The Jesuit Mathletes Packet

Jesuit Mathletes Packet (ver. 1)

Christopher Eur

October 23, 2011

This packet is created for training mathletes at Jesuit High School. Most of the problemsare from various competitions and books, including Challenging Problems in Algebra, ChallengingProblems in Geometry, A Shorter Way Mathematical Olympiad, Logos Series in Mathematics, etc.

Answer keys are available upon request; the rationale behind omitting the answers to this packetis to promote perseverence and the ability to think by oneself without relying on the answer keys.For solutions or errors, please email [email protected]. It is my sincere hope that this packetwill help anyone who is interested in mathematics and competitions but lacks the resources. Bestwishes to all future mathematicians!

Contents

1 Diagnostic Test 3

2 Basic Skills 52.1 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Factoring to make clean numbers . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Using variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Formulas to be memorized . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Few useful tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Quantitative Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Odds and evens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Box principle (or Pigeonhole principle) . . . . . . . . . . . . . . . . . . . . . 72.3.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Arithmetics 93.1 Means / Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.1 Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1

4 Intermediate Algebra 114.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1.1 Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.2 Introductory Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.1.3 Intermediate Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.1.4 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Algebraic Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.1 Introductory Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.2 Intermediate Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2.3 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3.1 Introductory Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3.2 Intermediate Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3.3 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4.1 Introductory Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4.2 Intermediate Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4.3 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Elementary Number Theory 205.1 Base Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1.1 Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2 Divisibility: Primes and Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.1 Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Trigonometry and Analysis 236.1 Exponents and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.1.1 Useful Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2.1 Useful Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2

1 Diagnostic Test

created by Keone Hon from AoPS Community

1. The measure of an angle is 3 times the measure of its complement. Find the measure of theangle in degrees.

2. Three calculus books weigh as much as two geometry books. Seven geometry books weigh asmuch as nine algebra books. Six trigonometry books weigh as much as eleven calculus books.How many algebra books weigh as much as 21 trigonometry books?

3. Notice that the product of the digits of 124 is 8. For many other three-digit positive integersis the product of the digits equal to 8?

4. Steven takes his favorite number and adds 7, multiplies the resulting number by 6, squaresthe resulting number, and divides by 9. The final result of these operations is 16. Given thatSteven’s favorite number is not 5, what is his favorite number?

5. Given that a, b, and c are all positive, and that ab = 21, ac = 18, and bc = 42, find a+ b+ c.

6. Ifx

4y=

y

4z, find the value of 3y4 − 3(xz)2 + 3.

7. Given the sequence 2, 4, 6, 10, 16, 26, . . . , find the 12th term.

8. Bill will be x years old in the year x2. If Bill was born between 1900 and 2000, in what yearwas he born?

9. Five fair coins are tossed. What is the probability that exactly two heads and three tails turnup? Express your answer as a decimal.

10. Define a “strange number” to be a number where each digit (other than the leftmost two)is equal to the sum of the two digits to the left. For instance, 11235 is a strange numberbecause 2 = 1 + 1, 3 = 1 + 2, and 5 = 2 + 3. How many four-digit strange numbers are there?

11. Blues, Inc. sells jeans at the following prices: $15 per pair if you buy 1 − 10 pairs, $13 perpair if you buy 11− 40 pairs, $10 if you buy 41− 70 pairs, and $8 per pair if you buy 71 ormore pairs. For how many values of n is it cheaper to buy a number of pairs greater than nthan it is to buy exactly n pairs?

12. For what value of k will the equation x3 − 9x2 + kx have exactly two solutions?

13. In a group of 200 students, 170 are taking history, 190 are taking math, 160 are taking english,and 135 are taking art. What is the minimum number of students that must be taking allfour of these classes?

14. Find the smallest value of x that satisfies√x2 − 3x+

√x2 − 1 = 2.

15. At a cookie shop, four different kinds of cookies are sold: chocolate chip, macaroon, peanutbutter, and white chocolate. In how many different ways can a person choose eight cookies?(Assume that the shop has an ample supply of each variety of cookie.)

3

16. The number√

8− 2√

15 can be expressed in the form√x−√y, where x and y are positive

integers. Find 2x+ y.

17. Bob is throwing turtles at a dartboard with two regions. If he hits the smaller region, he gets15 points; if he hits the larger region, he gets 7 points. What is the largest score that Bobcannot get?

18. A tennis tournament starts out with 140 players. Each game is played between two players;at the end of the game, the winner advances and the loser is knocked out. There are noties. If the tournament has a minimum number of byes, how many games must be played todetermine a single winner?

19. How many positive integers less than or equal to 1000 are multiples of 2 or 3?

20. Find x2 + y2 + z2 if

x+ y + z = 3

2x− y + z = 5

3x+ 2y − z = 16

21. Find the length of the longest median of a triangle with sides of length 4, 7, and 9.

22. If x+1

x= 8, find the value of x4 +

1

x4.

23. If the prime factorization of positive integer N is 24 ·39 ·11121, then how many positive integerfactors does N have?

