Transcript NIT FINALS - The Math Forum @ Drexel

Quiz Bowl
All eight students will solve problems as part of a quiz
bowl.
Students will work together to answer questions
and compete head to head against other teams.
Teams will be seated at tables.
A moderator will ask a question using a
microphone.
Each team will write an answer to each question on
the paper provided.
Team captains will hand one answer to the table
runner.
The team captain should be seated in middle of
the team.
Teams should write answers clearly and neatly so that
judges can read answers. Teams with unclear
answers will not receive points.
Time will begin once the moderator finishes
reading the question.
Teams should NOT include computational or scratch
work on the paper; only answers should be written on
the paper.
Once a team turns in their paper, the team may
NOT change their answer.
Teams will be provided with scratch paper and pencils
as well as their answer sheets.
NO books, notes, calculators, or electronic
devices, such as cell phones, may be used.
Cell phones must be turned off.
Quiz Bowl
Round 1: All Groups Can Answer
Round 1 will consist of 10
questions.
All teams that provide the correct
answer to a question posed by a
moderator will earn 5 points.
Bear is not next to Rabbit
Deer is not next to Owl
Squirrel is next to Bear
Owl is not next to Squirrel
Starting with Bear, in what order do Bear,
Rabbit, Deer, Owl, and Squirrel sit around the
campfire?
Rules
Bear is not next to Rabbit
Deer is not next to Owl
Squirrel is next to Bear
Owl is not next to Squirrel
Solution
Start with Bear
•Put Squirrel next to Bear
Owl
•That leaves 3 open spots
•Owl can’t go in the middle of those 3 spots
because that would force Deer to sit next
to Owl
•Owl can’t go next to Squirrel either
•So put Owl next to Bear
•Then put Deer next to Squirrel (not next to
Owl)
•That leaves Rabbit between Deer and Owl
•Final answer: Bear, Squirrel, Deer, Rabbit,
Owl OR Bear, Owl, Rabbit, Deer, Squirrel
Bear
Squirrel
Rabbit
Deer
Find the area of isosceles triangle ABC
B
5 cm
5 cm
A
6 cm
C
Solution
B
To quickly find the area
we need to find the height
of the triangle.
Because the triangle is
isosceles I know the
altitude bisects the base,
creating a 3-4-5 right
triangle.
Therefore, the height is
4, and the Area is:
0.5 * 6 * 4 = 12 cm2
5 cm
5 cm
4 cm
3 cm
A
6 cm
C
Solve for x:
5(x + 2) – 3(x + 2) + x
=
2(x + 2) + 10
Solution
• 5(x + 2) – 3(x + 2) + x = 2(x + 2) + 10
• Therefore, 2(x + 2) + x = 2(x + 2) + 10
• Therefore, x = 10 (by subtraction property
of equality)
The diagonal of a
square measures 10
miles. Find the area of
the square.
Solution
• Use the Pythagorean Theorem:
• side2 + side2 = 102
• 2 * side2 = 100
• side2 = 50
10mi
• Do I have to keep
solving? No!
• Area = side2 = 50
• Area = 50 square miles
Seven small, congruent rectangles form a large
rectangle as shown. The area of each of the 7 small
rectangles is 48 u2. Find the indicated length.
Solution
• I notice 4 of the widths = 3 of the
lengths of the small rectangles
• Length * width = 48
• 4W = 3L, LW = 48
• Guess W = 6, L = 8
• 4*6 = 3*8, 6 * 8 = 48
• ? = 6 + 6 + 6 + 6 = 24
Your “friend” offers to share some
Bertie Bott’s Every Flavour Beans with
you. He’s got a pile of 12 identical
beans and explains that 2 are barfflavored and 10 are peachflavored. He offers to let
you randomly take
5 beans. What is the
probability that you will
get 5 peach-flavored
beans?
Solution
• When you choose the first bean, you have
a 10/12 chance of getting peach-not bad.
