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

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 30-60 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.

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

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

ANSWER History of Math - $100 (B) Calculus

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

ANSWER History of Math - $200 (C) Descartes

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

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

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

ANSWER History of Math - $400 (A) Euler

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

History of Math - $500 (D) Indians

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

ANSWER Fun Number Facts - $100 (A) 8

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

ANSWER Fun Number Facts - $200 (B) 𝜔

Fun Number Facts - $300 In base 2015, the number 2015 2015 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

ANSWER Fun Number Facts - $300 (D) 2016

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

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

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 23.14 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 8 2 +8 3 5 5 +5 6 2× 2.14 3.14 2× 3.14 2.14 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

Staff: award $250 per correct answer. Award $0 per incorrect answer. Fun Number Facts - $500 (D) 2× 3.14 2.14 (E) 𝑒 𝜋 Staff: award $250 per correct answer. Award $0 per incorrect answer.

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

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 10 100 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

Math Games - $??? Fermi Problem! Staff: Score as shown in the following table. Source: Wolfram Alpha 2015 Answer Points Earned 10 27 or 10 28 $700 10 26 or 10 29 $600 10 25 or 10 30 $500 10 24 or 10 31 $400 10 22 , 10 23 , 10 32 , or 10 33 $300 10 20 , 10 21 , 10 34 , or 10 35 $200 Answer Points Earned 10 17 , 10 18 , 10 19 , 10 36 , 10 37 , or 10 38 $100 Anywhere from 10 10 to 10 45 inclusive but not in any of the previous categories $50 All other answers $0

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

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.

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

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

(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!!!

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

ANSWER Computations - $100 (C) 7056

NO CALCULATORS ALLOWED Computations - $200 Evaluate: 8! NO CALCULATORS ALLOWED 5040 40320 362880 3628800 39916800 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

ANSWER Computations - $200 (B) 40320

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

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

NO CALCULATORS ALLOWED Computations - $400 What is the value of: 3 262144 ? 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

ANSWER Computations - $400 (B) 64

NO CALCULATORS ALLOWED Computations - $500 Which of the following is closest to 2015 ? 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

ANSWER Face-off - $500 (F) 44.889

NO CALCULATORS ALLOWED Computations - $600 Evaluate: 5 5 + 6 5 + 7 5 + 8 5 + 9 5 + 10 5 + 11 5 + 12 5 + 13 5 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

Computations - $600 ANSWER (A) 3003

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 17190 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

Applied Math - $100 ANSWER (C) 12.5%

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

Applied Math - $200 ANSWER (D) Circle

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

Applied Math - $300 ANSWER (C) 2050

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

Applied Math - $400 ANSWER (F) Polynomial

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

Applied Math - $500 ANSWER (D) Ellipse

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

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

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

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

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

Math Theorems - $200 ANSWER (A) 1

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

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

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

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

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

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

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

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

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

Weird Stuff - $200 ANSWER (A) Real

Weird Stuff - $300 6 1 2 + 6 2 2 + 6 3 2 + 6 4 2 +…= ? (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

Weird Stuff - $300 ANSWER (B) 𝜋 2

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

Weird Stuff - $400 ANSWER (E) 5

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

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

CONGRATULATIONS!