Park Forest Math Team
Meet #5
Self-study Packet
Problem Categories for this Meet (in addition to topics of earlier meets):
1.
2.
3.
4.
5.
Mystery: Problem solving
Geometry: Solid Geometry (Volume and Surface Area)
Number Theory: Set Theory and Venn Diagrams
Arithmetic: Combinatorics and Probability
Algebra: Solving Quadratics with Rational Solutions, including word problems
Important Information you need to know about NUMBER THEORY:
Set Theory, Venn Diagrams
The symbol
stands for intersection. If you see the notation
the elements that are in Set A AND Set B.
The symbol
stands for union. If you see the notation
elements that are in Set A OR Set B.
, it means all
, it means all the
For example:
Solve sets using the normal order of operations. Do what’s in parentheses first, and
then work from left to right.
For example,
Prime Numbers
( Numbers with 3 as a digit
Numbers Less than 50)
First, you would find all the numbers that have 3 as a digit AND are less than 50.
{3, 13, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 43}
Then, you would find all the numbers that are prime AND are in the set you just
found. This would be your final answer.
{3, 13, 23, 31, 37, 43}
VENN DIAGRAMS
A
1, 5
B
7, 11
9
2, 4
3
12
C
6
A Venn diagram includes two or more
intersecting circles. The overlapping
areas are the intersections of the sets.
To the left is where you place the
following numbers if the elements of
each set are as follows:
A: {1, 3, 5, 7, 9, 11}
B: {1, 2, 3, 4, 5, 6}
C: {3, 6, 9, 12}
Meet #5 March 2011 Category 3 ± Number Theory
1. Set ‫ ܣ‬ contains ͳͲ elements. The union ‫ ܤ ڂ ܣ‬ contains ʹͲ elements, while the intersection ‫ ܤ ځ ܣ‬ contains Ͷelements. How many elements are the in the set ‫ ?ܤ‬ 2. In a general Venn diagram of ʹ sets, we can have ͵ different areas, as shown in this diagram: 1 2 3 Where the common area in the middle represents the intersection of the two sets. How many different areas are possible in a Venn diagram ofͶ sets? 3. How many subsets of the set ሼ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬ǡ ‫ܦ‬ǡ ‫ܧ‬ǡ ‫ܨ‬ሽ contain at least one vowel? Answers
1. _______________
2. _______________
3. _______________
www.imlem.org Meet #5 March 2011 Solutions to Category 3 ± Number Theory Answers
1. 14
number of elements in each one, minus the number of elements in 2. 15
their intersection. Denoting the number of elements in a set ‫ ܣ‬ as ȁ‫ܣ‬ȁ, 3. 48
1. The number of elements in a union of two sets is the sum of the we can write this as: ቚ‫ ܣ‬ራ ‫ܤ‬ቚ ൌ ȁ‫ܣ‬ȁ ൅ ȁ‫ܤ‬ȁ െ ቚ‫ ܣ‬ሩ ‫ܤ‬ቚ Using the numbers in our case we get ȁ‫ܤ‬ȁ ൌ ʹͲ ൅ Ͷ െ ͳͲ ൌ ͳͶ. We can see the number of elements in this Venn diagram: A 6 4 B 10 2. There are Ͷ areas containing only one set each. There are ͸ areas containing the intersections of exactly two sets each ሺ‫ܤܣ‬ǡ ‫ܥܣ‬ǡ ‫ܦܣ‬ǡ ‫ܥܤ‬ǡ ‫ܦܤ‬ǡ ‫ܦܥ‬ሻ, Ͷ areas of intersection of exactly ͵ sets ሺ‫ܥܤܣ‬ǡ ‫ܦܤܣ‬ǡ ‫ܦܥܣ‬ǡ ‫ܦܥܤ‬ሻ, and one area of the intersection of ‫ܦܥܤܣ‬. Overall ͳͷ distinct areas. This is basically the number of subsets of ሼ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬ǡ ‫ܦ‬ሽ without the empty set. 3. The set ሼ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬ǡ ‫ܦ‬ǡ ‫ܧ‬ǡ ‫ܨ‬ሽ has ͸ elements and therefore ʹ଺ ൌ ͸Ͷ subsets. Removing the ʹ vowels ሼ‫ܣ‬ǡ ‫ܧ‬ሽ ZH¶UHOHIWZLWKͶ elements that make ʹସ ൌ ͳ͸ subsets with no vowel in them. So the remaining Ͷͺ subsets must contain at least one vowel. Note we were careful to count the empty subset with the no-­vowel group. www.imlem.org You may use a calculator today.
Category 3
Number Theory
Meet #5, March 2009
1.
At the summer math camp there are 50 kids taking math classes. Thirty-five
kids took the Number Theory class and 24 kids took the Probability class. If 7
kids took neither of the two classes, how many kids took both classes?
