The First TST for the JBMO Satu Mare, April 6, 2018


 MargaretMargaret Bailey
 5 years ago
 Views:
Transcription
1 The First TST for the JBMO Satu Mare, April 6, 08 Problem. Prove that the equation x +y +z = x+y +z + has no rational solutions. Solution. The equation can be written equivalently (x ) + (y ) + (z ) = 7. If (x, y, z) would be a solution of this equation with rational components, denoting x = a b, y = a b, z = a 3 b 3, one would have integers a, b, a, b, a 3, b 3 satisfying the equality (a b b 3 ) + (b a b 3 ) + (b b a 3 ) = 7(b b b 3 ). This would lead to the existence of four integers a, b, c, d such that a + b + c = 7d. If the greatest common divisor of a, b, c, d is k, dividing by k one would obtain a solution (a, b, c, d) Z of the equation a + b + c = 7d with g.c.d.(a, b, c, d) =. Then a, b, c, d can not all be even. But a perfect square is congruent to either 0,, or modulo 8, so the left hand side can only be,, 3, 5 or 6 modulo 8, while the right hand side, 7d, is 0, or 7 modulo 8. In conclusion, the equality can not hold. Problem. Let a, b, c be positive real numbers such that a + b + c = 3. Prove that When does the equality hold? Solution. The inequality can be written a + 3 b + 5 c a + 3b + c. a + 3 b + 5 c + b + c (a + b + c ) or a + b + c + b + b + c + c. This follows from the following inequalities by using the AMGM inequality: a + b + c abc Marius Stănean because On the other hand, 3 = a + b + c 3 3 a b c = abc. b + b = b + b + b 3 3 b b b = 3 c + c = c + c + c 3 3 b b b = 6.
2 The equality holds if a = b = c =. Remark. One can apply directly the AMGM inequality for the following numbers: a, b, b, b, c, c, c, c, c, b, c, c. Solution. As x 3 3x + 0, x 0 (equivalent to (x ) (x + ) 0; alternatively, the previous inequality follows from AMGM: x x 3 = 3x; the equality holds whence x = ). We deduce that x 3 x. Writing this for a, b, c and multiplying these inequalities by, 3, and 5, respectively, we obtain by summing a + 3 b + 5 c 7 a + 3b 3 + 5c. Using 7 = 9(a + b + c ) one obtains the conclusion. The equality holds if and only if a = b = c =. Problem 3. Let ABC be a triangle with AB > AC. Point P (AB) is such that ACP = ABC. Let D be the reflexion of P into the line AC and let E be the point in which the circumcircle of BCD meets again the line AC. Prove that AE = AC. Vlad Robu Solution. Let Q be the point in which the circumcircle of BCD meets again the line AB. Then QEA = QBC = ECP, hence EQ P C. Moreover, ECP = ECD implies QD is parallel to EC, hence EQ = CD = CP. It follows that EQCP is a parallelogram, which leads to the conclusion. Solution. (given in the contest) If {O} = BD AC, it is easy to prove that BA and BO are isogonal in the angle EBC, therefore, in order to show that BA is the median, one has to prove that BO is the simedian. This follows readily by computation, using Steiner s Theorem. Indeed, EBA = BAC BEA = 80 ABC ACB BDC = 80 ECD
3 ECB BDC = CBD. Triangles OEB and ODC are similar, hence EO OD = EB CD. () Triangles OCB and ODE are also similar, therefore OC OD = BC ED. () We have ADC = AP C = P BC + P CB = ACB = EDB, which shows that the EB AD CD AC triangles ACD and EBD are similar. Hence ED = and, as AD = AC BC EB CD (from ACB AP C ADC), it follows that ED = BC. From () and () we obtain EO ( ) EB ED EB = OC CD BC =, whih shows that BO is indeed BC the simedian. Solution 3. (given in the contest by Andrei Mărginean) Let a = EBA, x = ABC. We have that DCE = ACP = x and EBC = a + x. As B, C, D, E are concyclic, it follows that CDE = 80 a x, DEC = 80 EDC DCE = a = EBA. But AE is the perpendicular bisector of [P D], hence P EA = DEC = EBA, which shows that the triangles AP E and AEB are similar. It follows that AE = AB AP. But triangles ACP and ABC are also similar, hence AC = AP AB = AE, and the conclusion follows immediately.. Problem. What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.) Alexandru Mihalcu Solution. We say that there can be two types of rooks on the chessboard: a rook of type T is a rook that is attacked from two perpendicular lines; a rook is of type T if it is attacked by two rooks situated on the same