Transcript www.damien

WELCOME TO THE 4TH ANNUAL
DAMIEN HIGH SCHOOL
MATH COMPETITION
Schedule
8:00 – 8:30
8:30 – 8:45
8:45 – 9:15
9:15 – 9:30
9:30 – 10:00
10:00 – 10:15
10:15 – 11:45
11:45 – 12:00
Check-In
Greeting from the Principal
Math Medley Exam
Break
Individual Subject Competition
Break
Super Quiz Bowl and Solutions
Results and Awards
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.
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

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
+
+
x
9
x
+
= 20
÷
+
= 15
+
x
+
+ 4 – 6 =
=
=
=
15
72
14
3
Q1: 9 GAPS NUMBER SQUARES- SOLUTION
+
1
2
3
5
7
8
+
x
9
x
+
= 20
÷
+
= 15
+
x
+
+ 4 – 6 =
=
=
=
15
72
14
3
Q2: MID-POINT: PROBLEM

C is the midpoint of segment 𝐴𝐵, D is the
midpoint of segment 𝐴𝐶, E is the midpoint
of segment 𝐴𝐷, F is the midpoint of
segment 𝐸𝐷, G is the midpoint of segment
𝐸𝐹, and H is the midpoint of segment 𝐷𝐵. If
segment 𝐷𝐶 is 16 feet long, then segment
𝐺𝐻 is how many feet long?
Q2: MID-POINT: SOLUTION







C is the midpoint of segment 𝐴𝐵
D is the midpoint of segment 𝐴𝐶,
E is the midpoint of segment 𝐴𝐷
F is the midpoint of segment 𝐸𝐷,
G is the midpoint of segment 𝐸𝐹
H is the midpoint of segment 𝐷𝐵.
Segment 𝐷𝐶 is 16 feet long
●
A
● ●●
E16’
GF
8’
●
D
16’
●
C
2’ 2’
●
H 32’
48’
8’
4’
segment 𝐺𝐻 is ____
30 feet long
4’
24’
●
B
Q3: NUMBER MASTER MIND- INTRODUCTION



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.
Number
Guess
Places Correct
Correct
Correct Answer

62
31
75
0
1
1
0
0
0
47
17
1
2
1
2
We can play this game with any size number.
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
14
86
0
1
1
0
0
1
Guess
Number
Correct
Places
Correct
153
982
532
471
0
1
1
2
0
1
1
0
Guess
Number
Correct
Places
Correct
310
630
735
1
2
1
0
0
1
Q3: NUMBER MASTER MIND- SOLUTIONS

Each clue tells us something about our number
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
14
86
0
1
1
0
0
1
8
X 1
Y
Guess
Number
Correct
Places
Correct
153
982
532
471
0
1
1
2
0
1
1
0
A B
7
4 C
2
Guess
Number
Correct
Places
Correct
310
630
735
1
2
1
0
0
1
If 6 & 0  _06 or 06_
If 6 & 3 _63
0 N
7
L M
6
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)
b)
c)
d)
e)
f)
g)
h)
i)
Use two numbers to create 11
Use two numbers to create 3
Use three numbers to create 9
Use three numbers to create 7
Use three numbers to create –9
Use four numbers to create –9
Use four numbers to create 17
Use four numbers to create 8
Use five numbers to create 12
1 + 4
– 8 –
6 + 5
Q5: QUENTO: SOLUTIONS
Two numbers to create 11
6 + 5
Three numbers to create 7
Two numbers to create 3
8 –
5
Three numbers to create –9
Three numbers to create 9
1 + 4
– 8 –
6 + 5
Four numbers to create –9
1 + 4
– 8
6 + 5
– 8 –
6
5
Four numbers to create 17
Four numbers to create 8
1 + 4

1 +
– 8 –
6
Five numbers to create 12
1 + 4
– 8
– 8 –
– 8 –
6 +
6
5
6 + 5
Answers can vary and still be correct. This shows just one
solution for each.
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 = 106
100 + (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

