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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 BowlRound 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.

You have 6 black socks, 12 white socks, and 8 pink socks. It’s pitch dark and you’re packing for a trip. You reach into your drawer, blindly choose an individual sock, and pack it.

How many socks do you need to choose to be sure you packed at least one pair of pink socks?

Solution

- Think of the worst case scenario: choosing no pink socks for the 1st, 2nd, 3rd, 4th, 5th, 6th, etc. socks. How long can you keep up that string of bad luck?
- The worst case is you pull all 18 black and white socks first.
- After that, 2 more socks will guarantee at least one pair of pink socks.
- So if you pack 20 socks, you get at least one pink pair.

given that ABC is a right triangle,

and ABFG and ACDE are both squares.

Solution

- According to the Pythagorean theorem, the sum of the areas of squares constructed on the legs of right triangles equals the area of the square constructed on the hypotenuse.
- A2 + B2 = C2
- 9 + 25 = 34
- No square roots needed!

Solution

- 3x – 5y = 17
- y = 12 + 2x
- Therefore, 3x – 5(12 + 2x) = 17
- Distributing: 3x – 60 – 10x = 17
- Combining Like Terms: -7x – 60 = 17
- Addition Property of Equality: -7x = 77
- Division Property of Equality: x = -11
- Substitution: y = 12 + 2(-11)
- y = -10

Two of the angles in a triangle measure 48º and 84º. What type of triangle is this? Be as specific as you can.

Solution

- If two of the angles are 48º and 84º, then the third is 180º - 48º - 84º = 180º - 132º = 48º
- Since the triangle has two congruent angles (the two 48º angles), it’s isosceles.
- The triangle is not equilateral or right, so the most specific name is an isosceles triangle.

Working together, Gertrude and Bertha can clear their entire driveway in 30 minutes. Working alone it takes Gertrude 40 minutes to shovel the whole driveway. How long does it take Bertha to shovel the whole driveway on her own?

Solution

Gertrude working alone:

Driveway Gertrude shovels in 40 minutes

Both working together

Driveway Bertha shovels in 30 minutes

Driveway Gertrude shovels in 30 minutes

So Bertha working alone would take 30 * 4 = 120 minutes

If you roll two identical fair dice, what is the probability of rolling “snake eyes” – in other words rolling a 2?

Solution

- With one fair die, the chance of rolling a 1 is 1 in 6 possibilities.
- With two independent events, the chance of both happening is the product of their probabilities.
- 1/6 * 1/6 = 1/36
- Or, list all the possibilities in a table:

A serving of grapes is 2/3 of a cup. How many servings are in a 6 ½ cup package of grapes?

Give your answer as a mixed number of servings.

Solutions

- 6 ½ ÷ 2/3 = 13/2 * 3/2 = 39/4 = 9 ¾ servings
- 3 servings of grapes is 2 cups (2/3 + 2/3 + 2/3 = 2), so 6 ½ cups is 9 servings plus another ½ cup. A ½ cup of grapes is 3/6 of a cup, and 2/3 of a cup is 4/6 of a cup, so 3/6 of a cup is ¾ of what is needed to make a full serving. So there are 9 full servings and another ¾ of a serving.

Solution

- Clearly -1.125 and -7/5 are smallest
- -7/5 = -14/10 = -1.4
- -1.4 < -1.125
- 2/3 = 0.666666666…
- 0.6 < 0.666666666…
- 10/12 is 2/12 or 1/6 away from 1
- 2/3 is 1/3 away from 1, therefore 10/12 > 2/3
- -7/5, -1.125, 0.6, 2/3, 10/12

3|-2x – 4| > 12

Solution

- 3|-2x – 4| > 12
- Therefore, |-2x – 4| > 4
- So -2x – 4 has to be greater (to the right of) than 4 or less than -4 (to the left of)
- -2x – 4 > 4 -2x > 8 x < -4
- -2x – 4 < -4 -2x < 0 x > 0

What is the sum of the whole numbers from 1 to 100? In other words, 1 + 2 + 3 + 4 + 5 + … + 98 + 99 + 100 = ???

Solution

- 1 + 100 = 101. 2 + 99 = 101. 3 + 98 = 101. 4 + 97 = 101…. 50 + 51 = 101.
- There are 50 pairs of numbers that add up to 101.
- 50 * 101 = 5050
- The sum of all the numbers from 1 to 100 is 5,050.

Quiz BowlRound 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.

Mel, Pat, Ash, and Sal meet for the first time and all shake hands.

List all the handshakes.

Solution

- Mel – Pat, Mel – Ash, Mel – Sal
- Pat – Ash, Pat – Sal
- Ash – Sal

List all the prime numbers less than 100 that are the sum of 3 (not necessarily distinct) perfect cubes.

