WELCOME TO THE 4TH ANNUAL DAMIEN HIGH SCHOOL

MATH COMPETITION

If your school has not checked in, please send one student and one adult to the registration table at the front of the Activity Center.

Schedule8:00 – 8:30 Check-In8:30 – 8:45 Greeting from the Principal8:45 – 9:15 Math Medley Exam 9:15 – 9:30 Break9:30 – 10:00 Individual Subject Competition10:00 – 10:15 Break10:15 – 11:45 Super Quiz Bowl and Solutions 11:45 – 12:00 Results and Awards

SUPER QUIZ BOWL• This is a group competition in which many

problems must be solved through collaboration.

• Scratch paper is available at your tables, but the final answer for each problem must be written legibly in the box on the provided answer forms.

• The final solution must be true for all the given clues.

• Each question is worth different amounts of points based off of the complexity, but partial points may be awarded on some problems.

Q1: 9 GAPS NUMBER SQUARES- PROBLEM

+ 9 + = 20

+ x ÷x + = 1

5+ x +

+ 4 – 6 = 3= = =15

72

14

Numbers 1- 9 are filled in to make the rows and columns equal the value indicated using the given operations. Numbers 4, 6, and 9 have been filled in. Fill in the remaining values:

1, 2, 3, 5, 7, 8

Q1: 9 GAPS NUMBER SQUARES- SOLUTION

+ 9 + = 20+ x ÷

x + = 15+ x +

+ 4 – 6 = 3= = =15 72 14

5

2

8

1

7

3

Q2: MID-POINT: PROBLEM

C is the midpoint of segment , D is the midpoint of segment , E is the midpoint of segment , F is the midpoint of segment , G is the midpoint of segment , and H is the midpoint of segment . If segment is 16 feet long, then segment is how many feet long?

24’4’

2’

Q2: MID-POINT: SOLUTION C is the midpoint of segment D is the midpoint of segment , E is the midpoint of segment F is the midpoint of segment , G is the midpoint of segment H is the midpoint of segment . Segment is 16 feet long segment is ____ feet long

●A

●B

●C

●D

●E

●F

●G

●H16’16’ 32’

48’

24’

8’

4’4’

2’

8’

2’

30

Q3: NUMBER MASTER MIND- INTRODUCTION

Guess Number Correct

Places Correct

62 0 0

31 1 0

75 1 0

47 1 1

17 2 2

I’m thinking of a 2-digit number. The number does not begin with a 0 and none of

the digits in the number are the same. (ex: 77 is not allowed).

If you guess what my number is, I’ll tell you if any of the numbers in it are correct, and then I’ll tell you if any of them are in the correct place.

We can play this game with any size number.

Correct Answer

Q3: NUMBER MASTER MIND- PROBLEM Guess the numbers based on the

information given in each table. Remember, no number begins with 0 and no number has digits that repeat .

Problem A Problem B Problem C

Guess a two digit number

Guess a three digit number

Guess a three digit number

Guess

Number

Correct

Places

Correct

24 0 014 1 086 1 1

Guess

Number

Correct

Places Correc

t

153 0 0982 1 1532 1 1471 2 0

Guess

Number

Correct

Places

Correct

310 1 0

630 2 0

735 1 1

7 6047 28 1

Q3: NUMBER MASTER MIND- SOLUTIONS Each clue tells us something about our

number

YXBA C ML N

If 6 & 0 _06 or 06_ If 6 & 3 _63

Guess

Number

Correct

Places

Correct

310 1 0

630 2 0

735 1 1

Problem A Problem B Problem C

Guess a two digit number

Guess a three digit number

Guess a three digit number

Guess

Number

Correct

Places

Correct

24 0 014 1 086 1 1

Guess

Number

Correct

Places Correc

t

153 0 0982 1 1532 1 1471 2 0

Q4: TETRIS PIECES: PROBLEM If you re-assemble the pieces of the

four compositions below by moving pieces, rotating pieces, or flipping pieces, three of them will be the same shape, and one will not. Which is the odd one out? What shape is formed with the other three?

Q4: TETRIS PIECES: SOLUTION

A, C, and D can be re-assemble each to form a square. B can not be re-assembled to become a square.

Q5: QUENTO: INTRODUCTION Quento is a game played on a 3x3 grid with 5

numbers and four operations. Players draw line segments that connect a designated number of values and any arithmetic operations between them to create target values

For instance, if your goal was to create the number 7 with two numbers, you could use either the 4 + 3 combination or the 8 – 1 combination

4 + 8+ 3 – 2 + 1

4 + 8+ 3 – 2 + 1

Q5: QUENTO: INTRODUCTION CONTINUED If your goal was to create the number 6 with a

three number combination, you could use 4 + 3 – 1 or 8 – 3 + 1 or even a clever use of negative signs, like -1 + 3 + 4.

Keep in mind that order matters. 8 – 3 is not equal to 3 – 8. Since order matters, you need to indicate the direction that you travel.