line, but opposite directions. Suppose one can place m rooks of type T and n of type T, with m + n = x. Each rook of type T determines two lines from which it is attacked (out of the 6 of the chessboard: 8 horizontal and 8 vertical ones), while exactly one other rook of type T is situated on each of these two lines. Notice that a rook of type T can not be attacked from any other line than the one it is already attacked from, hence, for each of these m rooks, there are m lines on which there can be no other rooks. In total, we could have at most n + m = n + m lines. Hence 6 m + n = x. To prove that 6 is indeed the desired maximum, it is sufficient to exhibit an example of 6 rooks that attack exactly two other ones. 3
4 ele sunt atacate din direcţii perpendiculare, respectiv de tipul T dacă ele sunt atacate din aceeaşi direcţie, dar sens contrar. Presupunem că putem plasa maxim x ture, m de tipul T şi n de tipul T. Observăm că orice tură de tipul T determină linii de pe care este atacată, iar exact o altă tură de tipul T se află pe oricare dintre aceste două linii. Pentru turele de tipul T, putem observa că ele nu pot fi atacate din altă direcţie decât cea din care este atacată deja, deci pentru fiecare din cele m ture, există m linii care nu mai pot fi ocupate de alte ture. În total am putea avea n + m = n + m linii. Deci 6 m + n = x. Problema se termină când găsim un exemplu cu 6 ture. 8 RSRSRSRS S0Z0Z0ZR a b c d e f g h Remark. Other elegant examples can be obtained by putting rooks on the two diagonals or, another example, regrouping squares into a square and placing rooks in each unit square of such squares that cover one of the diagonals. Solution. (given in the contest by Andrei Mărginean) We prove by induction after n that on an n n board one can place at most n rooks with the above restrictions. For n = the statement is obvious. For the inductive step, assume the statement above to hold for an arbitrary n and let us prove it for n +. Assume that on an (n + ) (n + ) one could place at least n + 3 rooks such that each of them attacks exactly two other ones. From the Pigeonhole Principle it follows that there exists at least one horizontal line containing at least 3 rooks and, also, a vertical line with 3 or more rooks. On a horizontal line with 3 or more rooks there exists at least one rook that is attacked by two rooks situated on the same horizontal line with it. Therefore this rook must stand alone on the vertical line it occupies. Similarly, there must be a horizontal line containing exactly one rook. Eliminating these two lines, we obtain an n n board with n + rooks that still satisfies the condition that each rook attacks exactly two other rooks. This contradicts our inductive hypothesis for n. Again, an example with 6 rooks finishes the proof. (Andrei gave the one from the first solution.) Remark. By induction after n + m one can prove that the maximum number of rooks one can place on an m n board such that each rook attacks exactly two other rooks is m + n. Solution 3. (given in the contest by Iustinian Constantinescu) Let us consider a configuration of rooks attacking each other. A rook can attack on rays, by opposite. On exactly of these rays one must have rooks, on each of the other two rays the rook... attacks one of the 3 segments that constitute the margins of the chessboard. Each of these 3 segments is attacked by a different rook, and each rook attacks two such segments, therefore one can have at most 6 rooks on the board. An example with 6 rooks on the board finishes the proof. (Iustinian placed the rooks on the two diagonals.)
5 5
12th Bay Area Mathematical Olympiad
2th Bay Area Mathematical Olympiad February 2, 200 Problems (with Solutions) We write {a,b,c} for the set of three different positive integers a, b, and c. By choosing some or all of the numbers a, b and
More informationDo Now: Do Now Slip. Do Now. Lesson 20. Drawing Conclusions. Quiz Tomorrow, Study Blue Sheet. Module 1 Lesson 20 Extra Practice.
Lesson 20 Drawing Conclusions HW Quiz Tomorrow, Study Blue Sheet Do Now: Do Now Slip Oct 20 1:03 PM Do Now 1. CB is parallel to DE 2.