Goal
210
Numbers
1, 2, 4, 10, 25, 100
117
230
387
3, 4, 9, 10, 25, 50
1, 2, 8, 10, 75, 100
3, 4, 5, 12, 50, 100
945
1, 2, 8, 9, 50, 100
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.
Q6: 6 NUMBERS: SOLUTIONS

This is just one solution. Others may be possible.
Goal Numbers
One Solution
210 1, 2, 4, 10, 25, 100 (100 x 2) + 10
117 3, 4, 9, 10, 25, 50
(10 + 3) x 9
230 1, 2, 8, 10, 75, 100 (75 x 2) + (10 x 8)
387 3, 4, 5, 12, 50, 100 (100+ 5 + (6 x 4)) x 3
945 1, 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:
1
8+
6x
2-
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.
1
1
8+

6x
3
2
1
3
2
2
2–
3
1
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.
Q7: MATH-DOKU: PROBLEM

Goal: Fill in the numbers missing from each grid.
Using numbers 1, Using numbers 1, 2, Using numbers 1, 2,
2, and 3, fill in the and 3, fill in the 3 x 3, and 4 fill in the 4 x
3 x 3 grid below.
3 grid below
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 options
Using numbers 1, Using numbers 1, 2, Using numbers 1, 2,
2, and 3, fill in the and 3, fill in the 3 x 3, and 4 fill in the 4 x
3 x 3 grid below.
3 grid below
4 grid below
3 2 1
3
2
1 3 2
1 3
1 3 2
3 2 1
3 1 2
4 2 1
3 4
2 4
4
3
2
1
Using numbers 1, 2,
3, 4, and 5 fill in the
5 x 5 grid below
4
2
1
3
5
5
1
4
2
3
2 3
5
5 2
4 1
1 4
1
4
3
5
2
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).
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
How many are right triangles? Example

How many are acute triangles? Example

How many are obtuse triangles? Example
Q8: CONGRUENT TRIANGLES: SOLUTIONS

Be organized. A triangle has 3 points. Start with the 1st point,
pick the 2nd, then pick all possible 3rd points.
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Increasing Areas
Triangles with
Area = ½
Triangles with Area = 1
Triangle with
Area =
Triangles with Area = 2
3 2
2
8 total triangles: 4 are right (Yellow), 2 Acute (light green), 2 Obtuse (dark green)
Q9: WHAT IS THIS?- PROBLEM

Another Classic Riddle:
Take half of this, and add one more
Then 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 more
Then triple that, and add on four.
But just the same result you’d see by
adding this to twenty three.
𝑥
3 +1 +4
2
𝑥𝑥
30.5𝑥
𝑥
+
1
3
3
∙
+
3 ∙+14
1.5𝑥
𝑥
+
1
3 21𝑥
222
= x + 23
=
x
+
19
16
16
=
x
+
=
x
+
23
=
32
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!
Q10: A CLOSE SECRET- SOLUTION
Her Age
4
X 2
Y
“Reverse my age“ 2Y 4X
“divide by three” (YX)/3
24/3
“add thirty-four” [(YX)/3]
8 + 34
+ 34
4
X 2
Y
“My age.”
Q11: NUMBER BROKEN INTO 4 PARTSPROBLEM
 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 PARTSSOLUTION

This is difficult to do algebraically




80 = a + b + c + d
𝑑
4
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
𝑑
8=
4

8 = 4c
c=2

Sum of a, b, c, d = 50. The Ratio of our goal to our
80
8
8
guess is = . Our guess is off by a factor of . If we
50
5
8
5
 d = 32
multiple our guess of 8 by we get 12.8.
5
Q11: NUMBER BROKEN INTO 4 PARTSSOLUTION

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


𝑑
4

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
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
4
LR
Tetris
2
1
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
42 min.
23 min.
3pts / problem (X 3 problems)
1 solution (2 solutions) 2 points each
+2 pts trap, +2 pts value, 3pts a, b, c, d
9 points
4 points
7 points
76 points
Presenter