4 + 8+ 3 – 2 + 1

4 + 8+ 3 – 2 + 1

4 + 8+ 3 – 2 + 1

Q5: QUENTO: PROBLEMS Today’s Grid + Goals

a) Use two numbers to create 11b) Use two numbers to create 3c) Use three numbers to create 9d) Use three numbers to create 7e) Use three numbers to create –9 f) Use four numbers to create –9 g) Use four numbers to create 17h) Use four numbers to create 8i) Use five numbers to create 12

1 + 4– 8 – 6 + 5

Q5: QUENTO: SOLUTIONS

Answers can vary and still be correct. This shows just one solution for each.

1 + 4

– 8

6 +

Two numbers to create 11

Two numbers to create 3 Three numbers to create 9

Three numbers to create 7

Three numbers to create –9

Four numbers to create –9

Four numbers to create 17

Four numbers to create 8 Five numbers to create 121 +

– 8 –

6 5

1 + 4

– 8 –

6 + 5

– 8

6 + 5

6 + 5

– 8 –

6 5

8 –

5

1 + 4

– 8 –

6

1 + 4

– 8 –

6 + 5

Q6: 6 NUMBERS: INTRODUCTION The game 6 NUMBERS allows

players to use addition signs, subtraction signs, multiplication signs, division signs, (parentheses by grouping) and 6 numbers to write an expression equivalent to a target value. You do not need to use all the numbers, nor all the signs. Though you may repeat signs, but you may not use a number twice.

Q6: 6 NUMBERS: INTRODUCTION CONTINUED

Example: Create 106 from 2, 4, 3, 6, 10, and 100

Here are just some of the ways you can achieve your goal. You only need one.

100 + 6 = 106100 + (2 x 3) = 106(100 + 10) – (6 / 3) – (4 / 2) = 106

No number is used twice but signs can be repeated.

Q6: 6 NUMBERS: PROBLEMS

Using addition signs, subtraction signs, multiplication signs, division signs, parentheses, and only the 6 numbers given in each set, write an expression equivalent to each target value. Remember, you do not need to use all the numbers or all the signs. You can repeat signs, but you may not use a number twice.

Goal Numbers

210 1, 2, 4, 10, 25, 100

117 3, 4, 9, 10, 25, 50

230 1, 2, 8, 10, 75, 100

387 3, 4, 5, 12, 50, 100

945 1, 2, 8, 9, 50, 100

Q6: 6 NUMBERS: SOLUTIONS This is just one solution. Others may be

possible.Goal

Numbers One Solution

2101, 2, 4, 10, 25, 100

(100 x 2) + 10

1173, 4, 9, 10, 25, 50

(10 + 3) x 9

2301, 2, 8, 10, 75, 100

(75 x 2) + (10 x 8)

3873, 4, 5, 12, 50, 100

(100+ 5 + (6 x 4)) x 3

9451, 2, 8, 9, 50, 100

(50+9) x (8 x 2) + 1

Q7: MATH- DOKU: INTRODUCTION In variations of the popular puzzle

game, Sudoku, each row and column in an n x n square has the numbers 1 to n listed only once in each row and column. For example, on a 3 x 3 grid, the numbers 1, 2, & 3 are listed once in each row and once in each column. For instance.

1 3 2

2 1 3

3 2 1

Q7: MATH- DOKU: INTRODUCTION CONTINUED In Math Doku+ (also known as Ken-Ken), the

same principal is true, except a layer of math is added onto it. Each colored grouping of numbers has an operation and a value in the upper left hand corner of that group. The goal is to use that operation with the numbers in that group to get that value. For instance:

8+ 2-

6x1

Q7: MATH- DOKU: INTRODUCTION CONTINUED The Sum (addition) of the numbers in the red grouping is

8 The difference (subtraction) of the two numbers in the

yellow group is 2 (here there’s no need for a negative) The product (multiplication) of the numbers in the green

group is 6 For this blue group, the number 1 goes inside.

Remember, each row and column has the numbers 1, 2 and 3 with no repeats. This can be played on square grids as large as you want.

1 3 2

2 1 3

3 2 1

8+ 2–

6x1

Q7: MATH-DOKU: PROBLEM Goal: Fill in the numbers missing from

each grid.Using numbers 1, 2, and 3, fill in the 3 x 3 grid below.

Using numbers 1, 2, and 3, fill in the 3 x 3 grid below

Using numbers 1, 2, 3, and 4 fill in the 4 x 4 grid below

Using numbers 1, 2, 3, 4, and 5 fill in the 5 x 5 grid below

Q7: MATH-DOKU: SOLUTIONS Use logic to find one piece, and fill in

the rest as that piece eliminates other optionsUsing numbers

1, 2, and 3, fill in the 3 x 3 grid below.

Using numbers 1, 2, and 3, fill in the 3 x 3 grid below

Using numbers 1, 2, 3, and 4 fill in the 4 x 4 grid below

Using numbers 1, 2, 3, 4, and 5 fill in the 5 x 5 grid below

21

3 2

2

1

33

33

31

1

122

2

3

43

3

1

14 2

42124 1

1

22 35

5

5

3

4 143

31

55

24

214

42

Q8: CONGRUENT TRIANGLES: PROBLEM How many non-congruent (different shaped)

triangles can you create by connecting any three dots from the 3 by 3 grid of dots in the figure below. Draw each triangle in the grid provided. Please draw them smallest to largest (it helps with grading but doesn’t affect your score).