More informationProject Maths Geometry Notes
The areas that you need to study are: Project Maths Geometry Notes (i) Geometry Terms: (ii) Theorems: (iii) Constructions: (iv) Enlargements: Axiom, theorem, proof, corollary, converse, implies The exam
More informationPermutations and Combinations
Permutations and Combinations NAME: 1.) There are five people, Abby, Bob, Cathy, Doug, and Edgar, in a room. How many ways can we line up three of them to receive 1 st, 2 nd, and 3 rd place prizes? The
More information65 P R OV I N G R H O M B U S E S, R E C TA N G L E S, A N D S Q UA R E S
65 P R OV I N G R H O M B U S E S, R E C TA N G L E S, A N D S Q UA R E S Workbook page 261, number 13 Given: ABCD is a rectangle Prove: EDC ECD A D E B C Statements Reasons 1) ABCD is a rectangle 1)
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationSOLUTIONS TO PROBLEM SET 5. Section 9.1
SOLUTIONS TO PROBLEM SET 5 Section 9.1 Exercise 2. Recall that for (a, m) = 1 we have ord m a divides φ(m). a) We have φ(11) = 10 thus ord 11 3 {1, 2, 5, 10}. We check 3 1 3 (mod 11), 3 2 9 (mod 11), 3
More informationSolutions to the 2004 CMO written March 31, 2004
Solutions to the 004 CMO written March 31, 004 1. Find all ordered triples (x, y, z) of real numbers which satisfy the following system of equations: xy = z x y xz = y x z yz = x y z Solution 1 Subtracting
More information3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm.
1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify
More information25 C3. Rachel gave half of her money to Howard. Then Howard gave a third of all his money to Rachel. They each ended up with the same amount of money.
24 s to the Olympiad Cayley Paper C1. The twodigit integer 19 is equal to the product of its digits (1 9) plus the sum of its digits (1 + 9). Find all twodigit integers with this property. If such a
More informationSolutions for the Practice Questions
Solutions for the Practice Questions Question 1. Find all solutions to the congruence 13x 12 (mod 35). Also, answer the following questions about the solutions to the above congruence. Are there solutions
More informationLESSON 4 COMBINATIONS
LESSON 4 COMBINATIONS WARM UP: 1. 4 students are sitting in a row, and we need to select 3 of them. The first student selected will be the president of our class, the 2nd one selected will be the vice
More informationLESSON 2: THE INCLUSIONEXCLUSION PRINCIPLE
LESSON 2: THE INCLUSIONEXCLUSION PRINCIPLE The inclusionexclusion principle (also known as the sieve principle) is an extended version of the rule of the sum. It states that, for two (finite) sets, A
More information0809ge. Geometry Regents Exam Based on the diagram below, which statement is true?
0809ge 1 Based on the diagram below, which statement is true? 3 In the diagram of ABC below, AB # AC. The measure of!b is 40. 1) a! b 2) a! c 3) b! c 4) d! e What is the measure of!a? 1) 40 2) 50 3) 70
More information61. Angles of Polygons. Lesson 61. What You ll Learn. Active Vocabulary
61 Angles of Polygons What You ll Learn Skim Lesson 61. Predict two things that you expect to learn based on the headings and figures in the lesson. 1. 2. Lesson 61 Active Vocabulary diagonal New Vocabulary
More information14th Bay Area Mathematical Olympiad. BAMO Exam. February 28, Problems with Solutions
14th Bay Area Mathematical Olympiad BAMO Exam February 28, 2012 Problems with Solutions 1 Hugo plays a game: he places a chess piece on the top left square of a 20 20 chessboard and makes 10 moves with
More informationUAB MATH TALENT SEARCH
NAME: GRADE: SCHOOL NAME: 20172018 UAB MATH TALENT SEARCH This is a two hour contest. There will be no credit if the answer is incorrect. Full credit will be awarded for a correct answer with complete
More informationOne of the classes that I have taught over the past few years is a technology course for
Trigonometric Functions through Right Triangle Similarities Todd O. Moyer, Towson University Abstract: This article presents an introduction to the trigonometric functions tangent, cosecant, secant, and
More informationSolutions of problems for grade R5
International Mathematical Olympiad Formula of Unity / The Third Millennium Year 016/017. Round Solutions of problems for grade R5 1. Paul is drawing points on a sheet of squared paper, at intersections
More informationCarmen s Core Concepts (Math 135)
Carmen s Core Concepts (Math 135) Carmen Bruni University of Waterloo Week 7 1 Congruence Definition 2 Congruence is an Equivalence Relation (CER) 3 Properties of Congruence (PC) 4 Example 5 Congruences
More informationBRITISH COLUMBIA SECONDARY SCHOOL MATHEMATICS CONTEST, 2006 Senior Preliminary Round Problems & Solutions
BRITISH COLUMBIA SECONDARY SCHOOL MATHEMATICS CONTEST, 006 Senior Preliminary Round Problems & Solutions 1. Exactly 57.4574% of the people replied yes when asked if they used BLEUOUT face cream. The fewest
More information0810ge. Geometry Regents Exam y # (x $ 3) 2 % 4 y # 2x $ 5 1) (0,%4) 2) (%4,0) 3) (%4,%3) and (0,5) 4) (%3,%4) and (5,0)
0810ge 1 In the diagram below, ABC! XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements
More informationModular Arithmetic. Kieran Cooney  February 18, 2016
Modular Arithmetic Kieran Cooney  kieran.cooney@hotmail.com February 18, 2016 Sums and products in modular arithmetic Almost all of elementary number theory follows from one very basic theorem: Theorem.