24. If 216x2−4x+7 = 16x2+2x+1, what is x?

25. For certain integers n, n2 − 3n − 126 is a perfect square. What is the sum of all distinctpossible values of n?

26. The function f(x) is a cubic polynomial of the form ax3 + bx2 + cx + d. Given that f(0) =7, f(1) = 10, f(2) = 15, and f(3) = 28, find a+ 2b+ 3c+ 4d.

27. Suppose that f(x) = 13 + 23 + . . . + x3 and g(x) = 1 + 2 + . . . + x. Compute the value off(1)

g(1)+f(2)

g(2)+ . . .+

f(99)

g(99).

4

2 Basic Skills

2.1 Computation

Computation is fundamental to mathematics. It is many times overlooked nowadays with theadvent of calculators, but the ability to compute quickly and efficiently is one of the essential skillas a mathematician. Here we introduce few crucial tricks that have been directly or indirectly usedin Mathletes problems.

2.1.1 Factoring to make clean numbers

• 2× 5 = 10

• 7× 11× 13 = 1001

• you need to know the powers of 2 and 5

Example Problems

1. 2× 5× 7× 8× 11× 13× 125

2. 99999× 22222 + 33333× 33334

3. 75× 256× 125

4. 99999× 77778 + 33333× 66666

2.1.2 Using variables

1. 3694× 3692− 36932

2. 1990× 20002000− 2000× 19901990

3. 1991× 1999− 1990× 2000

4.9876543210

(9876543210)2−9876543210×9876543212

2.1.3 Fractions

• some useful knowledge:

1

(n)(n+ 1)=

1

n− 1

n+ 1m

(n)(n+ 1)=m

n− m

n+ 1

Example Problems

1.1

1× 2+

1

2× 3+

1

3× 4+ . . .+

1

1990× 1991

2.2

3+

2

15+

2

35+ . . .+

2

4004000

5

2.2 Factorization

Some say that being skillful in factorization for algebra is equivalent of knowing the multiplicationtable for arithmetics. It is important to be skillful with all types of factorizations. Of course, thebest way to become efficient at factorizing polynomial is through school curriculum; for a faithfulmath student, this section should mostly be a breeze.

2.2.1 Formulas to be memorized

• a2 ± 2ab+ b2 = (a± b)2

• a2 − b2 = (a+ b)(a− b)

• x2 + (p+ q)x+ pq = (x+ p)(x+ q), abx2 + (aq + bp)x+ pq = (ax+ p)(bx+ q)

• a3 + b3 = (a+ b)(a2 − ab+ b2), a3 − b3 = (a− b)(a2 + ab+ b2)

• a3 + 3a2b+ 2ab2 + b3 = (a+ b)3, a3 + 3a2b+ 2ab2 + b3 = (a+ b)3

• a2 + b2 + c2 + 2ab+ 2bc+ 2ca = (a+ b+ c)2

• a3 + b3 + c3 − 3abc = (a+ b+ c)(a2 + b2 + c2 − ab− bc− ca)

Example Problems

1. a(x− y) + b(y − x) =

2. ab+ b2 − ac− bc =

3. 4x2 + 20xy + 25y2 =

4. −x2 − 4y2 + 4xy =

5. x4 − y4 =

6. 1− x2 − y2 + 2xy =

7. x2 + 6x− 16 =

8. x2 − 14x+ 48 =

9. 6x2 + 5x− 6 =

10. 4x2 + x2y2 − 2y4 =

11. x3 + 8y3 =

12. x6 + y6 =

13. x3 − y3 + x2 + y2 − 2xy =

14. x3 + y3 − 1 + 3xy = (hint: (−1)3 = −1)

15. (a− b)3 + (b− c)3 − (c− a)3 =

16. a(b2 − c2) + b(c2 − a2) + c(a2 − b2)

6

2.2.2 Few useful tricks

substitution

1. x4 − 4x2 + 3 =

2. (x2 + 5x+ 4)(x2 + 5x+ 6)− 24 =

3. x(x+ 1)(x+ 2)(x+ 3)− 15 =

synthetic division

1. x4 + 2x3 − 9x2 − 2x+ 8 =

2. 3x5 − 3x4 − 13x3 − 11x2 − 10x− 6 =

3. challenge: If 4x3 − 5x2 − 7x + 3 = a(x − 2)3 + b(x − 2)2 + c(x − 1) + d, find the constantsa, b, c, d

add-and-subtract

1. x4 + 4 =

2. a4 + 4b4 =

3. x4 − 23x2y2 + y4 =

2.3 Quantitative Sense

Having a good sense of numbers is always helpful, in finding patterns, in guessing the right numbers,etc. Here we take a look at few introductory ideas.

2.3.1 Odds and evens

Many problems espescially regarding ones that ask about existence or possibility can be solvedvery easily just using this concept of odd and even. The odd and even character of number seemsentirely obvious and innocent, but becomes highly useful in many cases.

• odd× odd = odd, even× any number = even

• odd± odd = even, odd± even = odd, even± even = even

2.3.2 Box principle (or Pigeonhole principle)

• Principle I: If n + 1 objects are put into n many boxes, at least one box has two or moreobjects.