How many are right triangles? Example

How many are acute triangles? Example

How many are obtuse triangles? Example

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Q8: CONGRUENT TRIANGLES: SOLUTIONS

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Increasing AreasTriangles with Area = ½

Triangles with Area = 1 Triangle with Area =

Triangles with Area = 2

8 total triangles: 4 are right (Yellow), 2 Acute (light green), 2 Obtuse (dark green)

Be organized. A triangle has 3 points. Start with the 1st point, pick the 2nd, then pick all possible 3rd points.

Q9: WHAT IS THIS?- PROBLEM

Another Classic Riddle:Take half of this, and add one moreThen treble that, and add on four.But just the same result you’d see

by adding this to twenty three.So what is this? You’ll have to say! Your fun with figures for today.

Find what number represents This

Hint: “Treble” means multiple by 3

Q9: WHAT IS THIS?- SOLUTION

=

Take half of this, and add one moreThen triple that, and add on four.But just the same result you’d see by adding this to twenty three.𝑥2

+13 =

0 .5 𝑥 =1 .5𝑥 =32𝑥 =32𝑥+3 =3 ∙𝑥2+3 ∙1 =

𝑥2

+13 =𝑥2

+13 =

Q10: A CLOSE SECRET- PROBLEM

A classic riddle“My age?” She smiled. “You’ll have to guess. Just let me think. Ah that’s it yes. Reverse my age: divide by three: add thirty-four. My age you’ll see.” That’s what she said. So can you say how old she must have been that day?

Determine the woman’s age in the poem.

Hint: She’s Double Digits Years old!

24

2 4

24/3

8 + 34

24

Q10: A CLOSE SECRET- SOLUTION

Her Age“Reverse my age““divide by three” “add thirty-four” “My age.”

YX

Y X(YX)/

3[(YX)/3] + 34 YX

Q11: NUMBER BROKEN INTO 4 PARTS- PROBLEM

The number 80 is the sum of four positive numbers a, b, c, d such that a increased by 4, b decreased by 4, c multiplied by 4, and d divided by 4 all equal the same number.

Determine what this same number is (hint: it is a decimal) 2 Points for finding 2 consecutive positive

integers that the number is trapped between.

2 Points for finding the number 3 Points for finding the values of a, b, c,

and d.

Q11: NUMBER BROKEN INTO 4 PARTS- SOLUTION This is difficult to do algebraically

80 = a + b + c + d a + 4 = b – 4 = 4c = = “number”

What’s the “number”? Guess at it. Try 8. If we try 8, we would know what a, b, c, and d would be

8 = a + 4 a = 4 8 = b – 4 b = 12 8 = 4c c = 2 8 = d = 32 Sum of a, b, c, d = 50. The Ratio of our goal

to our guess is . Our guess is off by a factor of . If we multiple our guess of 8 by we get 12.8.

Q11: NUMBER BROKEN INTO 4 PARTS- SOLUTION Let’s check this answer of 12.8. 12.8 = a + 4 a = 8.8 12.8 = b – 4 b = 16.8 12.8 = 4c c = 3.2 12.8 = d = 51.2 Sum a + b + c + d = 80

Thus the number is between 12 and 13.

CREDITS• Students Ambassadors• AP Calculus Students• Members of Math Department• Parents, Teachers, and Principals• Participants

SOURCES• Many of these problems, as past

problems from math Competitions, were inspired by apps from smart phones. If you’re going play on your phone, play smart!

NineGaps 6 Numbers Math Doku+ Quento

SCORING, ORDER, POINTS

# Type

Problem Time Work

(in min)

Time Explain

(in min)

Scoring Total Points total

Presenter

1 AL 9 Gaps Square 3 1 1 solution (1pt - 1st, 1pt - 2nd, 2pts- rest)

4 points

2 G Midpoint 3 2 1 solution (only one solution) 5 points

3 Ar Master Mind 4 3 3pts / problem (X 3 problems) 9 points

4 LR Tetris 2 1 1 solution (2 solutions) 2 points each

4 points

5 Ar Quento 4 2 1pt / problem X (9 problems) 9 points

6 AL 6 Numbers 4 2 2 pts / problem (X 5 problems) 10 points

7 Ar Math-Doku 5 2 2 pts / problem (x 4 problems) 8 points

8 G Triangles 4 3 1pt / triangle (X 7 triangles)+1 #right Δ, +1 #Acute Δ, +1 Obtuse Δ

10 points

9 AL What is This 4 2 1 solution (5 points) 5 points

10

AL Close Secret 4 2 1 solution (5 points) 5 points

11

LR Number in 4 Parts

5 3 +2 pts trap, +2 pts value, 3pts a, b, c, d

7 points

42 min. 23 min. 76 points