Solution

Cubes: 1, 8, 27, 64, 125…

- 1 + 1 + 1 = 3 prime!
- 1 + 1 + 8 = 10
- 1 + 8 + 8 = 17 prime!
- 1 + 1 + 27 = 29 prime!
- 1 + 27 + 27 = 55
- 1 + 8 + 27 = 36
- 1 + 1 + 64 = 66
- 1 + 64 + 64 = too big
- 1 + 8 + 64 = 73 prime!
- 1 + 27 + 64 = even
- 8 + 8 + 8 = 24
- 8 + 8 + 27 = 43 prime!
- 8 + 27 + 27 = even
- 8 + 27 + 64 = 99
- 8 + 8 + 64 = multiple of 8
- 8 + 64 + 64 = too big
- 27 + 27 + 27 = mult. of 3
- 27 + 27 + 64 = too big
- 27 + 64 + 64 = too big
- 64 + 64 + 64 = too big

A triangle has 3 sides, the lengths of which are whole numbers of centimeters. Two sides of the triangle are 3 cm and 5 cm, respectively. List all the possible lengths for the 3rd side.

Solution

If 5 is the longest side length, then

x + 3 > 5 and x ≤ 5, so x = 3, 4, 5.

If x is the longest side length, then x > 5 and 3 + 5 > x, so x = 6, 7.

x is any integer from 3 to 7 (including 3 and 7)

Quiz BowlRound 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: A super tripledon is found by taking an integer raising it to the 3rd power, then multiplying it by 3 and then adding 3 to the result.

What is the super tripledon of 3?

Solution

- 33 * 3 + 3 = 27 * 3 + 3 = 81 + 3 = 84

Solution

- x3 * 3 + 3 = 81,003
- x3 * 3 = 81,000
- x3 = 27,000
- x = 30

Q: Determine the slope-intercept equation of a line that passes through points (2,-3) & (5, 5).

Solution

- Slope:
- Change in y: 5 - -3 = 8
- Change in x: 5 – 2 = 3
- Ratio: 8/3
- Intercept:
- Going left 2 units results in going down 2 * (8/3) units = 16/3 units
- Going left 2 and down 16/3 from (2, -3) yields a y-intercept of (0, -25/3)
- y = (8/3)x – 25/3

F: What is the slope of a line that passes through the origin and is perpendicular to the previous line?

Solution

- Slopes of perpendicular lines are opposite reciprocals.
- The opposite reciprocal of 8/3 is -3/8
- The slope is -3/8

Q: How many distinguishable arrangements are there for the letters in the word “Drexel”?

Solution

- 6 * 5 * 4 * 3 * 2 * 1 / 2 = 360 arrangements.
- If all the letters were distinct it would be 6 * 5 * 4 * 3 * 2 * 1 because there would be 6 choices for the first letter, times 5 for the second, times 4 for the thirds, etc.
- But we divide by 2 because there are two ways to arrange the indistinguishable e’s in each of the words, so we double-counted, for example xldree and xldree.

Solution

- 1 * 5 * 4 * 3 * 2 * 1 / 2 = 60 arrangements.
- One choice for the first letter (D), then 5 choices for the second, 4 for the third, etc.
- We again have to divide by 2 because we are still double counting words like dlrxee and dlrxee

Q: A right triangle has side lengths of x, 5, and 12. What are the 2 possible exact values for x?

Solution

Let’s say 5 and 12 are the legs, then

52 + 122 = x2.

I recognize that as a 5, 12, 13 triangle, but could also solve: 25 + 144 = x2, so 169 = x2.

x = 13

The other case is that 12 is the hypotenuse. 52 + x2 = 122, or 25 + x2 = 144.

x2 = 144 – 25 = 119

x = √119

F: A triangle has sides 7, 9, 12. Classify the triangle as acute, right or obtuse.

Solution

- We can use the Pythagorean Theorem:
- 72 + 92 = 49 + 81 = 130
- 122 = 144
- 130 < 144
- To be a right triangle 72 + 92 would have to equal 122 but 122 is too big… so the “hypotenuse” is too big for a right triangle
- That means this triangle is obtuse.

Q: Is the number of 2-digit numbers in which both 2 digits are odd greater than, less than, or the same as the number of 2-digit numbers in which both 2 digits are even?

Solution

- Half of the 2-digit numbers in the 10s, 30s, 50s, 70s, and 90s have both digits odd
- Half of the 2-digit numbers in the 20s, 40s, 60s, and 80s have both digits even.
- Because 00, 02, 04, 06, and 08 are not considered 2-digit numbers, the number of 2-digit numbers with both digits odd is greater.

Solution

- Half of the 2-digit numbers in the 20s, 40s, 60s, and 80s have both digits even.
- 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88
- 20 two-digit numbers exist such that both digits are even.

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