What you eed to ow already: What sequeces are. Basic termiology ad otatio for sequeces. What you ca lear here: What a ifiite series

### CS3203 #5. 6/9/04 Janak J Parekh

CS3203 #5 6/9/04 Jaak J Parekh Admiistrivia Exam o Moday All slides should be up We ll try ad have solutios for HWs #1 ad #2 out by Friday I kow the HW is due o the same day; ot much I ca do, uless you

### Arithmetic Sequences and Series Sequences and Series Preliminary Maths

Arithmetic Sequeces ad Series Arithmetic Sequeces ad Series Sequeces ad Series Prelimiary Maths www.primeeducatio.com.au Arithmetic Sequeces ad Series Sequeces ad Series 1 Questio 1 The first 5 terms of

### An Application of Assignment Problem in Laptop Selection Problem Using MATLAB

Applied themtics d Scieces: A Itertiol Jourl (thsj ), Vol., No., rch 05 A Applictio of Assigmet Problem i ptop Selectio Problem Usig ATAB ABSTRAT Ghdle Kirtiwt P, uley Yogesh The ssigmet selectio problem

### Counting on r-fibonacci Numbers

Claremot Colleges Scholarship @ Claremot All HMC Faculty Publicatios ad Research HMC Faculty Scholarship 5-1-2015 Coutig o r-fiboacci Numbers Arthur Bejami Harvey Mudd College Curtis Heberle Harvey Mudd

### c The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?

3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem

### Lecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Leture 16 Double integrls Dn Nihols nihols@mth.umss.edu MATH 233, Spring 218 University of Msshusetts Mrh 27, 218 (2) iemnn sums for funtions of one vrible Let f(x) on [, b]. We n estimte the re under

### POWERS OF 3RD ORDER MAGIC SQUARES

Fuzzy Sets, Rough Sets ad Multivalued Operatios ad Applicatios, Vol. 4, No. 1, (Jauary-Jue 01): 37 43 Iteratioal Sciece Press POWERS OF 3RD ORDER MAGIC SQUARES Sreerajii K.S. 1 ad V. Madhukar Mallayya

### You Think You ve Got Problems? Marc Brodie Associate Professor of Mathematics, WJU

You Thik You ve Got Problems? Marc Brodie Associate Professor of Mathematics, WJU Itroductio. My life, like that of ay other s, has its share of problems. I cosider myself fortuate, however, to have more

### Geometric quantities for polar curves

Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions

### Procedia - Social and Behavioral Sciences 128 ( 2014 ) EPC-TKS 2013

Available olie at www.sciecedirect.com ScieceDirect Procedia - Social ad Behavioral Scieces 18 ( 014 ) 399 405 EPC-TKS 013 Iductive derivatio of formulae by a computer Sava Grozdev a *, Veseli Nekov b

### SECOND EDITION HOME CONNECTIONS GRADE

SECOND EDITION HOME COECTIONS GRADE 4 Bridges in Mthemtics Second Edition Grde 4 Home Connections Volumes 1 & 2 The Bridges in Mthemtics Grde 4 pckge consists of: Bridges in Mthemtics Grde 4 Techers Guide

### Samantha s Strategies page 1 of 2

Unit 1 Module 2 Session 3 Smnth s Strtegies pge 1 of 2 Smnth hs been working with vriety of multiplition strtegies. 1 Write n expression to desribe eh of the sttements Smnth mde. To solve 18 20, I find

### METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin

METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN Inventor: Brin L. Bskin 1 ABSTRACT The present invention encompsses method of loction comprising: using plurlity of signl trnsceivers to receive one or

### MATH 118 PROBLEM SET 6

MATH 118 PROBLEM SET 6 WASEEM LUTFI, GABRIEL MATSON, AND AMY PIRCHER Section 1 #16: Show tht if is qudrtic residue modulo m, nd b 1 (mod m, then b is lso qudrtic residue Then rove tht the roduct of the

### MDM 4U MATH OF DATA MANAGEMENT FINAL EXAMINATION

Caadia Iteratioal Matriculatio rogramme Suway Uiversity College MDM 4U MTH OF DT MNGEMENT FINL EXMINTION Date: November 28 th, 2006 Time: 11.30a.m 1.30p.m Legth: 2 HOURS Lecturers: lease circle your teacher

### Chapter (6) Discrete Probability Distributions Examples

hapter () Discrete robability Distributios Eamples Eample () Two balaced dice are rolled. Let X be the sum of the two dice. Obtai the probability distributio of X. Solutio Whe the two balaced dice are

### MEI Core 2. Logarithms and exponentials. Section 2: Modelling curves using logarithms. Modelling curves of the form y kx

MEI Core 2 Logarithms ad eoetials Sectio 2: Modellig curves usig logarithms Notes ad Eamles These otes cotai subsectios o: Modellig curves of the form y = k Modellig curves of the form y = ka Modellig

### Math 7 Flipped Mastery Self Tester Worksheet Name: Class:. Chapter 1 (Unit 1) Patterns and Relationships - Accommodated 1.1 Patterns In Division /36

