# 1. By how much does 1 3 of 5 2 exceed 1 2 of 1 3? 2. What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3.

Save this PDF as:

Size: px
Start display at page:

Download "1. By how much does 1 3 of 5 2 exceed 1 2 of 1 3? 2. What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3."

## Transcription

1 1 By how much does 1 3 of 5 exceed 1 of 1 3? What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3 A ticket fee was \$10, but then it was reduced The number of customers increased by 50%, but the amount of money received only increased by 0% How many dollars was the reduced ticket price? 4 If 6x + 7y = 007 and 7x + 6y = 700, then what is the value of x + y? 5 What is the sum of the digits of ? 6 There are three consecutive positive integers such that the square of the second minus twelve times the first is three less than twice the third What is the smallest of the three integers? 7 How many perfect cubes greater than 1 are divisors of 9 9? 8 The harmonic mean of two numbers is defined to be the reciprocal of the average of the reciprocals of the numbers Find x such that the harmonic mean of 4 and x is 6 9 What is the smallest possible value of x 1 + y x + y 007 for any real numbers x and y? 10 What is the value of log (log (log (16)))? 11 Two fair 6-sided dice, colored red and white, are tossed What is the probability that the number on the red die is at least as large as that on the white die? 1 What is the smallest positive number which is equal to the cube of one positive integer and also is equal to the fourth power of a different positive integer? 13 What value of c occurs in a solution of the system of equations a+b+c = 14, ab = 14, and c = a + b? 14 If 11 positive integers a 1 a a 11 have the property that no three of them are the sides of a triangle, what is the smallest possible value of a 11 /a 1? 15 A cube and sphere have the same surface area What is the ratio of the volume of the sphere to that of the cube? 16 A rhombus ABCD with sides of length 5 lies in the first quadrant with A at the origin If AD has slope 1/ and AB has slope, then what are the coordinates of C? 17 For how many primes p is p + 3p 1 also prime? 18 A large container, labeled R, is partially filled with 4 quarts of red paint Another large container, labeled W, is partially filled with 5 quarts of white paint A small empty can is completely filled with red paint taken from R, and the contents of the can then emptied into W After thorough mixing of the contents of W, the can is completely filled with some of this mixture from W, and the contents of the can then emptied into R The ratio of red paint to white in R is now 3:1 What is the size of the can, in quarts? 1

2 19 What is the largest number A such that the graphs of x = y and (x A) + y = 1 intersect? 0 A number is written with one 1, followed by three 3 s, then five 5 s, then seven 7 s, then nine 9 s, then eleven 11 s, etc Thus it begins , and just as it gets to the 13 s it has What is its 999th digit? 1 What is the radius of a circle inscribed in an isosceles right triangle whose legs have length 1? In what base b is the equation = 73 valid? Here all three numbers are base-b numbers, and b must be a positive integer 3 A strip of rubber is initially 80 cm long, and after every minute it is instantaneously and uniformly stretched by 40 cm An ant moves at a rate of 4 cm per minute, starting at one end Each time the strip is stretched, the ant is moved to a position on the modified strip proportional to its position on the strip before the stretching took place How many minutes does it take the ant to cross the strip? 4 The first term of an infinite geometric series is 10, and the sum of the series lies between 9 and 11, inclusive What is the range of values for the common ratio r between terms of this series? 5 If the roots of x bx + c = 0 are sin(π/9) and cos(π/9), then express b in terms of c (Don t write b = ; just write the expression involving c) 6 Let R denote the set of points (x, y) satisfying x 4 x + y What is the area of R? 7 The sum of the 3-digit numbers 35x and 4y7 is divisible by 36 Find all possible ordered pairs (x, y) 8 What is the ratio of the area of a regular 10-gon to that of a regular 0-gon inscribed in the same circle? Express your answer using a single trig function with its angle in degrees 9 Let k > 0 and let A lie on the curve y = k x so that the vertical and horizontal segments from A to the x- and y-axes are sides of a square with the origin at the vertex opposite A Let B be the vertex of this square lying below A on the x-axis Let C be the point on the same curve between the origin and A such that the vertical segment from C to the x-axis and the horizontal segment from C to AB is a square with B at the vertex opposite C What is the ratio of the side length of the second square to that of the first? 30 What is the smallest multiple of 999 which does not have any 9 s among its digits? (If the number is n = 999 d, write n, not d) 31 Let P (n) denote the product of the digits of n, and let S(n) denote the sum of the digits of n How many positive integers n satisfy n = P (n) + S(n)?

