1. Jeopardy! Vancouver Math Olympiad Competition #4: Spring Event Game #1
Hosted by: Burnaby South Secondary School

2. Jeopardy Instructions
32 of the Jeopardy questions are multiple choice questions. Each question has 6 choices, of which only 1 of them is correct (unless otherwise specified). There are also 5 math games, for which further instructions will be given when each question is shown.
No calculators are allowed for the Jeopardy round. You will need lined paper and other writing utensils for the Jeopardy round.
Your team has seconds for every question, depending on the automatic timer on every slide. The timer begins as soon as the slide is flashed, and the host will announce the end of the time period. Occasionally, problems will be granted extra time, as needed.
Teams will alternate (as selected by VMO Staff) selecting questions from the Jeopardy board. Every team can answer every question, by reporting your answer to your assigned staff member.
Your team will lose 50% of the value of the question for every incorrect response to a Multiple Choice question. Blank responses will not be awarded nor be deducted any credit.

3. History of Math
Fun Number Facts
Math Games
Applied Math
Math Theorems
Computations
Weird Stuff
$100
$100
$???
$100
$100
$100
$100
$200
$200
$200
$200
$???
$200
$200
$300
$300
$300
$300
$???
$300
$300
$400
$400
$400
$400
$???
$400
$400
$500
$500
$500
$500
$???
$500
$500
$600
$600

4. History of Math - $100
What area of mathematics is Newton and Leibniz both credited with inventing?
Number Theory
Calculus
Statistics
Non-Euclidean Geometry
String Theory
None of the above 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

5. ANSWER
History of Math - $100
(B) Calculus

6. History of Math - $200
This man was the first to use the Cartesian plane, and is referred to as the father of analytic geometry. He is also known as the father of modern philosophy.
Pythagoras
Euclid
Descartes
Leibniz
Napoleon
None of the above 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

7. ANSWER
History of Math - $200
(C) Descartes

8. History of Math - $300
This mathematician is considered the greatest mathematician of antiquity. He calculated the area under the arc of a parabola, investigated buoyancy of an object, and gave remarkably accurate approximations of 𝜋.
Plato
Pythagoras
Sun Tzi
Euclid
Da Vinci
None of the above 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

9. ANSWER
History of Math - $300
(F) None of the above

10. History of Math - $400
Which of the following mathematicians appears on a Bank of CSSMA note?
Euler
Galois
Fermat
Cantor
Beethoven
None of the above 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

11. ANSWER
History of Math - $400
(A) Euler

12. History of Math - $500
Which culture is credited with the invention of “0” (zero)?
Egyptians
Greeks
Chinese
Indians
Cavemen
None of the above 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

13. History of Math - $500
(D) Indians

14. Fun Number Facts - $100
The only perfect cube other than 1 in the Fibonacci sequence is: 8 27
343
729
1728
There are more than one perfect cubes other than 1 in the Fibonacci sequence 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

15. ANSWER
Fun Number Facts - $100
(A) 8

16. Fun Number Facts - $200
The number larger than all natural numbers, or the smallest ordinal number, is equal to which of the following?
(A) 𝜀
(B) 𝜔
(C) 𝛾
(D) ∞
(E) 𝜏
(F) This number does not exist 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

17. ANSWER
Fun Number Facts - $200
(B) 𝜔

18. Fun Number Facts - $300
In base 2015, the number has how many digits? 1 2
2015
2016
2017
None of the above 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

19. ANSWER
Fun Number Facts - $300
(D) 2016

20. Fun Number Facts - $400
The icosahedron has 12 vertices, 30 sides, and how many faces, and of what type?
12 pentagonal faces
12 triangular faces
16 pentagonal faces
20 triangular faces
20 square faces
This shape does not exist 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

21. ANSWER
Fun Number Facts - $400
(D) 20 triangular faces

22. NO CALCULATORS ALLOWED
Fun Number Facts - $500
Isn’t it interesting how Valentine’s day can be expressed as 2.14 and π day can be expressed as 3.14 (both are in the form a .14)? Two of the numbers listed below have as the first four digits of their decimal representation.
Which two numbers are these?
($250 per correct answer—NO PENALTY for incorrect answers)
NO CALCULATORS ALLOWED 2× 2×
𝑒 𝜋
𝜋 𝑒
𝜑 2.14×3.14
𝜑 2 𝑒𝜋 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

23. Staff: award $250 per correct answer. Award $0 per incorrect answer.
Fun Number Facts - $500
(D) 2×
(E) 𝑒 𝜋
Staff: award $250 per correct answer. Award $0 per incorrect answer.