More informationb. Draw a line and a circle that intersect at exactly one point. When this happens, the line is called a tangent.
61. Circles can be folded to create many different shapes. Today, you will work with a circle and use properties of other shapes to develop a threedimensional shape. Be sure to have reasons for each
More informationG.SRT.B.5: Quadrilateral Proofs
Regents Exam Questions G.SRT.B.5: Quadrilateral Proofs www.jmap.org Name: G.SRT.B.5: Quadrilateral Proofs 1 Given that ABCD is a parallelogram, a student wrote the proof below to show that a pair of its
More information3. Given the similarity transformation shown below; identify the composition:
Midterm Multiple Choice Practice 1. Based on the construction below, which statement must be true? 1 1) m ABD m CBD 2 2) m ABD m CBD 3) m ABD m ABC 1 4) m CBD m ABD 2 2. Line segment AB is shown in the
More informationBmMT 2013 TEAM ROUND SOLUTIONS 16 November 2013
BmMT 01 TEAM ROUND SOLUTIONS 16 November 01 1. If Bob takes 6 hours to build houses, he will take 6 hours to build = 1 houses. The answer is 18.. Here is a somewhat elegant way to do the calculation: 1
More informationMATHEMATICS ON THE CHESSBOARD
MATHEMATICS ON THE CHESSBOARD Problem 1. Consider a 8 8 chessboard and remove two diametrically opposite corner unit squares. Is it possible to cover (without overlapping) the remaining 62 unit squares
More informationMATH CIRCLE, 10/13/2018
MATH CIRCLE, 10/13/2018 LARGE SOLUTIONS 1. Write out row 8 of Pascal s triangle. Solution. 1 8 28 56 70 56 28 8 1. 2. Write out all the different ways you can choose three letters from the set {a, b, c,
More informationLECTURE 3: CONGRUENCES. 1. Basic properties of congruences We begin by introducing some definitions and elementary properties.
LECTURE 3: CONGRUENCES 1. Basic properties of congruences We begin by introducing some definitions and elementary properties. Definition 1.1. Suppose that a, b Z and m N. We say that a is congruent to
More information6.1 Warm Up The diagram includes a pair of congruent triangles. Use the congruent triangles to find the value of x in the diagram.
6.1 Warm Up The diagram includes a pair of congruent triangles. Use the congruent triangles to find the value of x in the diagram. 1. 2. Write a proof. 3. Given: P is the midpoint of MN and TQ. Prove:
More informationThe problems in this booklet are organized into strands. A problem often appears in multiple strands. The problems are suitable for most students in
The problems in this booklet are organized into strands. A problem often appears in multiple strands. The problems are suitable for most students in Grade 7 or higher. Problem C Retiring and Hiring A
More informationBuilding Blocks of Geometry
Practice A Building Blocks of Geometry Write the following in geometric notation. 1. line EF 2. ray RS 3. line segment JK Choose the letter for the best answer. 4. Identify a line. A BD B AD C CB D BD
More informationGeometry  Chapter 6 Review
Class: Date: Geometry  Chapter 6 Review 1. Find the sum of the measures of the angles of the figure. 4. Find the value of x. The diagram is not to scale. A. 1260 B. 900 C. 540 D. 720 2. The sum of the
More informationI.M.O. Winter Training Camp 2008: Invariants and Monovariants
I.M.. Winter Training Camp 2008: Invariants and Monovariants n math contests, you will often find yourself trying to analyze a process of some sort. For example, consider the following two problems. Sample
More informationGEOMETRY. Workbook Common Core Standards Edition. Published by TOPICAL REVIEW BOOK COMPANY. P. O. Box 328 Onsted, MI
Workbook Common Core Standards Edition Published by TOPICAL REVIEW BOOK COMPANY P. O. Box 328 Onsted, MI 492650328 www.topicalrbc.com EXAM PAGE Reference Sheet...i January 2017...1 June 2017...11 August
More informationChapter 9. Q1. A diagonal of a parallelogram divides it into two triangles of equal area.