2.
The set A contains only the vowels a, e, i, o, u and y. How many subsets of set
A contain the letter y?
3.
In the Venn diagram below, the four ovals represent 4 sets of numbers as
described below. How many numbers fall in the shaded regions?
W = the set of positive even integers less than 50
X = the set of prime numbers less than 50
Y = the positive multiples of 3 less than 50
Z = the perfect squares less than 50
Answers
1. _______________
2. _______________
3. _______________
Solutions to Category 3
Number Theory
Meet #5, March 2009
Answers
1. 16
2. 32
1. 50 – 7 = 43 kids took at least one of the two classes. A nice
way to think about this is that there are 35 Number Theory books
being used and 24 Probability books being used for a total of 59
books. If each of the 43 kids had one book there would be 16 left
over books. So 16 of the kids must take a second book and are
taking both classes.
3. 5
2. Subsets of set A can contain anywhere from none of the
elements of A up to containing all of the elements of A. Since we
are looking for the subsets that contain “y”, we just need to
determine whether or not the other letters are in a subset. There
are 5 other letters and each can either be in the subset or not be in
the subset resulting in 2 possibilities for each letter. That makes
for            possible subsets.
3. The five shaded areas represent the overlap between X and Y, between Y and Z,
between Z and W, between X and W, and the region where all 4 overlap.
X and Y contain just 3
Y and Z contain just 9
Z and W contain 4 and 16
X and W contain just 2
The region where all circles overlap would be empty
That’s a total of 5 numbers in the shaded regions.
Category 3
Number Theory
Meet #5, March 2007
You may use a calculator.
1. Set A is all the positive whole number multiples of 13 that are less than 100.
Set B is all the positive whole numbers that are one more than a multiple of 5 and
less than 100. How many numbers are there in A ∪ B ? Reminder: ∪ means union. Also,
we will include the number 1 in set B.
2. A guessing game has 24 cards with pictures of different people. Five people in
the pictures have white hair, five people wear glasses, and 10 people wear hats.
Two of the people with white hair wear glasses and two of the people with white
hair wear hats. Nobody who wears glasses wears a hat. How many of the 24
different people do not have white hair or wear glasses or wear a hat?
3. Venn diagrams work well for two or three sets, but not as well for four or more
sets. The noodle-shaped region
Multiples of 4
Perfect Squares
below shows one way that a fourth
set might be included in a Venn
diagram. If the natural numbers 1
through 72 inclusive are placed in
A
the appropriate regions, what is
B
the sum of the numbers in regions
A, B, and C?
Answers
1. _______________
2. _______________
3. _______________
Perfect Cubes
C
www.imlem.org
Factors of 72
Solutions to Category 3
Number Theory
Meet #5, March 2007
Answers
1. 25
2. 8
3. 137
1. There are 7 multiples of 13 less than 100: 13, 26, 39, 52, 65,
78, and 91. There are 20 numbers less than 100 that are one
more than a multiple of 5: 1, 6, 11, 16, 21, 26, 31, 36, 41, 46,
51, 56, 61, 66, 71, 76, 81, 86, 91, and 96. The two numbers that
occur on both lists are in bold. Thus there are 7 + 20 – 2 = 25
numbers less than 100 that are either multiples of 13 or one
more than a multiple of 5.
2. If the sets were mutually exclusive, there would be 5 + 5 +
10 = 20 people in our three sets. But two of the people with
white hair wear glasses and two of the people with white hair
wear a hat, so there must be only 20 – 2 – 2 = 16 people in our
three sets. There are 24 people in the game, so there must be 24
– 16 = 8 people who do not have white hair or wear glasses or
wear a hat.
3. We need only consider the factors of 72, but the diagram below shows all the
numbers 1 through 72. Region A has 12, 24, and 72, which adds up to 108. There
are no numbers in region B. Region C has 2, 3, 6, and 18, which adds up to 29.
The desired total is 108 + 0 + 29 = 137.
Multiples of 4
Perfect Squares
Ten other
multiples of 4
1
A 242
72
49
4
25
36
9
16
B
8
64
1
27
Perfect Cubes
C
18
2
3
6
www.imlem.org
Factors of 72
Category 3
Number Theory
Meet #5, March/April 2005
You may use a calculator
1. Set A = {a, b, c, d}. How many subsets are there for set A? Note: A subset of
set A is a set containing all, some, or none of the elements in set A.
2. Set A is the multiples of 7.
Set B is the natural numbers that leave a remainder of 2 when divided by 5.
Set C is the multiples of 13.
Set D is the natural numbers that leave a remainder of 3 when divided by 5.
Set E is the natural numbers less than 50.