3 3 For each point in the plane, consider the sum of the squares of its distances from the four points ( 3, 1), ( 1, 0), (1, ), and (1, 3) What is the smallest number achieved as such a sum? 33 What is the remainder when x 3 + x 6 + x 9 + x 7 is divided by x 1? 34 Find k so that the solutions of x 3 3x + kx + 8 = 0 are in arithmetic progression 35 In triangle ABC, D lies on BC so that AC = 3, AD = 3, BD = 8, and CD = 1 What is the length of AB? 36 What is the ninth digit from the right in 101 0? 37 Two circles of radius 1 overlap so that the center of each lies on the circumference of the other What is the area of their union? 38 The first two positive integers n for which n is a perfect square are 1 and 8 What are the next two? 39 In the diagram below, which is not drawn to scale, the circles are tangent at A, the center of the larger circle is at B, CD = 4, and EF = 4 What are the radii of the circles? E F A B C D 40 What is the side length of an equilateral triangle ABC for which there is a point P inside it such that AP = 6, BP = 8, and CP = 10? 3

4 Solutions to 007 contest Numbers in brackets are the number of people who answered the question correctly, first out of the 8 people who scored at least 0, and then out of the other 8 people 1 /3 [8, 193] The difference is /5 [7, 1385] The fraction is (4 3 )/5 = 7/5 3 8 [4, 136] If the new fee is x and the original number of customers was N, then 1 10N = 15Nx Thus x = 1/15 = [8, 105] Add the equations to get 13(x+y) = 9009 Thus x+y = 9009/13 = [, 81] The number has 53 9 s followed by the digits 45 Thus the answer is [8, 141] If x is our desired number, then (x + 1) 1x = (x + ) 3 Thus x + x + 1 1x = x + 1, and hence x = 0 or [5, 46] They are 3 3, 3 6, 3 9, 3 1, 3 15, and [4, 140] The harmonic mean of x and y equals 1/( 1 ( 1 x + 1 xy y )) = x+y We have 6 = 8x/(x + 4) We obtain x = [5, 130] This value will be achieved whenever 1 x y 007 The sum must be 006 by the general fact that a b + b c a c 10 1 [8, 138] log (16) = 4 log (4) = log () = /1 [55, 1015] There is 1/6 chance that they are equal, and then 1 (1 1 6 ) that the red die is greater The desired answer is [6, 66] This is 1 which equals 16 3 and 8 4 Note that 1 is ruled out since it is not the cube and fourth power of different integers To see that there is nothing smaller, one can easily check that none of 4,, 7 4 is a cube 13 6 [5, 37] We have c = (a + b) 8 = (14 c) 8 We obtain 8c = 196 8, hence c = [4, 1] We may set a 1 = 1 To make the numbers as small as possible, we should choose a = a 1 and a n = a n + a n 1 for n 3 Thus we obtain the Fibonacci numbers 1, 1,, 3, 5, 8, 13, 1, 34, 55, and /π [155, 3] If s is the side of the cube and r the radius of the sphere, then 6s = 4πr and so (r/s) = 6/(4π) The ratio which we desire is 4 3 π(r/s)3 = 4 3 π(6/4π)3/ = 6/π 16 (3 5, 3 5) [3, 49] The vector AC is the sum of AD and AB, which are, respectively, 5, 1 and 5 1, 17 1 [, 58] For p = 3, we obtain the prime 17 Every prime p 3 is of the form 3k ± 1 for some k and hence has p 1 a multiple of 3 Thus p + 3p 1 is also a multiple of 3 4