• Principle II: If pn + r objects are put into n many boxes, at least one box has p+1 or moreobjects.

• Examples:

– If 50 students randomly board 49 buses, at least one bus has 2 or more students.

– If John has 21 pens, no matter how he puts them in 4 boxes, one of them will have 6pens or more. (Note that 21 = 5× 4 + 1)

7

2.3.3 Problems

The best to develop these skills is to do many problems. These use ideas introduced above andvary greatly in difficulty.

1. John multiplied three consecutive numbers and got 693 and immediately knew it was wrong.How did he know?

2. The sum of two integers a, b is 11. Show that the sum of the squares of the two integerscannot be 60.

3. The sum of four integers a, b, c, d is 9. show that the sum of the cubes of the four integerscannot be 100.

4. There are n many integers. If their sum is 0, and their product is n, prove that n is divisibleby 4.

5. 121 people took a mathematics examination consisting of 30 problems. Each person’s scoreis determined as follows: 15 points are given as a free point for participation; for everycorrect answer, 5 additional points are given; for a wrong answer, -1 point; for an unansweredquestion, 1 point. What is the sum of scores of the 121 people?

6. Suppose there are 6 pairs of blue socks all alike, and 6 paris of black socks all alike, scrambledin a drawer. How many socks must be drawn out, all at once (in the dark), to be certain ofgetting a matching pair?

(a) Suppose the drawer now contains 3 black pairs of socks, 7 green pairs, and 4 blue pairs,scrambled. How many socks must be drawn out to be certain of getting a matchingpair?

(b) Suppose there are 6 different pairs of cuff links scrambled in a box. How many linksmust be drawn out, all at once (in the dark), to be certain of getting a matching pair?

7. Prove that, given 7 random integers, one can select two numbers such that either their sumor difference is a multiple of 10.

8. People shook hands with each other in a meeting; prove that the number of people who shookhands odd number of times is even.

8

3 Arithmetics

This section covers wide variety of problems found especially in the arithmetics. It is broken intothree parts; Means, Proportions, and Operations.

3.1 Means / Averages

3.1.1 Key Concepts

• Arithmetic mean (AM) of n numbers is their total sum divided by n. More precisely, A.M.of n numbers a1, a2, ...an−1, an is:

AM =

n∑i=1

ai

n=a1 + a2 + ...+ an−1 + an

n

• AM × n = sum of the numbers.

• One trick for finding AM quickly is to use an estimate and then calculate the error.

- For example, suppose we have numbers {12, 15, 17, 23, 25, 31}, and we are to find the AM. It isnot hard to intuit that AM should be around 20. Then we find the differences between 20 andthen numbers given: {12−20, 15−20, 17−20, 23−20, 25−20, 31−20} = {−8,−5,−3, 3, 5, 11}.Finding the AM of these errors is much easier: −8−5−3+3+5+11 = −8+11 = 3, 3÷6 = 0.5Adding the error to the original estimate 20 + 0.5 = 20.5 we have that AM=20.5. Note that(12 + 15 + 17 + 23 + 25 + 31)÷ 6 = 20.5 also.

• Geometric mean (GM) of n numbers is their total product to the 1/n th power. More precisely,G.M. of n numbers a1, a2, ...an−1, an is:

GM =

(n∏i=1

ai

)1/n

= n√a1a2...an−1an

• There is a trick for finding GM in a likewise way as we did for AM. It is not very usefulhowever. See if you can find the trick as a challenge.

• A useful fact: AM ≥ GM as long as ai > 0. In other words, AM = GM only whena1 = a2 = ...an; otherwise, AM > GM

3.1.2 Problems

1. The arithmetic mean (AM) of a set of 50 numbers 32. The AM of a second set of 70 numbersis 53. Find the AM of the numbers in the sets combined.

(a) Suppose now that there is a third set of 40 numbers with AM 36. What is the AM ofall three sets combined?

(b) In the original question, change ”AM” to ”GM” and solve. Leave the answer as a”calculator-ready” form. You do not need to (and cannot) calculate the result by hand.

9

2. Find the point-average of a student with A in mathematics, A in physics, B in chemistry,B in English, and C in history–using the scale: A, 5 points; B, 4 points; C, 3 points; D, 1point–when

(a) the credits for the courses are equal

(b) the credits for the courses are mathematics, 4; physics, 4; chemistry, 3; English, 3; andhistory, 3.

3. John’s four test scores were 87, 92, 97, and 76. What score must John get on his final test tohave average (AM) of 90 or higher if

(a) all five test scores are weighed?

(b) the final test is worth twice more than the first four tests?

4. Let x, y, and z be consecutive even integers. If the product of 3 and y is 32 more than thesum of x and z, what is the AM of x, y, and z?

5. Three part question. (Hint: be careful!)

(a) Given n numbers each equal to 1 + 1n, and two numbers each equal to 1, find their AM.

(b) Given n numbers each equal to 1 + 1n, and one number equal to 1, find their AM.

(c) Which of (a) or (b) is larger?

6. Express the difference of the squares of two consecutive even integers in terms of their arith-metic mean (AM).

7. The difference between AM and GM of two numbers is 4. If one of the numbers is 2, what isthe value of the other number?