How many elements are in the set ((A ∩ B) ∪ (C ∩ D))∩ E ?
3. Of the 125 members of the “Divers Down” scuba club, 70 have been to the
coral reefs, 80 have explored the shipwreck, and 60 have visited the underwater
caves. Thirty-eight have been to both the coral reefs and the shipwreck, 42 have
been to both the shipwreck and the caves, and 34 have been to the coral reefs and
the caves. Only 23 divers have been to all three sites. How many members of the
“Divers Down” scuba club have not been to any of these sites?
Answers
1. _______________
2. _______________
3. _______________
www.Imlem.org
Solutions to Category 3 Average team got 12.73 points, or 1.1 questions correct
Number Theory
Meet #5, March/April 2005 Average number of correct answers: 1.06 out of 3
Answers
1. Each elements in set A can either be included or excluded
when making a subset. This means we have two choices for
each of the four letters in set A, so there are 2 4 = 2 ⋅ 2 ⋅ 2 ⋅ 2 = 16
possible subsets. They are: {}, {a}, {b}, {c}, {d}, {ab}, {ac},
{ad}, {bc}, {bd}, {cd}, {abc}, {abd}, {acd}, {bcd}, and
{abcd}. The empty brackets, {}, indicate the empty set or null
set. The symbol Ø is also used for the null set.
1. 16
2. 3
3. 6
2. Set A is {7, 14, 21, 28, 35, 42, 49, ...}.
Set B is {2, 7, 12, 17, 22, 27, 32, 37, 42, 47, ...}.
Set C is {13, 26, 39,...}
Set D is {3, 8, 13, 18, 23, 28, 33, 38, 43, 48, ...}
A intersect B and is everything that is in both A and B, so
(A ∩ B) is {7, 42,...}. Similarly, (C ∩ D) is {13,...}. The union
of (A ∩ B) and (C ∩ D) is {7, 13, 42, ...}. We can end our
search here, since the last step is to intersect with set E, the
numbers less than 50. Thus, the set ((A ∩ B) ∪ (C ∩ D))∩ E
has just 3 elements.
Coral
Reefs
70
21
34
Shipwreck
80
23
38
15
11
23
19
7
60
Caves
42
3. Since 23 divers have been to all three places and 38 have
been to both the coral reefs and the shipwreck, we can
determine that 38 – 23 = 15 divers have been to both the coral
reefs and the shipwreck but not to the caves. Similarly, 19
people have been to the shipwreck and the caves but not to the
coral reefs, and 11 have been to the coral reefs and the caves
but not to the shipwreck. From here we can find out the
numbers of divers that have been to only one of the three sites
since we know the totals for each site. If we add up all seven
regions of the Venn diagram shown at left, we find that only
119 divers have been to at least one site. That means there are
125 – 119 = 6 members of the club who haven’t been to any of
the three sites. We can get this result directly as follows
125 – 70 – 80 – 60 + 38 + 42 + 34 – 23 = 6.
www.Imlem.org
Category 3
Number Theory
Meet #5, April 2003
You may use a calculator
today!
1. A set of attribute blocks includes every possible
combination of three shapes (circles, squares, and
triangles), three colors (red, green, and blue), and two
sizes (small and large) with no duplicate blocks. Set S
contains all the small blocks and set R contains all the
red blocks. How many blocks are in S ‰ R but not in
S ˆ R ?
S R 2. Students at Yippee I. A. Middle School held three spirit days recently. Of the
40 7th graGHUVDWWKHVFKRROSDUWLFLSDWHGLQ³&UD]\+DLU'D\´SDUWLFLSDWHGLQ
³&ODVK'D\´DQGSDUWLFLSDWHGLQ³)XQQ\+DW'D\´)LYHRIWKHVWXGHQWVZKR
SDUWLFLSDWHG LQ ³&UD]\ +DLU 'D\´ DOVR SDUWLFLSDWHG LQ ³&ODVK 'D\´ 7HQ RI WKH
students who participatHG LQ ³&ODVK 'D\´ DOVR SDUWLFLSDWHG LQ ³)XQQ\ +DW 'D\´
6HYHQ RI WKH VWXGHQWV ZKR SDUWLFLSDWHG LQ ³)XQQ\ +DW 'D\´ DOVR SDUWLFLSDWHG LQ
³&UD]\+DLU 'D\´7KUHHVWXGHQWVSDUWLFLSDWHGLQDOOWKUHHVSLULWGD\V+RZPDQ\
7th graders at Yippee I. A. Middle School did not participate at all in the spirit
days?
3. Set A is all prime numbers.
Set B is all odd numbers.
Set C is all natural numbers that leave a remainder of 1 when divided by 6.
Set D is all natural numbers less than 40.
Set E is A ˆ D ˆ B ˆ D ‰ C ˆ D .