Chapter 1 (Uit 1) Patters ad Relatioships - Accommodated 1.1 Patters I Divisio /36 Divisibility Rule Cheats; A whole umber is divisible by 2 if it is a eve umber A whole umber is divisible by 4 if the

### LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY

LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY 1. Bsic roerties of qudrtic residues We now investigte residues with secil roerties of lgebric tye. Definition 1.1. (i) When (, m) 1 nd

Table Of Cotets Blues Turarouds Turaroud #1 Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Blues Turarouds Blues Soloig Masterclass Week 1 Steve Stie A Blues Turaroud

### The Math Learning Center PO Box 12929, Salem, Oregon Math Learning Center

Resource Overview Quntile Mesure: Skill or Concept: 300Q Model the concept of ddition for sums to 10. (QT N 36) Model the concept of sutrction using numers less thn or equl to 10. (QT N 37) Write ddition

### Room Design [ HOW TO SET UP YOUR EVENT SPACE ]

Room Desig [ HOW TO SET UP YOUR EVENT SPACE ] There are so may compoets of plaig a evet ad so may decisios! I this article you will lear about some factors that will help you choose the best space for

### MEASURE THE CHARACTERISTIC CURVES RELEVANT TO AN NPN TRANSISTOR

Electricity Electronics Bipolr Trnsistors MEASURE THE HARATERISTI URVES RELEVANT TO AN NPN TRANSISTOR Mesure the input chrcteristic, i.e. the bse current IB s function of the bse emitter voltge UBE. Mesure

### ON THE FUNDAMENTAL RELATIONSHIP BETWEEN THE ACHIEVABLE CAPACITY AND DELAY IN MOBILE WIRELESS NETWORKS

Chapter ON THE FUNDAMENTAL RELATIONSHIP BETWEEN THE ACHIEVABLE CAPACITY AND DELAY IN MOBILE WIRELESS NETWORKS Xiaoju Li ad Ness B. Shroff School of Electrical ad Computer Egieerig, Purdue Uiversity West

### Patterns and Algebra

Student Book Series D Mthletis Instnt Workooks Copyright Series D Contents Topi Ptterns nd funtions identifying nd reting ptterns skip ounting ompleting nd desriing ptterns numer ptterns in tles growing

### arxiv: v2 [math.co] 15 Oct 2018

THE 21 CARD TRICK AND IT GENERALIZATION DIBYAJYOTI DEB arxiv:1809.04072v2 [math.co] 15 Oct 2018 Abstract. The 21 card trick is well kow. It was recetly show i a episode of the popular YouTube chael Numberphile.

### Series. Teacher. Numbers

Series B Techer Copyright 2009 3P Lerning. All rights reserved. First edition printed 2009 in Austrli. A ctlogue record for this book is vilble from 3P Lerning Ltd. ISBN 978-1-921860-17-1 Ownership of

### Regular languages can be expressed as regular expressions.

Regulr lnguges cn e expressed s regulr expressions. A generl nondeterministic finite utomton (GNFA) is kind of NFA such tht: There is unique strt stte nd is unique ccept stte. Every pir of nodes re connected

### AP Calculus BC. Sample Student Responses and Scoring Commentary. Inside: Free Response Question 6. Scoring Guideline.

208 AP Calculus BC Sample Studet Resposes ad Scorig Commetary Iside: Free Respose Questio 6 RR Scorig Guidelie RR Studet Samples RR Scorig Commetary College Board, Advaced Placemet Program, AP, AP Cetral,

### 7. Counting Measure. Definitions and Basic Properties

Virtual Laboratories > 0. Foudatios > 1 2 3 4 5 6 7 8 9 7. Coutig Measure Defiitios ad Basic Properties Suppose that S is a fiite set. If A S the the cardiality of A is the umber of elemets i A, ad is

### PERMUTATIONS AND COMBINATIONS

Chapter 7 PERMUTATIONS AND COMBINATIONS Every body of discovery is mathematical i form because there is o other guidace we ca have DARWIN 7.1 Itroductio Suppose you have a suitcase with a umber lock. The

### Join a Professional Association

Joi a Professioal Associatio 1. The secret job resource: professioal orgaizatios. You may ot kow this, but the career field you re i, or plaig to work i i the future, probably has at least oe professioal

### GRADE SECOND EDITION HOME CONNECTIONS ANSWER KEY

HOME CONNECTIONS SECOND EDITION ANSWER KEY GRADE 4 Bridges in Mthemtics Second Edition Grde 4 Home Connections Volumes 1 & 2 The Bridges in Mthemtics Grde 4 pckge consists of: Bridges in Mthemtics Grde

### E X P E R I M E N T 13

E X P E R I M E N T 13 Stadig Waves o a Strig Produced by the Physics Staff at Colli College Copyright Colli College Physics Departmet. All Rights Reserved. Uiversity Physics, Exp 13: Stadig Waves o a