• If you get the first bean right, you have a
9/11 of getting second bean right too.
• You have an 8/10 chance of getting the 3rd
right, a 7/9 chance of getting the 4th, and
a 6/8 chance of getting the 5th
• All together, you have a 10/12 * 9/11 * 8/10
* 7/9 * 6/8 = 42/132 or 7/22 chance of
getting all good beans.
You are trying to bring your
quiz average up to a 95%. So
far, you’ve had 9 quizzes and
averaged a 92%. Is it
possible to bring your
average up to a 95% by
th
acing the 10 quiz? The
maximum possible score is
100% -- no extra credit!
Solution
• If you averaged 92% on 9 quizzes, you
earned a total of 92 * 9 percentage points on
those 9 quizzes.
• 92 * 9 = 828 percentage points
• To have a 95% average on 10 quizzes, you
need to have 950 total percentage points
• 950 – 828 = 122, more than is possible.
• Alternatively, assume you get the top score,
100, on your last quiz. That brings you up to
938 points/10 quizzes, a 94%
Logan, Sam, and Pat share 3 large pizzas. Logan eats:
• 1/3 of the pepperoni pizza,
• ¼ of the cheese pizza,
• 1/6 of the everything pizza.
Sam eats
• 1/6 of the pepperoni pizza,
• 1/3 of the cheese pizza,
• ½ of the everything pizza.
Pat eats
• ½ the pepperoni pizza,
• 1/6 of the cheese pizza,
• none of the everything pizza (Pat hates anchovies).
Are the leftovers more than or less than one whole
pizza?
Solution
• Look pizza by pizza:
– Amount of pepperoni eaten: 1/3 + 1/6 + ½ = 1
– Amount of cheese eaten: ¼ + 1/3 + 1/6 =
3/12 + 4/12 + 2/12 = 9/12
– Amount of everything pizza eaten: 1/6 + ½ =
2/12 + 6/12 = 8/12
• Total amount eaten: 12/12 + 9/12 + 8/12 =
29/12. 29/12 is more than 2 pizzas
(24/12) so there is less than one whole
pizza left over.
Points A, B, C, D, and E are marked
between 1 and 1 billion on a number
line. Which of A, B, C, D, and E is
closest to where 1 million would be
marked?
A B
1
C
D E
1 billion
Solution
•
•
•
•
1 million is 1,000,000
1 billion is 1,000,000,000
1 million is one one-thousandth of a billion
Therefore, 1 million is one one-thousandth
of the way from one to 1 billion
• 1 million is closest to point A
Find the value of x:
x°
100°
135°
Solution
x°
100°
135°
Quiz Bowl
Round 2: Multiple Answers–1 Minute
Round 2 will consist of 3 questions
that each have multiple answers.
Each team will earn 1 point for
each correct answer and 1 bonus
point if the team provides all of the
possible answers.
Max’s special: scrambled-egg
sandwiches.
Toast options: white or wheat
Cheese options: American or none
Meat options: ham, bacon, or none.
List all possible sandwich orders
Solution
•
•
•
•
•
•
•
•
•
•
•
•
White toast, American cheese, ham
White toast, American cheese, bacon
White toast, American cheese, no meat
White toast, no cheese, ham
White toast, no cheese, bacon
White toast, no cheese, no meat
Wheat toast, American cheese, ham
Wheat toast, American cheese, bacon
Wheat toast, American cheese, no meat
Wheat toast, no cheese, ham
Wheat toast, no cheese, bacon
Wheat toast, no cheese, no meat
The partitions of a number are the
ways to write that number as a sum
of positive integers. For example,
the partitions of 3 are:
•3
•1+2
•1+1+1
List all the partitions of 6.
Solution
•
•
•
•
•
•
•
•
•
•
•
1+1+1+1+1+1
2+1+1+1+1
2+2+1+1
3+1+1+1
2+2+2
3+2+1
4+1+1
3+3
4+2
5+1
6
A probability
experiment
consists of
flipping three quarters to
see if they land HEADS or
TAILS. List all the equally
likely outcomes of the
experiment.