How many elements are there in set E?
Answers
1. _______________
2. _______________
3. _______________
Solutions to Category 3
Number Theory
Meet #5, April 2003
Answers
1. 9
2. 4
3. 13
1. There are 3 u 3 u 2 18 blocks total. Set S has 3 u 3 u 1 9 small
blocks and set R has 3 u 1 u 2 6 red blocks. There are 3 blocks in
the intersection S ˆ R , the small red circle, the small red square,
and the small red triangle.
To find the number of blocks in
the
union, we subtract 3 from the sum 9 + 6 to get 9 6 3 12 blocks
in S ‰ R . This avoid double counting. If we want to exclude those
three blocks entirely from our count, we must subtract 3 again to get
12 ± 3 = 9 blocks in S ‰ R but not in S ˆ R .
2. The Venn diagram at right can help us to keep Crazy Hair Clash track of the participants. The easiest number
14 19 to
2 place first is the 3 in the center which corresponds
7 5 to the three students who participated in all three
3 spirit days. The five students who participated in
4 7 ³&UD]\+DLU'D\´DQG³&ODVK'D\´DUHWKHVHWKUHH
students and two others and so forth. When all the
8 22 numbers are placed correctly, we see that a total of
Funny Hat 36 students participated in the spirit days, which
means that 40 ± 36 = 4 did not participate at all.
Editor note March 2011: Thanks to Leena Chacrone for pointing out a wording error in this question. The question is now fixed. 3.
The intersection A ˆ D is the prime numbers less than 40:
^2, 3,5, 7,11,13 ,17 ,19 ,23 ,29 , 31, 37 ` . The intersection B ˆ D is the odd numbers less
than 40: ^1, 3,5, 7,9,... 37 , 39 `.
The intersection of these two sets,
A ˆ D ˆ B ˆ D , is just the odd prime numbers less than 40 (excluding only the
A ˆ D above): ^3,5, 7,11,13 ,17 ,19 ,23 ,29 , 31, 37 `. The intersection
number 2 from
C ˆ D is all natural numbers less than 40 that leave a remainder of 1 when divided
by 6: ^1, 7,13 ,19 ,25 , 31, 37 `. Set E is the union of the odd primes less than 40 with
the natural numbers less than
40 that leave a remainder of 1 when divided by 6.
This adds only
the numbers 1 and 25 to the set of odd primes less than 40, giving
us: E ^1, 3,5, 7,11,13 ,17 ,19 ,23 ,25 ,29 , 31, 37 ` . Set E has 13 elements.
Category 3
Number Theory
Meet #5, April 2001
You may use a
calculator today!
1. In set notation, A means “the number of elements in set A”. For
example, if A = {8,9,10} , then A = 3 . For this problem, set A is all the
multiples of 2 between 0 and 100, set B is all the multiples of 3 between 0
and 100, set C is all the perfect squares between 0 and 100, and set D is all
the perfect cubes between 0 and 100. Find ( A  B) ( C  D ) .
2. Of the 25 students in Linda’s class, 13 went skiing over vacation, 15 saw
a movie, and 9 worked on math problems. Three students did all of these
things; three students did none of these things; three students skied and did
math problems, but did not see a movie; three students did math problems
and saw a movie, but did not ski; and three students saw a movie and skied,
but did not do math problems. How many of the students in Linda’s class
did exactly one of these activities over vacation?
3. During the holidays, the 43 students on math team had a party and the 58
students in chorus had a party. Nine students got to go to both parties! Jill
decided to have a party for the students in math team who are not in chorus
and for the students in chorus who are not on math team. If every student
who qualifies comes, how many students will be at the party?
Answers
1. _____________
2. _____________
3. _____________
Solutions to Category 3
Number Theory
Meet #5, April 2001
Answers
1. 8
2. 10
3. 83
1. The elements of ( A  B) ( C  D ) are either
perfect squares or perfect cubes that are either
multiples of 2 or multiples of 3, between 0 and
100. The resulting set is {4,8,9,16 ,27,36 ,64,81} ,
which has 8 elements, thus
( A  B ) ( C  D ) = 8 .
2. This problem is best solved using Venn
diagrams, as shown below.
Skiing (13)
3
4
3
3
3
0
25 Students total;
3 did none of the
above activities.
Movie (15)
6
Math (9)
There were 4 + 6 + 0 or 10 students who did
exactly one of the three activities over vacation.
3. Using Venn diagrams, we have:
Chorus (58)
Math Team (43)
34
9
49
There are 34 + 49 or 83 students who are on the
math team or in chorus but not both. Another
way to arrive at this result is to add the numbers
of students in math team and chorus (43 + 58 =
101) and subtract twice the number who do both
(101 − 2 × 9 = 101 − 18 = 83 ).