6 0 sin(9) cos(9) By the double angle formula, this latter equals 10 sin(18) Thus the ratio is cos(18) 9 ( 5 1)/ [7, 1] A lies on the intersection of the curve with the line y = x and so its x coordinate satisfies x = k x, hence x = k The side length s of the second square satisfies s = k k s Thus s + k s k 4 = 0 and hence s = 1 ( k + k 4 + 4k 4 ) The desired ratio s/k equals 1 ( ) [15, 1] The multiples of 999 have the form 1000d d If d 100, this has a 9 in the 100 s position If 101 d 110, it has a 9 in the 10 s position If d = 111, it has a 9 in the 1 s position If d = 11, it equals [16, 9] Clearly no 1-digit number works If n = 10a + b is a -digit number satisfying the required property, then 10a + b = ab + a + b and so b = 9, yielding the nine numbers 19, 9,,99 If n = 100a + 10b + c is a 3-digit number, we would need 100a + 10b + c = abc + a + b + c and so 99a + 9b = abc Since bc 81, this has no solutions A similar argument will show that there will be no solutions if n has more than three digits 3 1 [9, 3] For the point (x, y), this sum is (x+3) +(y+1) +(x+1) +y +(x 1) +(y ) +(x 1) +(y 3) = 4x +4x+1+4y 8y +14 = (x+1) +11+4(y 1) +10 The minimum value of this occurs when x = 1/ and y = 1, and is x + 1 [4, 5] The remainder must be a first-degree polynomial ax + b We have x 3 + x 6 + x 9 + x 7 = q(x)(x 1) + ax + b Letting x = 1 yields 4 = a + b, while letting x = 1 yields = a + b We solve these equations, obtaining a = 3 and b = [13, 6] Let the roots be a d, a, and a + d The sum of the roots is the coefficient of x ; hence a = 1 The negative of the product of the roots equals the constant term Thus d 1 = 8 Hence the roots are, 1, and 4 Then k is the sum of products of pairs of roots, so equals [15, 4] Let AE be the altitude from A, and let h denote its length Note that triangle ACD is isosceles, and so ED = EC = 1/ We have h = 3 ( 1 ), and AB = h + ( 17 ) Thus AB = = 81 B A D E C 6

7 36 6 [8, 4] We expand ( ) 0 = ( 0 ) ( ) ( ) Neither the first three terms nor the omitted terms at the end will affect the ninth digit from the right The fourth and fifth terms are and The desired digit is [, 1] 4π Their overlap is twice the area A to the right of the vertical line in the diagram below This vertical line is at radius 1/ Thus the indicated angle is 10 o, and so the area of the wedge subtended by the angle is π/3 Then A equals π/3 minus the area of the indicated triangle, which is 1 3 Thus the area of overlap is π 3 3 Finally the area of the union is π minus the overlap and 88 [9, 1] We need that n(n + 1) is times a perfect square Since n and n + 1 contain no common divisors, one must be a square and the other twice a square The square must be odd We examine the first few odd squares m to see whether m 1 or m + 1 is times a square For m 1, we obtain 8, 4, 48, 80, 10, 168, 4, and 88 For m + 1, we obtain 10, 6, 50, 8, 1, 170, 6, and 90 Of these 8, 50, and 88 are times a perfect square The pairs (n, n + 1) are (8, 9), (49, 50), and (88, 89) 39 75, 96 [11, 1] Let R (resp r) denote the radius of the large (resp small) circle Then R = r + 1 Let O denote the center of the small circle Referring to the diagram on your exam sheet, the right triangle OF B implies r = (R 4) + 1, and hence r = (r 3) + 441, which implies 6r = [3, 0] Let s be the desired length, and place the triangle with A = (s 3, s ), B = (s 3, s ), and C = (0, 0) If P = (x, y), we have x + y = 100, (x 3 s) + (y + s ) = 64, and (x 3 s) + (y s ) = 36 The second and third imply sy = 14 Substitute this and the first into the second and get sx = (s + 50)/ 3 Multiply the first equation by s and substitute our expressions for sx and sy, obtaining an equation which reduces to s 4 00s = 0 Get s =