Solution
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Quiz Bowl
Round 3: Speed Round with
Follow Up Questions
Round 3 will consist of 5 questions.
The first team to give a correct answer
to the runner will get 3 points.
The first team with the correct answer
will then receive a follow-up question
for 2 bonus points.
Q: Let the operation be
defined for all real numbers a
and b as follows.
a
b = a + ab
Find (3
17) + (1
-1)
Solution
• (3 17) + (1 -1) =
• (3 + 3*17) + (1 + 1*-1) =
• 3 + 51 + 1 + -1 = 54
F: If x 7 = 32, find x.
Solution
•If x 7 = 32,
•then x + x*7 = 32, so
•8x = 32
•x = 4
• Check: 4 7 = 4+4*7 = 32
Q: Find the equation of
line BC
Solution
• The slope of BC can be found by
(4 – 0) / (3 - -9) = 4/12 = 1/3
• The y-intercept of BC appears to be at 3,
we can check like this:
• Does y = (1/3)x + 3 work for (3, 4) and (9,0)?
• Check: 0 = (1/3)(-9) + 3? Yup!
• Check: 4 = (1/3)(3) + 3? Yup!
• y = (1/3)x + 3
F: A: (-11, 6), B: (-9, 0),
and C: (3, 4) are three
vertices of a rectangle.
th
Where is the 4 vertex?
Solution
14
12
10
A
6
2
–15
A: (–11.00, 6.00)
B: (–9.00, 0.00)
C: (3.00, 4.00)
D: (1.00, 10.00)
8
4
–20
D
B
–10
–5
C
5
–2
–4
–6
–8
–10
–12
10
15
20
B
A
The picture shows a 4-block path from A to B, with
no shortcuts or backtracking. How many 4-block
paths from A to B (no shortcuts or backtracking!)
exist?
Solution
• There are 6 paths. Here is one way to
count them:
B
A
F:
How many 6-block paths are there
from A to B (no shortcuts or
backtracking)?
B
A
Solution
• Another way to count is to think of
permutations.
• Every path consists of 6 blocks, traveling
either North or East (otherwise we’d be
backtracking).
• Three blocks have to be North and three
blocks have to be East, or we fall off the map.
• Therefore, the answer is the number of ways
to arrange 3 N’s and 3 E’s:
• 6! / (3!*3!) = 20
Q: Find the sum of
123 + 123 + 123 + 123 +
246 + 246 + 246
Solution
246 = 123 + 123, so the problem can be
rewritten as
123 + 123 + 123 + 123 + 123 +
123 + 123 + 123 + 123 + 123
or 123 * 10
123 * 10 is just 1230
F: Find the sum of the
first 100 odd numbers:
1+3+5+7+…+
193 + 195 + 197 + 199
Solution
•
•
•
•
•
•
•
•
Look at some patterns
1=1
1+3=4
1+3+5=9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + … + (2n - 1) = n2
1 + 3 + … + 197 + 199 = 1002 = 10,000
Q: (x – 2)(2x + 1) = 0
Find all the values of x that
make this statement true.
Solution
• I know the product will be 0 when either
one of the numbers being multiplied is
equal to 0.
• The product will be 0 when either x – 2 = 0
or 2x + 1 = 0
• x – 2 = 0 implies x = 2
• 2x + 1 = 0 implies x = -1/2
• The statement is true when x = 2 and
when x = -1/2
F:
2
x
+ 20x + 51 = 0
Find all the values of x
that make this
statement true.
Solution
• We can rewrite x2 + 20x + 51 = 0 in the
same useful format as the original
question:
– (x + __)(x + __)
– The __’s must multiply to 51 and add to 20
– 51 = 3 * 17
– x2 + 20x + 51 = 0 is equivalent to
(x + 3)(x + 17) = 0
– This is true when x = -3 and x = -17