3.2 Proportions

3.2.1 Key Concepts

• Ratio between two quantities is an intuitive and convenient tool in mathematics.

• Suppose we have two similar polygons P and Q. Suppose the ratio of a length of a side of Ato its corresponding side on B is p : q, then the following holds true:

– Perimeter length of P : that of Q = p : q

– Area of P to Q = p2 : q2

• Proportionality can be tricky at times and fine attention must be paid to the dimensionality.For example, in the case we’ve just seen, the ratio of the areas is no longer p : q but p2 : q2.

10

4 Intermediate Algebra

This section covers most topics in algebra broken into five parts: basic set theory, algebraicmanipulations, equations, inequalities, and functions. Throughout these sections, word problemsneed be emphasized when working on the problems, since math competitions will usually go forword problems that requires good problem solving skills.

Because basic concepts of algebra is covered well in high school curiculum, the best way tobecome better in competition math algebra is through solving many problems. Each section hasproblems divided into three levels of difficulty, and important theorems and concepts are imbeddedin the problems themselves.

4.1 Set Theory

4.1.1 Key Concepts

• Definition: A Set (S) is a group of clearly definable elements, notated S ={”elements”}For example,

– The following are NOT sets: {group of tall people}, {group of small numbers}. Thenotion of ”tall” and ”small” is not a precisely defined term.

– The following are sets: {group of people with height over 6‘}, {non-negative integers lessthan or equal to 3}.

• Notation: Sets are notated in two different ways. The first is simply listing all the elementsin brackets, and the second is providing the condition for the element.To show that an element e belongs to a set S, we write e ∈ S.A null set (set without any elements) is denoted ø.

e.g. 1. A={1,2,3,4,5}, B={2, 4, 6, 8, 10, 12, ... }e.g. 2. A={x: x is natural number less than 6}, B={2n, n ≥ 1, n ∈ N}e.g. 3. 2 ∈ A, 8 /∈ A, 26 ∈ B

• Definition: A Subset S ′ of a set S is a set in which everyone element in S ′ belongs to S aswell. In this way, the subset S ′ belongs to S and is notated S ′ ⊂ S. N.B. A null set is alwaysa subset of the original set. For example, {1, 2, 3} subset of {1, 2, 3, 4, 5}.

• Definition: A Union of two sets A and B consists of the elements that belong to A orB, and is denoted A ∪ B. A Intersection of two sets A and B consists of the elements thatbelong to both A and B, and is denoted A ∩ B. For example, given that A={1, 2, 3, 4, 5},and B={2, 4, 6, 8, 10, 12, ... },

– A ∪B = {1, 2, 3, 4, 5, 6, 8, 10, 12, ...}– A ∩B = {2, 4}

• Notation: Given a set S, n(S) or |S| is used to denote the number of elements in the setS. For example, given A = {1, 2, 3, 4, 5}, n(A) = 5.

• Note: Van Diagram is a highly useful tool in set theory.

11

4.1.2 Introductory Problems

1. Which of the following are real sets in the sense we defined above, and which are not?

(a) A set of tallest people from each class.

(b) A set of tall buildings in Boston

(c) A set of months with average temperature above 92 degrees.

(d) A set of natural numbers less than 1

(e) A set of great books.

2. Express the given set in a different way; that is, if given a set with a list, give a condition. Ifgiven a condition, make it into list.

(a) {x— x is factor of 6}(b) { . . . ,−3,−1, 1, 3, 5, . . . }

3. Given X = {1, 2}, find the number of pair of sets (Y, Z) such that X ⊃ Y ⊃ Z.

4. Given sets A,B,C and that n(A) = 13, n(B) = 22, n(C) = 19, n(A∪B) = 27, n(B∪C) = 29,n(C ∪ A) = 25, n(A ∩B ∩ C) = 5, find the value of n(A ∪B ∪ C).

4.1.3 Intermediate Problems

4.1.4 Advanced Problems

4.2 Algebraic Manipulations


1. Find values of a, b, c where (x3ya)4 = xby8 and(xc

x2

)3

= 1

2. Find all values that the expression (−1)a−1a+1a+1 +(−1a+1) can have where a is an integer.

3. Foil these expressions:

(a) (−a+ 2b)(a+ 2b)

(b) (2a− b)2

(c) (x− 1)(x− 2)(x− 3)

(d) (x+ y)3

4. Find the value of x2 + y2 if

(a) x+ y = 3, xy = 2

(b) x+ y = 6, xy = 4

(c) x3 + y3 = 45, x+ y = 3

12

5. Rationalize the denominators for the following: (a)1√

5 +√

3(b)

1

3 +√

7

6. x =5 +√

21

2, y =

5−√

21

2, find the value of x2 − y2


1. If a+ b = 91, ab = 7, find the value of1

a+

1

b

2. If a+ b = 3, ab = 2, find the value of a3b+ ab3

3. Two of the factors of 248 − 1 is between 60 and 70. Find these two factors.

4. a = 1, b = 10, c = 100, d = 1000, find the value of (a+ b+ c− d) + (a+ b− c+ d) + (a− b+c+ d) + (−a+ b+ c+ d)

5. If the ratio between 3x− 2 and y + 15 is constant, and x = 2 and y = 3, find the value of xif y becomes 12.

6. Show that for positive real numbers a and b,√

(a+ b)± 2√ab =

√a ±√b. Then, simplify

the following:

(a)√

5 + 2√

6

(b)√

20− 4√

21

(c)√

6 +√

35

7. Find the value of a, b, c if the equation (ax+ b)(x+ c)− bx = −12(4x+ 7)−2(x+ 1)(x−2) +x

is true for all x.