24. Math Games - $???
You will play a game called “Chicken” with a randomly chosen other team in this VMO competition.
Your team is driving down the road in a car, and a randomly chosen other team is also driving down the road in a car. However, you two are on collision course: one team must swerve, or both may die in the crash, but if one driver swerves and the other does not, the one who swerved will be called a “chicken,” meaning a coward.
The potential outcomes of this game is shown below:
After you’ve made a pick, write it on paper with your team name on it. 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 30 38 43 44 39 42 41 40
You swerve
You drive on
They swerve
Both teams get $0
You: +$300
Them: -$200
They drive on
You: -$200.
Them: +$300
Both teams lose $400

25. Math Games - $???
Fermi Problem!
How many atoms are there in the human body? Assume that the average human body has a mass of 70kg.
Give your answer to the nearest power of 10, so an answer like atoms would be of the right form.
Teams will be ranked based on how close they are to the best answer as given by the computer. You will not be penalized if you give an absurd answer. 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

26. Math Games - $??? Fermi Problem!
Staff: Score as shown in the following table.
Source: Wolfram Alpha 2015
Answer
Points Earned or
$700 or
$600 or
$500 or
$400
10 22 , ,
10 32 , or
$300
10 20 , , , or
$200
Answer
Points Earned
10 17 , , , , , or
$100
Anywhere from to inclusive but not in any of the previous categories
$50
All other answers $0

27. Math Games - $???
Nim
There are three piles of berries: one pile of 8 berries, one pile of 9 berries, and one pile of 10 berries. Your team is playing a game with the VMO staff called Nim where your team and the VMO staff alternate turns taking berries from the three piles. The rules are:
On every turn, at least one berry must be taken from one of the piles.
As many berries is allowed to be taken from any of the piles on any given turn, but no berries can be taken from more than one pile on any given turn.
Your goal is to be the person/team to take the last berry. What is your winning move, if your team goes first?
Note: there are multiple winning moves from this position. Your team only need to determine one of them. You will not be penalized for a wrong answer. 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

28. Math Games - $???
Nim
The three possible winning moves are:
Take 9 berries from the pile of 10 berries
Take 7 berries from the pile of 9 berries
Take 5 berries from the pile of 8 berries
Staff: Award $500 for any of the above three answers. Do not deduct points for other answers.

29. Math Games - $???
Mind Game!
Your team may choose any positive integer, and write that number down on a slip of paper and hand that piece of paper in to your assigned staff member..
Teams will be ranked from the least number chosen to the greatest number chosen. The team in place 𝑛 will receive $50 more than the team in place 𝑛+1. The top team will receive $500.
If your team chooses a number that is the same as the number chosen by at least one other team, your team will lose $100. 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

30. Math Games - $???
Random Walks
Starting from 0, the VMO staff has flipped a coin 1000 times. Each time that a head appears, the VMO staff moves right by 1 unit on a number line. Each time that a tail appears, the VMO staff moves left by 1 unit on a number line.
For example, a sequence of HTHHTH where H stands for Heads and T stands for Tails would result at a final position of 2 on the number line.
Suppose the final position on the number line is 𝑛, your team’s goal is to predict the value of |𝑛|, or the positive distance from 𝑛 to the 0 mark on the number line.
Your team’s score will be based on how close you are to an experimental trial conducted by the VMO staff. 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

31. (Most probable sum as selected by team)
Math Games - $???
Random Walks
Staff:
Please award points
as shown the table below
Random Walk generated by Wolfram|Alpha:
Team Bet
(Most probable sum as selected by team)
Points Earned
24, 25
$500
22, 23, 26, 27
$400
21, 22, 28, 29
$300
17-20, 30-33
$200
12-16, 34-38
$100
6-11, 39-44 $0
All other numbers
-$100
Contrary to what many may think, the theoretical expected value of |𝑛| is NOT 0!!!

32. NO CALCULATORS ALLOWED
Computations - $100
Evaluate: 84 2
NO CALCULATORS ALLOWED
5476
5776
7056
7396
8836
This problem cannot be solved 00 15 14 16 12 13 18 21 20 19 11 17 10 03 02 01 00 04 05 09 08 07 06 22 24 39 38 37 36 40 41 45 44 43 42 35 34 28 27 26 25 29 30 33 32 31 23

33. ANSWER
Computations - $100
(C) 7056

34. NO CALCULATORS ALLOWED
Computations - $200
Evaluate: 8!
NO CALCULATORS ALLOWED
5040
40320
362880 8 00 15 14 16 12 13 18 21 20 19 11 17 10 03 02 01 00 04 05 09 08 07 06 22 24 39 38 37 36 40 41 45 44 43 42 35 34 28 27 26 25 29 30 33 32 31 23