### 4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?

1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of

### 4. How many integers between 2004 and 4002 are perfect squares?

5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

### 13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

### CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

### 2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2

00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

Advanced GMAT Math Questions Version Quantitative Fractions and Ratios 1. The current ratio of boys to girls at a certain school is to 5. If 1 additional boys were added to the school, the new ratio of

### Geometry A Solutions. Written by Ante Qu

Geometry A Solutions Written by Ante Qu 1. [3] Three circles, with radii of 1, 1, and, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to

### Factoring Polynomials

UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### 2000 Solutions Fermat Contest (Grade 11)

anadian Mathematics ompetition n activity of The entre for Education in Mathematics and omputing, University of Waterloo, Waterloo, Ontario 000 s Fermat ontest (Grade ) for The ENTRE for EDUTION in MTHEMTIS

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### 1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

### Pre-Calculus Review Problems Solutions

MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry Pre-Calculus Review Problems Solutions Problem 1. Give equations for the following lines in both point-slope and slope-intercept form. (a) The

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### 2014 Chapter Competition Solutions

2014 Chapter Competition Solutions Are you wondering how we could have possibly thought that a Mathlete would be able to answer a particular Sprint Round problem without a calculator? Are you wondering

### 0810ge. Geometry Regents Exam 0810

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 identify

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### Definitions, Postulates and Theorems

Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 20, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

### San Jose Math Circle February 14, 2009 CIRCLES AND FUNKY AREAS - PART II. Warm-up Exercises

San Jose Math Circle February 14, 2009 CIRCLES AND FUNKY AREAS - PART II Warm-up Exercises 1. In the diagram below, ABC is equilateral with side length 6. Arcs are drawn centered at the vertices connecting

### Specimen paper MATHEMATICS HIGHER TIER. Time allowed: 2 hours. GCSE BITESIZE examinations. General Certificate of Secondary Education

GCSE BITESIZE examinations General Certificate of Secondary Education Specimen paper MATHEMATICS HIGHER TIER 2005 Paper 1 Non-calculator Time allowed: 2 hours You must not use a calculator. Answer all

### SAT Math Facts & Formulas Review Quiz

Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

### SAT Subject Math Level 2 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

### The 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 11 or higher. Problem E Mathletes in Action Four

### Homework from Section Find two positive numbers whose product is 100 and whose sum is a minimum.

Homework from Section 4.5 4.5.3. Find two positive numbers whose product is 100 and whose sum is a minimum. We want x and y so that xy = 100 and S = x + y is minimized. Since xy = 100, x = 0. Thus we have

### Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions

Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.

### San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS

San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4

of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook

Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook Objectives Use the triangle measurements to decide which side is longest and which angle is largest.

### Induction Problems. Tom Davis November 7, 2005

Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily

Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

### 11-4 Areas of Regular Polygons and Composite Figures

1. In the figure, square ABDC is inscribed in F. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. Center: point F, radius:, apothem:,

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

### Chapter 8. Quadratic Equations and Functions

Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property

### Cadet Name: Date: A) 2 B) 5 C) 7 D) 14 E) 28. 2. (ACTMTPT:Q4) Find the value of x. A) 40 B) 50 C) 60 D) 75 E) 90

Cadet Name: Date: 1. (ACTMTPT:Q1) The average (arithmetic mean) of x and y is 5, the average of x and z is 8, and the average of y and z is 11. What is the value of z? A) 2 B) 5 C) 7 D) 14 E) 28 2. (ACTMTPT:Q4)

### 2009 Chicago Area All-Star Math Team Tryouts Solutions

1. 2009 Chicago Area All-Star Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total

### Solutions Manual for How to Read and Do Proofs

Solutions Manual for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve

### MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

### Park Forest Math Team. Meet #5. Algebra. Self-study Packet

Park Forest Math Team Meet #5 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number

### Solution Guide for Chapter 6: The Geometry of Right Triangles

Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E-. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab

### Triangles. (SAS), or all three sides (SSS), the following area formulas are useful.

Triangles Some of the following information is well known, but other bits are less known but useful, either in and of themselves (as theorems or formulas you might want to remember) or for the useful techniques

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### GEOMETRY (Common Core)

GEOMETRY (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY (Common Core) Wednesday, August 12, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The

### Lesson 19: Equations for Tangent Lines to Circles

Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### 2015 Mathematics. New Higher Paper 1. Finalised Marking Instructions

National Qualifications 0 0 Mathematics New Higher Paper Finalised Marking Instructions Scottish Qualifications Authority 0 The information in this publication may be reproduced to support SQA qualifications

### Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

### Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18

Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,

### Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

### Mth 95 Module 2 Spring 2014

Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

### 10-4 Inscribed Angles. Find each measure. 1.

Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXMINTION GEOMETRY Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### SAMPLE QUESTION PAPER (Set - II) Summative Assessment II. Class-X (2015 16) Mathematics. Time: 3 hours M. M. : 90. Section A

SAMPLE QUESTION PAPER (Set - II) Summative Assessment II Class-X (2015 16) Mathematics Time: 3 hours M. M. : 90 General Instructions: 1. All questions are compulsory. 2. The question paper consists of

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, June 19, :15 a.m. to 12:15 p.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 19, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE

MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### MATEMATIKA ANGOL NYELVEN

Matematika angol nyelven középszint 03 ÉRETTSÉGI VIZSGA 03. május 7. MATEMATIKA ANGOL NYELVEN KÖZÉPSZINTŰ ÍRÁSBELI ÉRETTSÉGI VIZSGA JAVÍTÁSI-ÉRTÉKELÉSI ÚTMUTATÓ EMBERI ERŐFORRÁSOK MINISZTÉRIUMA Instructions

### Geometry in a Nutshell

Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Chapter 8 Geometry We will discuss following concepts in this chapter.

Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

### GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT!

GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! FINDING THE DISTANCE BETWEEN TWO POINTS DISTANCE FORMULA- (x₂-x₁)²+(y₂-y₁)² Find the distance between the points ( -3,2) and

### SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses

CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### 1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft

2 MODULE 6. GEOMETRY AND UNIT CONVERSION 6a Applications The most common units of length in the American system are inch, foot, yard, and mile. Converting from one unit of length to another is a requisite

### ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

### Conjectures for Geometry for Math 70 By I. L. Tse

Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

### Geometry SOL G.11 G.12 Circles Study Guide

Geometry SOL G.11 G.1 Circles Study Guide Name Date Block Circles Review and Study Guide Things to Know Use your notes, homework, checkpoint, and other materials as well as flashcards at quizlet.com (http://quizlet.com/4776937/chapter-10-circles-flashcardsflash-cards/).

Graduate Management Admission Test (GMAT) Quantitative Section In the math section, you will have 75 minutes to answer 37 questions: A of these question are experimental and would not be counted toward

### Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

### A. 3y = -2x + 1. y = x + 3. y = x - 3. D. 2y = 3x + 3

Name: Geometry Regents Prep Spring 2010 Assignment 1. Which is an equation of the line that passes through the point (1, 4) and has a slope of 3? A. y = 3x + 4 B. y = x + 4 C. y = 3x - 1 D. y = 3x + 1

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### MATH 21. College Algebra 1 Lecture Notes

MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

### NCERT. Area of the circular path formed by two concentric circles of radii. Area of the sector of a circle of radius r with central angle θ =

AREA RELATED TO CIRCLES (A) Main Concepts and Results CHAPTER 11 Perimeters and areas of simple closed figures. Circumference and area of a circle. Area of a circular path (i.e., ring). Sector of a circle

### Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

### Transformational Geometry in the Coordinate Plane Rotations and Reflections

Concepts Patterns Coordinate geometry Rotation and reflection transformations of the plane Materials Student activity sheet Transformational Geometry in the Coordinate Plane Rotations and Reflections Construction

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Math 531, Exam 1 Information.

Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. Exam 1 will be based on: Sections 1A - 1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### The measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures

8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 2013-2014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to