8. Find the value of a− b if a and b are constants that make the equationa

10x − 1+

b

10x + 2=

2× 10x + 3

(10x − 1)(10x + 2)for all x.

9. When x is added to both the numerator and the denominator of ab, the result is c

d. Express

the value of x in terms of a, b, c, d.


1. Simplify: 1− 1

1− 1x−1

2. Given a+ a−1 = 2, find the value of a2 + a−2

3. If x =

√5−√

3 +√

2

2, y =

√5 +√

3−√

2

2, find the value of (x+ y)2 − (x− y)2

4. Given 0 < 2x < y, simplify the expression√

(y − x)2 +√

(2y − x)2 +√

(x− 3y)2

13

5. The volume of a box is 64, its surface area 112, and its dimensions a, b, c, satisfy b2 = ac.Find the value of a+b+c.

6. The sum of the surface area of two cubes is 204, and the sum of all the sides is 96. Find thesum of the volumes of the two cubes.

7. Given a+ b+ c =1

3and

1

a+

1

b+

1

c= 1, find the value of (a− 1)(b− 1)(c− 1).

8. Givena+ b

3=b+ c

4=c+ a

5= k 6= 0, find the value of

ab+ bc+ ca

a2 + b2 + c2

9. Define function p(n) for n a natural number as follows: p(n)=(product of all the digits of n).For example, p(24) = 2× 4 = 8. Find n ≤ 50 such that p(2n) = 2p(n) 6= 0.

10. Given that real numbers x, y, z satisfy the two conditions below, find the value of x2 +y2 +z2.

(a) x+ y + z = 3

(b) x2(

1

y+

1

z

)+ y2

(1

z+

1

x

)+ z2

(1

x+

1

y

)= −3

4.3 Equations

This section is arguably the heart of algebra. Make sure that you master this section well.


1. Solve for x or (x, y) in the following:

(a) 3x− 2{(3x+ 1)− 5(x+ 1)} = 5

(b) 3− x− 1

2=

3x+ 5

12+

6− x4

(c)

{2(x+ 1)− (y − 1) = 83(x+ 1) + 2(y − 1) = 5

(d)

1− x

5+ y = 2

1− y4

+ x = 0

2. The system of equations

{x+ y = 6x− 2y = a

has a solution (2, b). Find the values of a and b each.

3. Explain how D = b2 − 4ac (the expression inside the square-root in the quadratic formula)can be used to tell how many roots the equation has. More specifically, given ax2+bx+c = 0,

(a) D < 0: no real solution for x

(b) D = 0: one real solution for x

(c) D > 0: two real solutions for x

14

4. Using D = b2− 4ac from the quadratic formula, determine if the following equations have noreal solution, one real solution, or two real solutions.

(a) x2 + 2x+ 2 = 0

(b) x2 − 3x+ 2 = 0

(c) x2 + 4x+ 4 = 0

(d) x2 = 3x− 3

(e) x2 − 6x = 9

5. When the equation x2 − 3x+ 2a+ 1 = 0 has one real solution, find the value of a.

6. The sum of two solutions to x2 − px+ p+ 1 = 0 is 2. Find p.

7. Suppose α, β are two roots of ax2 + bx+ c = 0. Show that α + β = − ba

and αβ = ca

8. The product of two consecutive natural numbers is 30. What are the two numbers?

9. 100g of 5% salt water has the same amount of salt as a 2% salt water of volume V . What isthe value of V ?

10. Driving over an enormous hill, John drives 20km/hr when going uphill and 30km/hr goingdownhill. If John spent 6 hours travelling over and back a hill, what is the total distance hetravelled?

11. Solve for x: |x| − 1 = |2x| in two ways: (a) using graph; (b) dividing up into cases (x ≥0 and x < 0)


1. Find the solutions for the equations:

(a) x2 − (p+ pq + q)x+ p2q + pq2 = 0

(b) 4x2 − 4ax+ a2 − b2 = 0

2. Of the two roots of x2 + 2x− 48 = 0, the greater one is also a root of x2 − 3x+ a = 0. Findthe other root of the latter equation.

3. Find all solutions for x2 − 4|x|+ 3 = 0.

4. On a particular month, the sum of the dates of the second Thursday and the fourth Mondaywas 31. Find the date of second Thursday.

5. A two digit number has the following property: it is 7 times the sum of its digits; it is 36 lessthan the number when the two digits are switched. Find the two digit number.