Chapter 9 Q1. A diagonal of a parallelogram divides it into two triangles of equal area. Q2. Parallelograms on the same base and between the same parallels are equal in area. Q3. A parallelogram and a
More informationBilbo s New Adventures
Bilbo s New Adventures Problem 1 Solve the equation: x + x + 1 x + = 0 Solution The left side of the equation is defined for x 0 Moving x + to the right hand side of the equation one gets: x + x + 1 =
More informationTeam Round University of South Carolina Math Contest, 2018
Team Round University of South Carolina Math Contest, 2018 1. This is a team round. You have one hour to solve these problems as a team, and you should submit one set of answers for your team as a whole.
More informationDownloaded from
1 IX Mathematics Chapter 8: Quadrilaterals Chapter Notes Top Definitions 1. A quadrilateral is a closed figure obtained by joining four points (with no three points collinear) in an order. 2. A diagonal
More informationG.SRT.B.5: Quadrilateral Proofs
Regents Exam Questions G.SRT.B.5: Quadrilateral Proofs www.jmap.org Name: G.SRT.B.5: Quadrilateral Proofs 1 Given that ABCD is a parallelogram, a student wrote the proof below to show that a pair of its
More informationMATHEMATICS: PAPER II
NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 2017 MATHEMATICS: PAPER II EXAMINATION NUMBER Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of
More information2006 Pascal Contest (Grade 9)
Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario 2006 Pascal Contest (Grade 9) Wednesday, February 22, 2006
More informationSOLUTIONS FOR PROBLEM SET 4
SOLUTIONS FOR PROBLEM SET 4 A. A certain integer a gives a remainder of 1 when divided by 2. What can you say about the remainder that a gives when divided by 8? SOLUTION. Let r be the remainder that a
More information9.3 Properties of Chords
9.3. Properties of Chords www.ck12.org 9.3 Properties of Chords Learning Objectives Find the lengths of chords in a circle. Discover properties of chords and arcs. Review Queue 1. Draw a chord in a circle.
More informationc) What is the ratio of the length of the side of a square to the length of its diagonal? Is this ratio the same for all squares? Why or why not?
Tennessee Department of Education Task: Ratios, Proportions, and Similar Figures 1. a) Each of the following figures is a square. Calculate the length of each diagonal. Do not round your answer. Geometry/Core
More informationGeometry Unit 5 Practice Test
Name: Class: Date: ID: X Geometry Unit 5 Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. What is the value of x in the rectangle? Hint: use
More information2014 Edmonton Junior High Math Contest ANSWER KEY
Print ID # School Name Student Name (Print First, Last) 100 2014 Edmonton Junior High Math Contest ANSWER KEY Part A: Multiple Choice Part B (short answer) Part C(short answer) 1. C 6. 10 15. 9079 2. B
More informationSecondary 2 Unit 7 Test Study Guide
Class: Date: Secondary 2 Unit 7 Test Study Guide 20142015 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which statement can you use to conclude that
More informationSMML MEET 3 ROUND 1
ROUND 1 1. How many different 3digit numbers can be formed using the digits 0, 2, 3, 5 and 7 without repetition? 2. There are 120 students in the senior class at Jefferson High. 25 of these seniors participate
More informationMath 255 Spring 2017 Solving x 2 a (mod n)
Math 255 Spring 2017 Solving x 2 a (mod n) Contents 1 Lifting 1 2 Solving x 2 a (mod p k ) for p odd 3 3 Solving x 2 a (mod 2 k ) 5 4 Solving x 2 a (mod n) for general n 9 1 Lifting Definition 1.1. Let
More informationRosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples
Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples Section 1.7 Proof Methods and Strategy Page references correspond to locations of Extra Examples icons in the textbook. p.87,
More informationGeometry Ch 3 Vertical Angles, Linear Pairs, Perpendicular/Parallel Lines 29 Nov 2017
3.1 Number Operations and Equality Algebraic Postulates of Equality: Reflexive Property: a=a (Any number is equal to itself.) Substitution Property: If a=b, then a can be substituted for b in any expression.