35. ANSWER
Computations - $200
(B) 40320

36. NO CALCULATORS ALLOWED
Computations - $300
Evaluate: 9,621,576,521×11 
NO CALCULATORS ALLOWED
105,937,341,731
105,737,341,731
105,837,241,731
105,837,441,731
105,837,341,731
I give up 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

37. ANSWER
Computations - $300
(E) 105,837,341,731

38. NO CALCULATORS ALLOWED
Computations - $400
What is the value of: ?
NO CALCULATORS ALLOWED 54 64 74 84 94
The answer is not an integer 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

39. ANSWER
Computations - $400
(B) 64

40. NO CALCULATORS ALLOWED
Computations - $500
Which of the following is closest to ?
NO CALCULATORS ALLOWED
44.884
44.885
44.886
44.887
44.888
44.889 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

41. ANSWER
Face-off - $500
(F)

42. NO CALCULATORS ALLOWED
Computations - $600
Evaluate:
NO CALCULATORS ALLOWED
3003
3005
3007
3009
3011
None of the above 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

43. Computations - $600
ANSWER
(A) 3003

44. Applied Math - $100
The isotope 14 𝐶 has a half life of 5730 years. Roughly what percentage of 14 𝐶 remains in a specimen after a period of years?
50%
25%
12.5%
6.25%
3.125% 0% 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

45. Applied Math - $100
ANSWER
(C) 12.5%

46. Applied Math - $200
Given a set perimeter of a figure, which of the following shapes maximizes the area of the figure?
Square
Equilateral Triangle
Regular Hexagon
Circle
Golden Rectangle
Cube 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

47. Applied Math - $200
ANSWER
(D) Circle

48. Applied Math - $300
The national inflation rate is roughly 2% per year. A pound of carrots cost $1.00 today. In what year, if the national inflation rate stays constant indefinitely, will the carrot first cost at least $2.00?
2040
2045
2050
2055
2060
2065 00 15 14 16 12 13 18 21 20 19 11 17 10 03 02 01 00 04 05 09 08 07 06 22 24 39 38 37 36 40 41 45 44 43 42 35 34 28 27 26 25 29 30 33 32 31 23

49. Applied Math - $300
ANSWER
(C) 2050

50. Applied Math - $400
Which of the following is NOT a probability distribution?
Normal
Student’s 𝑡
Chi-squared
Poisson
Geometric
Polynomial 00 15 14 16 12 13 18 21 20 19 11 17 10 03 02 01 00 04 05 09 08 07 06 22 24 39 38 37 36 40 41 45 44 43 42 35 34 28 27 26 25 29 30 33 32 31 23

51. Applied Math - $400
ANSWER
(F) Polynomial

52. Applied Math - $500
Kepler’s Second Law states: “A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.”
The truth of the above statement is dependent upon the fact that the Sun is at the center or foci of which of the following conic sections, representing the planet’s orbit?
Parabola
Hyperbola
Circle
Ellipse
Bézout’s curve
Kepler’s Second Law is a lie 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

53. Applied Math - $500
ANSWER
(D) Ellipse

54. Applied Math - $600
Choose the correct answer, given that there is only one correct answer:
If (B) is correct, then (C) is wrong
If (A) is wrong, then (F) is wrong
(E) is wrong if and only if (D) is wrong
(A), (E), and (F) are all wrong
If (D) and (E) are wrong, then (A) is wrong
If (C) is correct, then (B) and (E) are both wrong 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

55. The VMO staff apologizes for making your brain hurt.
Applied Math - $600
ANSWER
(F)
The VMO staff apologizes for making your brain hurt.

56. Math Theorems - $100
The sum of the angles in a convex polygon with 𝑛 sides is:
180
360
180(𝑛−2)
360(𝑛−2)
360𝑛−360
There does not exist such a formula 00 20 21 19 17 16 22 18 24 27 28 26 25 15 23 13 04 05 03 02 00 01 06 07 12 29 11 10 08 09 14 31 51 52 50 49 47 48 53 54 58 59 57 56 55 46 45 36 37 35 34 32 33 38 39 43 44 42 41 40 30

57. Math Theorems - $100
ANSWER
(C) 180(𝑛−2)

58. Math Theorems - $200
𝑠𝑖𝑛 2 𝜃+ 𝑐𝑜𝑠 2 𝜃= ? 1 -1 2
𝑐𝑜𝑠 2𝑥
You cannot square the sine of any angle 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

59. Math Theorems - $200
ANSWER
(A) 1

60. Math Theorems - $300
The following theorem is called:
𝑐 2 = 𝑎 2 + 𝑏 2 −2𝑎𝑏 cos 𝐶
Law of Cosines
Triangle trigonometry theorem
Squared length theorem
Heron’s formula
Extended Law of Sines
None of the above 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