6. A frog can jump either 5 units forward or 3 units backwards per jump. The frog starts at theorigin and after 20 jumps arrives at +4. Find the number of forward jumps.

15

7. Given that α and β are the two solutions to 2x2 + 4x + 1 = 0, find the equation that hasα + 1 and β + 1 as the two solutions.

8. After a dinner, two brothers drank orange juice from a bottle. The elder drank 80ml andthen the younger drank 1

4of what was left. In the end, the amount of orange juice left was 2

3

the beginning. Find the amount of juice at first.

9. A train that is xm long spends 20 minutes to enter and completely exit a tunnel that is 600mwhile travelling at the speed of 1.6km. Find x.

10. A boat that travels 5km/hr on a currentless water travels upstream to a station and comesback in 8 hours. How fast is the current on the river?

11. Given that the equation (k+ 1)x2− 2(k+a)2x+ 2(k2 + b) = 0 has a solution x = 1 no matterwhat value k takes, find the value of a and b.


1. Given that α and β are the two roots of x2 + x + 1 = 0, we define Sn = αn + βn. Find thevalue of Sn + Sn+1 + Sn+2.

2. Given that x+ 1 = a+ b and x2 − 2ax+ 2x+ a2 − 2a− 3 = 3b, find the sum of values that bcan have.

3. When John looked at the clock the minute-hand and the second-hand coincided. Find theminimum amount of time past when this happens again.

4. Solve the system of equations:

{x2 + y2 = 13x2 − xy + y = 1

5. On a 13cm cube, John writes numbers 1 through 6, one number on each faces. Next, on a23cm cube, John writes numbers 7 through 12. Then, on a 33cm cube, he writes 13 through18. He keeps doing this until he is done writing on a n3cm cube. He finds that the sum of allthe numbers he wrote is 381. Find n.

6. A road starts at town A continues on to town B and ends at town C, spanning a total of125km. Normally, when Chris drives from A to B at 30km/hr and then from B to C at40km/hr, it takes 3 hours and 45 minutes. The lessen the travel time by 15 minutes in thiswhole course, Chris changes from 30km/hr to 40 km/hr at some point on the road betweenA and B. Find the distance Chris travels at 40km/hr as he travels from A to C now.

7. 24 cows can eat all the grass in a given grazing area in 6 days. 21 cows can eat all the grassin the same grazing area in 8 days. Determine the maximum number of cows allowed so thatthey can eat in the same grazing area forever. (Here we assume that the daily growth of thegrass is a constant, and that each cow consumes the same amount of grass per day.)

8. After a soccer game, the two teams line up and march inopposite directions saying ”goodgame” to each other. Team A has 20 players and team B has 18 players. Suppose that theteam A line is moving twice as fast as the team B line does and it takes any given B player

16

12 seconds to go through the whole team A line, how long does it take any given A player togo through the whole team B line?

9. Fifteen people sat around at circular table; each asked the two sitting next their favoritenumber and finds the average of the two, and announces it going around in the clockwiseorder starting from a designated person. The result was that starting from 0 at first, thenext was 1, the next 2, ..., the next and last 14. Find the favorite number of the person whoannounced 10.

10. On a straight road, an inspecting officer traveled from the rear to the front of an army column,and back, while the column marched forward its own length. If the officer and the columnmaintained steady speeds, what was the ratio of their speeds, faster to slower?

4.4 Inequalities


1. Given that a, b, c 6= 0 and ab < 0, bc < 0, a > 0, find the signs of b and c.

2. If the solution for the inequality ax− 1 < 0 is x > −1, what is the value of a?

3. How many integers satisfy the inequalities 2x− 1 ≤ 4 and x+ 2 ≥ 1?

4. Solve the following inequalities:

(a) 2x− 6 ≥ 5x− 3

(b)1− 3x

2− 5

6≥ 2x− 1

3+ 13

(c) 1 ≤ |x− 3| ≤ 6

5. Given −3 < a < 8, 2 < b < 5, find the range of a− b.


• Two primary ways of comparing two quantities are dividing and substracting.

• To compare A and B, we see whether A − B > 0 or A − B < 0, which implies A > B orA < B, respectively.

• To compare A and B when both are positive, we see whetherA

B> 1 or

A

B< 1, which implies

A > B or A < B, respectively.

1. Compare the quantities using either one of the methods introduced above. Choose yourmethod carefully.

(a) 7× 13× 17 + 3× 5× 91 and 7× 13× 16 + 3× 7× 91

(b) 7!3! and 6!4!

(c) a+ b and√a2 + b2

17

2. This problem will prove the well known theorem of AM ≥ GM ≥ HM for two elements,where AM,GM,HM are arithmetic, geometric, and harmonic means. Harmonic mean is sumof reciprocals over number of elements.