More information1.6 Congruence Modulo m
1.6 Congruence Modulo m 47 5. Let a, b 2 N and p be a prime. Prove for all natural numbers n 1, if p n (ab) and p  a, then p n b. 6. In the proof of Theorem 1.5.6 it was stated that if n is a prime number
More informationOlimpiad«Estonia, 2003
Problema s«pt«m nii 128 a) Dintro tabl«p«trat«(2n + 1) (2n + 1) se ndep«rteaz«p«tr«telul din centru. Pentru ce valori ale lui n se poate pava suprafata r«mas«cu dale L precum cele din figura de mai jos?
More informationThe congruence relation has many similarities to equality. The following theorem says that congruence, like equality, is an equivalence relation.
Congruences A congruence is a statement about divisibility. It is a notation that simplifies reasoning about divisibility. It suggests proofs by its analogy to equations. Congruences are familiar to us
More informationVMO Competition #1: November 21 st, 2014 Math Relays Problems
VMO Competition #1: November 21 st, 2014 Math Relays Problems 1. I have 5 different colored felt pens, and I want to write each letter in VMO using a different color. How many different color schemes of
More informationClass 5 Geometry O B A C. Answer the questions. For more such worksheets visit
ID : in5geometry [1] Class 5 Geometry For more such worksheets visit www.edugain.com Answer the questions (1) The set square is in the shape of. (2) Identify the semicircle that contains 'C'. A C O B
More informationPREJUNIOR CERTIFICATE EXAMINATION, 2010 MATHEMATICS HIGHER LEVEL. PAPER 2 (300 marks) TIME : 2½ HOURS
J.20 PREJUNIOR CERTIFICATE EXAMINATION, 2010 MATHEMATICS HIGHER LEVEL PAPER 2 (300 marks) TIME : 2½ HOURS Attempt ALL questions. Each question carries 50 marks. Graph paper may be obtained from the superintendent.
More information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 7 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 7 1 / 8 Invertible matrices Theorem. 1. An elementary matrix is invertible. 2.
More informationName Date Class Period. 5.2 Exploring Properties of Perpendicular Bisectors
Name Date Class Period Activity B 5.2 Exploring Properties of Perpendicular Bisectors MATERIALS QUESTION EXPLORE 1 geometry drawing software If a point is on the perpendicular bisector of a segment, is
More informationALGEBRA: Chapter I: QUESTION BANK
1 ALGEBRA: Chapter I: QUESTION BANK Elements of Number Theory Congruence One mark questions: 1 Define divisibility 2 If a b then prove that a kb k Z 3 If a b b c then PT a/c 4 If a b are two non zero integers
More informationUnit 10 Arcs and Angles of Circles
Lesson 1: Thales Theorem Opening Exercise Vocabulary Unit 10 Arcs and Angles of Circles Draw a diagram for each of the vocabulary words. Definition Circle The set of all points equidistant from a given
More informationUK Junior Mathematical Challenge
UK Junior Mathematical Challenge THURSDAY 28th APRIL 2016 Organised by the United Kingdom Mathematics Trust from the School of Mathematics, University of Leeds http://www.ukmt.org.uk Institute and Faculty
More information1. Write the angles in order from 2. Write the side lengths in order from
Lesson 1 Assignment Triangle Inequalities 1. Write the angles in order from 2. Write the side lengths in order from smallest to largest. shortest to longest. 3. Tell whether a triangle can have the sides
More informationTitle: Quadrilaterals Aren t Just Squares
Title: Quadrilaterals ren t Just Squares Brief Overview: This is a collection of the first three lessons in a series of seven lessons studying characteristics of quadrilaterals, including trapezoids, parallelograms,
More informationNumber Theory/Cryptography (part 1 of CSC 282)
Number Theory/Cryptography (part 1 of CSC 282) http://www.cs.rochester.edu/~stefanko/teaching/11cs282 1 Schedule The homework is due Sep 8 Graded homework will be available at noon Sep 9, noon. EXAM #1
More informationTilings with T and Skew Tetrominoes
Quercus: Linfield Journal of Undergraduate Research Volume 1 Article 3 1082012 Tilings with T and Skew Tetrominoes Cynthia Lester Linfield College Follow this and additional works at: http://digitalcommons.linfield.edu/quercus
More informationMistilings with Dominoes
NOTE Mistilings with Dominoes Wayne Goddard, University of Pennsylvania Abstract We consider placing dominoes on a checker board such that each domino covers exactly some number of squares. Given a board
More informationAssignment 2. Due: Monday Oct. 15, :59pm
Introduction To Discrete Math Due: Monday Oct. 15, 2012. 11:59pm Assignment 2 Instructor: Mohamed Omar Math 6a For all problems on assignments, you are allowed to use the textbook, class notes, and other
More informationModular arithmetic Math 2320
Modular arithmetic Math 220 Fix an integer m 2, called the modulus. For any other integer a, we can use the division algorithm to write a = qm + r. The reduction of a modulo m is the remainder r resulting
More informationCoding Theory on the Generalized Towers of Hanoi
Coding Theory on the Generalized Towers of Hanoi Danielle Arett August 1999 Figure 1 1 Coding Theory on the Generalized Towers of Hanoi Danielle Arett Augsburg College Minneapolis, MN arettd@augsburg.edu
More informationPermutations, Combinations and The Binomial Theorem. Unit 9 Chapter 11 in Text Approximately 7 classes
Permutations, Combinations and The Binomial Theorem Unit 9 Chapter 11 in Text Approximately 7 classes In this unit, you will be expected to: Solve problems that involve the fundamental counting principle.