61. Math Theorems - $300
ANSWER
(A) Law of Cosines

62. Math Theorems - $400
There is a theorem that states that every integer greater than 1 either is prime itself or is the product of prime numbers. This theorem also states that every integer greater than 1 has a unique prime factorization.
What is the name of this theorem?
The Fundamental Theorem of Algebra
Descartes’ Rule of Signs
Fermat’s Little Theorem
The Divergence of Prime Numbers Theorem
The Fundamental Theorem of Arithmetic
This theorem has been disproven 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

63. (E) The Fundamental Theorem of Arithmetic
Math Theorems - $400
ANSWER
(E) The Fundamental Theorem of Arithmetic

64. Math Theorems - $500
The proof of a theorem contains the following portion:
“…Consider the number 𝑝=𝑝 1 × 𝑝 2 × 𝑝 3 ×…× 𝑝 𝑛 +1 where 𝑝 1 , 𝑝 2 , 𝑝 3 ,…, 𝑝 𝑛 are primes. Clearly 𝑝 is prime. Contradiction.”
What does the above proof prove?
All primes are of the form 𝑝=6𝑛±1 for some integer 𝑛.
The infinitude of primes
The Fundamental Theorem of Arithmetic
If a number 𝑝 can be expressed in the form 𝑝=𝑝 1 × 𝑝 2 × 𝑝 3 ×…× 𝑝 𝑛 +1 where 𝑝 1 , 𝑝 2 , 𝑝 3 ,…, 𝑝 𝑛 are primes, then 𝑝 is a prime number.
The Twin Prime Conjecture
The Riemann Hypothesis 00 09 10 08 12 13 07 11 06 01 00 02 03 05 04 14 16 26 25 27 28 30 29 24 23 18 17 19 20 22 21 15

65. (B) The infinitude of primes
Math Theorems - $500
ANSWER
(B) The infinitude of primes

66. Weird Stuff - $100
How many positive integer solutions for (𝑥, 𝑦, 𝑧) are there to the equation:
𝑥 4 + 𝑦 4 = 𝑧 4 1 2 16
256 ∞
This is a troll question 00 15 14 16 12 13 18 21 20 19 11 17 10 03 02 01 00 04 05 09 08 07 06 22 24 39 38 37 36 40 41 45 44 43 42 35 34 28 27 26 25 29 30 33 32 31 23

67. (F) This is a troll question
Weird Stuff - $100
ANSWER
(F) This is a troll question

68. Weird Stuff - $200
What type of number is 𝑖 𝑖 , where 𝑖= −1 ?
Real
Imaginary
Complex non-real
Rational
Quaternion
This number does not exist 00 15 14 16 12 13 18 21 20 19 11 17 10 03 02 01 00 04 05 09 08 07 06 22 24 39 38 37 36 40 41 45 44 43 42 35 34 28 27 26 25 29 30 33 32 31 23

69. Weird Stuff - $200
ANSWER
(A) Real

70. Weird Stuff - $300
…= ?
(A) 𝑒 2
(B) 𝜋 2
(C) 𝜑 𝜑
(D) 6𝛿
(E) 𝑒 𝜋
(F) ∞ 00 15 14 16 12 13 18 21 20 19 11 17 10 03 02 01 00 04 05 09 08 07 06 22 24 39 38 37 36 40 41 45 44 43 42 35 34 28 27 26 25 29 30 33 32 31 23

71. Weird Stuff - $300
ANSWER
(B) 𝜋 2

72. Weird Stuff - $400
Which of the following can be “disproven”?
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
(F) None of the above can be “disproven” 00 15 14 16 12 13 18 21 20 19 11 17 10 03 02 01 00 04 05 09 08 07 06 22 24 39 38 37 36 40 41 45 44 43 42 35 34 28 27 26 25 29 30 33 32 31 23

73. Weird Stuff - $400
ANSWER
(E) 5

74. Weird Stuff - $500
The object depicted on right
is:
A 4th dimensional sphere
A sphere turned inside out
A torus in 4th dimensional space
Klein’s bottle
What the inside of a black hole is conjectured to look like
Ummm…what the heck? 00 15 14 16 12 13 18 21 20 19 11 17 10 03 02 01 00 04 05 09 08 07 06 22 24 39 38 37 36 40 41 45 44 43 42 35 34 28 27 26 25 29 30 33 32 31 23

75. (B) A sphere turned inside out
Weird Stuff - $500
ANSWER
(B) A sphere turned inside out

76. CONGRATULATIONS!