(a) For any real numbers a, b > 0, prove thata+ b

2≥√ab. (Hint: note that

√a2

= a)

(b) For any real numbers a, b > 0, prove that√ab ≥ 2

1a

+ 1b

- As noted before, AM ≥ GM ≥ HM is also true for cases with more than two elements,as long as the elements are positive real numbers. For example, with three elements, itwould be:

a+ b+ c

3≥ 3√abc ≥ 3

1a

+ 1b

+ 1c

3. Find the complete range of a such that ax+ 4 > 2x+ a is true for all x.

4. Find the condition for a and b under which 2x+ a ≥ ax− b has no solution.

5. A class 10 students, 6 male 4 female, is taking a group test, and in order to pass their averagemust be greater than or equal to 67. If the average of male students was 65, what is theminimum average score of the female students?

6. In planning the PACE Auction, Ms. Latko realizes that if 5 people sit per table, 9 peoplewon’t have a seat. If 6 people sit per table, 5 chairs will be empty. Find all possible numberof people coming to the auction.

7. Find the range of k for which x2 − 3x+ k + 2 = 0 does not have two distinct solutions.

8. Find the value of x for which x, x+ 2 and x+ 4 are lengths of three sides of a right triangle.


1. Adam, Bob, Charles, Danny, and Ethan arrived at school at different times one day. Adamarrived 4 minutes after Danny. Bob and Charles saw each other on highway, but Bob took awrong turn and arrived 7 minutes after Charles and 4 minutes after the bell. Danny arrived11 minutes before the bell. Ethan knew it takes 10 minutes to get to school, and he left hishouse to arrive at school 15 minutes before the bell but his watch was actually running 10minutes late. List the five guys in the order they arrived at school.

2. Pencils are being distributed in a class. If each student receives 3 pencils, then there will be37 pencils left over. If each student receives 5 pencils, the last one receiving will receive atleast one but not five. If the number of students is a and the number of pencils is b, find thevalue of a+ b. (The number of students is odd.)

3. Find the minimum value of n a natural number for which n2 + 4n > 106.

4. Given 0 < a < b < c < d, list from least to greatest:a

d,c

b,a+ c

b+ d,ac

bd

18

5. Prove that for all positive real numbers x, y, z, the following inequality x3+y3+z3−3xyz ≥ 0is always true in two ways:

(a) By using AM ≥ GM .

(b) By using the formula given in 2.2.1.

6. In the following problems, we introduce how to use AM ≥ GM property to solve problemsthat would normally require calculus.

(a) Prove that for positive real numbers a and b, a+b2

>√ab if a 6= b and a+b

2=√ab if

a = b. This implies that GM is always smaller than AM unless all the elements areequal; Moreover, GM is thus at maximum when it equals AM . This can be furtherextended to more than two elements.

(b) Using only the identity sin2 θ + cos2 θ = 1, prove that the maximum value of sin θ cos θis 0.5 given that 0 ≤ θ ≤ π

2

(c) A construction company is granted 40 feet of fences to enclose a rectangular shapedregion. If the length and the width of the region is x and y, find the values for x and ysuch that the area of the region enclosed is maximized.

(d) Of all the boxes that can be enclosed in a sphere of radius 3, find the surface area of theone that has the maximum volume.

7. For all positive real numbers a, b, c such that ab ≥ 1, bc ≥ 1, ca ≥ 1, prove that the followinginequality is always true. (Hint: The tools you need are #1 and #2 of the 4.4.2 IntermediateProblems, and as always, have fun.)

(a+ b+ c)(

1

1 + a2+

1

1 + b2+

1

1 + c2

)≥(

a2

1 + a2+

b2

1 + b2+

c2

1 + c2

)(1

a+

1

b+

1

c

)

8. Given that a, b, c are three sides of a triangle, prove that

8abc ≥ (a+ b− c)(a− b+ c)(−a+ b+ c)

4.5 Functions

19

5 Elementary Number Theory

This section provides an extensive coverage of concepts in number theory. It is broken into threeparts: Base Systems, Divisibility, GCD and LCM, Modulo Arithmetics.

5.1 Base Systems

5.1.1 Key Concepts

• a number N in base m is denoted

N = anan−1...a2a1a0(m) = an×mn + an−1×mn−1 + . . .+ a2×m2 + a1×m+ a0 (ai < m)

- For example, our decimal system (base 10) works in this way:

453679 = 4× 106 + 5× 105 + 3× 104 + 6× 103 + 7× 102 + 9× 100

• The number of zeros at the end of a number n is equal to the number of times n can bedivided by the base number.

- For example, the number 384000 (in decimal) can be divided by 10 evenly three times.(384000 ÷ 10 = 38400. 38400 ÷ 10 − 3840. 3840 ÷ 10 = 384). It has three zeros at theend. Likewise, if 432(= 63 × 2) is expressed in base 6, it has zeros; that is, 432(10) = 2000(6).

5.1.2 Problems

1. When 7 was written in base n, the result was 21(n). What is the value of n?

2. Find the base b such that:

(a) (a) 72(b) = 2(37(b))

(b) (b) 72(b) = 3(27(b))

3. How many zeros are at the end of 5! in base 6 number system?

4. When 10! is written in base 12, it has k zeros at the end. What is the value of k?

5. Let N1 = 0.888 . . ., written in base 9, and let N2 = 0.888 . . ., written in base 10. Find thevalue of N1 −N2 in base 9.