More information(A) Circle (B) Polygon (C) Line segment (D) None of them (A) (B) (C) (D) (A) Understanding Quadrilaterals <1M>
Understanding Quadrilaterals 1.A simple closed curve made up of only line segments is called a (A) Circle (B) Polygon (C) Line segment (D) None of them 2.In the following figure, which of the polygon
More informationPRMO Official Test / Solutions
Date: 19 Aug 2018 PRMO Official Test  2018 / Solutions 1. 17 ANSWERKEY 1. 17 2. 8 3. 70 4. 12 5. 84 6. 18 7. 14 8. 80 9. 81 10. 24 11. 29 12. 88 13. 24 14. 19 15. 21 16. 55 17. 30 18. 16 19. 33 20. 17
More informationCLASS NOTES. A mathematical proof is an argument which convinces other people that something is true.
Propositional Statements A mathematical proof is an argument which convinces other people that something is true. The implication If p then q written as p q means that if p is true, then q must also be
More information(A) Circle (B) Polygon (C) Line segment (D) None of them
Understanding Quadrilaterals 1.The angle between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the parallelogram is 60 degree. Find the angles of the parallelogram.
More informationMath 3 Geogebra Discovery  Equidistance Decemeber 5, 2014
Math 3 Geogebra Discovery  Equidistance Decemeber 5, 2014 Today you and your partner are going to explore two theorems: The Equidistance Theorem and the Perpendicular Bisector Characterization Theorem.
More informationTrigonometry. David R. Wilkins
Trigonometry David R. Wilkins 1. Trigonometry 1. Trigonometry 1.1. Trigonometric Functions There are six standard trigonometric functions. They are the sine function (sin), the cosine function (cos), the
More informationIndicate whether the statement is true or false.
MATH 121 SPRING 2017  PRACTICE FINAL EXAM Indicate whether the statement is true or false. 1. Given that point P is the midpoint of both and, it follows that. 2. If, then. 3. In a circle (or congruent
More information(1) 2 x 6. (2) 5 x 8. (3) 9 x 12. (4) 11 x 14. (5) 13 x 18. Soln: Initial quantity of rice is x. After 1st customer, rice available In the Same way
1. A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys
More informationFoundations of Math II Unit 3: Similarity and Congruence
Foundations of Math II Unit 3: Similarity and Congruence Academics High School Mathematics 3.1 Warm Up 1. Jill and Bill are doing some exercises. Jayne Funda, their instructor, gently implores Touch your
More informationTo Explore the Properties of Parallelogram
Exemplar To Explore the Properties of Parallelogram Objective To explore the properties of parallelogram Dimension Measures, Shape and Space Learning Unit Quadrilaterals Key Stage 3 Materials Required
More informationRegents Exam Questions by Topic Page 1 TOOLS OF GEOMETRY: Constructions NAME:
Regents Exam Questions by Topic Page 1 1. 060925ge, P.I. G.G.17 Which illustration shows the correct construction of an angle bisector? [A] 3. 060022a, P.I. G.G.17 Using only a ruler and compass, construct
More informationCS1800: Permutations & Combinations. Professor Kevin Gold
CS1800: Permutations & Combinations Professor Kevin Gold Permutations A permutation is a reordering of something. In the context of counting, we re interested in the number of ways to rearrange some items.
More informationThe Chinese Remainder Theorem
The Chinese Remainder Theorem 832014 The Chinese Remainder Theorem gives solutions to systems of congruences with relatively prime moduli The solution to a system of congruences with relatively prime
More informationOrganization Team Team ID# If each of the congruent figures has area 1, what is the area of the square?
1. [4] A square can be divided into four congruent figures as shown: If each of the congruent figures has area 1, what is the area of the square? 2. [4] John has a 1 liter bottle of pure orange juice.
More informationTable of Contents. Constructions Day 1... Pages 15 HW: Page 6. Constructions Day 2... Pages 714 HW: Page 15
CONSTRUCTIONS Table of Contents Constructions Day 1...... Pages 15 HW: Page 6 Constructions Day 2.... Pages 714 HW: Page 15 Constructions Day 3.... Pages 1621 HW: Pages 2224 Constructions Day 4....
More informationWinter Quarter Competition
Winter Quarter Competition LA Math Circle (Advanced) March 13, 2016 Problem 1 Jeff rotates spinners P, Q, and R and adds the resulting numbers. What is the probability that his sum is an odd number? Problem
More informationCounting Things. Tom Davis March 17, 2006
Counting Things Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles March 17, 2006 Abstract We present here various strategies for counting things. Usually, the things are patterns, or
More informationGeometry  Midterm Exam Review  Chapters 1, 2
Geometry  Midterm Exam Review  Chapters 1, 2 1. Name three points in the diagram that are not collinear. 2. Describe what the notation stands for. Illustrate with a sketch. 3. Draw four points, A, B,
More informationNumber Theory  Divisibility Number Theory  Congruences. Number Theory. June 23, Number Theory
 Divisibility  Congruences June 23, 2014 Primes  Divisibility  Congruences Definition A positive integer p is prime if p 2 and its only positive factors are itself and 1. Otherwise, if p 2, then p
More informationPublic Key Cryptography Great Ideas in Theoretical Computer Science Saarland University, Summer 2014
7 Public Key Cryptography Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 Cryptography studies techniques for secure communication in the presence of third parties. A typical
More information1. Answer: 250. To reach 90% in the least number of problems involves Jim getting everything
. Answer: 50. To reach 90% in the least number of problems involves Jim getting everything 0 + x 9 correct. Let x be the number of questions he needs to do. Then = and cross 50 + x 0 multiplying and solving
More information1 = 3 2 = 3 ( ) = = = 33( ) 98 = = =
Math 115 Discrete Math Final Exam December 13, 2000 Your name It is important that you show your work. 1. Use the Euclidean algorithm to solve the decanting problem for decanters of sizes 199 and 98. In
More information1. Answer: 250. To reach 90% in the least number of problems involves Jim getting everything
8 th grade solutions:. Answer: 50. To reach 90% in the least number of problems involves Jim getting everything 0 + x 9 correct. Let x be the number of questions he needs to do. Then = and cross 50 + x
More informationth Grade Test. A. 128 m B. 16π m C. 128π m
1. Which of the following is the greatest? A. 1 888 B. 2 777 C. 3 666 D. 4 555 E. 6 444 2. How many whole numbers between 1 and 100,000 end with the digits 123? A. 50 B. 76 C. 99 D. 100 E. 101 3. If the
More informationParallels and Euclidean Geometry
Parallels and Euclidean Geometry Lines l and m which are coplanar but do not meet are said to be parallel; we denote this by writing l m. Likewise, segments or rays are parallel if they are subsets of
More informationSuppose that two squares are cut from opposite corners of a chessboard. Can the remaining squares be completely covered by 31 dominoes?
Chapter 2 Parent Guide Reasoning in Geometry Reasoning is a thinking process that progresses logically from one idea to another. Logical reasoning advances toward a conclusion in such a way as to be understood
More information3. Rewriting the given integer, = = so x = 5, y = 2 and z = 1, which gives x+ y+ z =8.
2004 Gauss Contest  Grade Solutions Part A 1. 25% of 2004 is 1 4 of 2004, or 501. 2. Using a common denominator, + 3 5 = 4 + = 1 2 4 6 5 5 3. Rewriting the given integer, 00 670 = 00 000 + 600 + 70 =
More information