5.2 Divisibility: Primes and Composites

5.2.1 Key Concepts

Primes and Composites

• A prime number is defined as an integer that has exactly two factors, 1 and itself. Compositenumber is defined as an integer that can be expressed as a product of 2 or more primes.

• Fact: There are infinitely many prime numbers. Challenge: Prove it.

20

• Fact: For any integer, there exists a unique prime factorization. That is, any integer n can beexpress as the following where P1, P2, . . . , Pk are distinct primes and α1, α2, . . . , αk are wholenumbers:

n = Pα11 × Pα2

2 × . . .× Pαkk

1. Theorem 1: Number of positive divisors (factors) of n, denoted d(n), is:

d(n) = (α1 + 1)(α2 + 1) . . . (αk + 1)

2. Theorem 2: The sum of all positive divisors of n, denoted σ(n), is:

σ(n) = (1 + P1 + P 21 + . . .+ Pα1

1 )(1 + P2 + P 22 + . . .+ Pα2

2 ) . . . (1 + Pk + P 2k + . . .+ Pαk

k )

Divisibility

• For integers a and b, if there exists an integer k such that b = ka, we say that b is divisibleby a or a divides b, denoted a|b. Mathematically put,

For a, b ∈ Z,∃k ∈ Z such that b = ka ⇐⇒ a | b

• Properties of divisibility (try to see why these are true and try out examples):

1. If a|b, then a|bc where c is a non-zero integer.

2. If a|b and a|c, then a|k1b+ k2c where k1, k2 are integers.

3. Caution: the converses are not true:

(a) NOT: If a|bc, then a|b or a|c. (e.g. 6|12(= 3 × 4) but 6|3 and 6|4 is certainly nottrue)

(b) NOT: If a|k1b+ k2c, then a|b and a|c. (Can you find an example?)

4. If a|b and a | b+ c, then a|c.5. If a|bc and gcd(a, b) = 1, then a|c.6. If a± b = c± d and e|a, e|b, and e|c, then e|d.

7. It is important that you workout examples of these properties on your own

• Divisibility Tests:

– 2: any even number is a multiple of 2.

– 3: if the sum of the digits is a multiple of 3, it is divisible by 3.

– 4: if the last two digits are divisible by 4.

– 5: all numbers with unit digit (the ”one’s” digit) 0 or 5.

– 6: if divisible by 2 and 3.

– 8: if the last three digits are divisible by 8.

– 9: if the sum of the digits is a multiple of 9.

– 11: if the difference between sums of odd digits and even digits is a multiple of 11.

21

5.2.2 Problems

These problems will be difficult and may take quite some time since you are not familiar withnumber theory. It is important that you try to think everything out yourself, and as always,especially for proofs, trying out examples with numbers can get you a sense of how things work.

1. Prove that if ab+ cd is divisible by a− c, then ad+ bc is also divisible by a− c.

2. Of numbers that only consist of odd numbers, find the smallest six digit one that is divisibleby 125.

3. If the seven digit number 62xy427 is a multiple of 99, find x and y.

4. Using the fact that 782 + 8161 is a multiple of 57, prove that 783 + 8163 is also a multiple of 57.

5. If the sum of squares of a prime number and an odd number is 125, find the two numbers.

6. If the product three prime numbers p1, p2, and p3 is five times their sum, find the three primenumbers.

22

6 Trigonometry and Analysis

6.1 Exponents and Logarithms

6.1.1 Useful Formulae

(a) aα × aβ = aα+β, aα

aβ= aα−β, (aα)β = aαβ

(b) ax = y ⇐⇒ loga y = x

(c) logam bn = n

mloga b, aloga x = x, loge x = lnx

(d) logn a+ logn b = logn ab, logn a− logn b = lognab

(e)loga b

loga c= logc b,

1

loga b= logb a

6.1.2 Problems

1. Prove all the logarithm properties (c), (d), and (e) using (a) and (b)

2.

6.2 Trigonometry

6.2.1 Useful Formulae

For the formulae in the following, let4ABC be a triangle where a, b, c denote the sidesBC,CA,AB,respectively.Must-knows

• sin θ

cos θ= tan θ, sin(−θ) = − sin θ, cos(−θ) = cos θ

• sin2 θ + cos2 θ = 1, 1 + cot2 θ = csc2 θ, tan2 θ + 1 = sec2 θ

• Law of Sines 1: 4ABC is Given two sides a, b and the angle θ between them of a triangle,the area of the triangle is:

4ABC =1

2ab sin θ

.

• Law of Sines 2: Let, R be the radius of the circumcircle of 4ABC, then:

a

sinA=

b

sinB=

c

sinC= 2R

• Law of Cosines: c2 = a2 + b2 − 2ab cosC

Advanced

• sin(α± β) = sinα cos β ± cosα sin β, sin(2α) = 2 sinα cosα

23

• cos(α± β) = cosα cos β ∓ sinα sin β, cos(2α) = cos2 α− sin2 α= 2 cos2 α− 1= 1− 2 cos2 α

• tan(α± β) =tanα± tan β

1∓ tanα tan β

6.2.2 Problems

24

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