The Sixth Annual West Windsor-Plainsboro Mathematics Tournament

Transcription

1 The Sixth Annual West Windsor-Plainsboro Mathematics Tournament Saturday October 27th, 2018 Grade 8 Test RULES The test consists of 2 multiple choice problems and short answer problems to be done in 40 minutes. All multiple choice answers should be marked on the Scantron mark sheet. No marks on this paper will be graded. All short answers you write should be in exact, simplified form, unless otherwise stated in the question, following the rules below: All fractions must be simplified. All fractional answers must be expressed in the form a b where a and b share no common factors other than 1. For example, if the answer is, then (mixed fraction) or 10 are NOT ACCEPTABLE. Any question that involves the number π in the answer must be written using the symbol π. In particular, decimal approximations of π are NOT ACCEPTABLE. All radicals must be simplified. Radicals in the denominator of a final answer, a perfect n th power under a n th root, or fractions/decimals under a radical are NOT ACCEPTABLE. You can ask your proctor a question, but if they think answering your question will give you an unfair advantage on the test over other contestants, they don t have to respond. After the testing has completed, please go to your selected mini-event rotation (directions will be given for this after the main contest is over). DO NOT FLIP THIS PAGE TO BEGIN THE TEST UNTIL INSTRUCTED TO DO SO BY YOUR PROCTOR(S).

2 Multiple-Choice Questions Answers to this section of the test should go on the scantron multiple-choice form given to you. NOTHING WRITTEN ON THIS TEST PAPER WILL BE SCORE 1. What is ( + + )/? A. 100 C E David and his dad share their birthday on the same day! David just turned 17, and his dad is now exactly 3 times David s age. How old was his dad when David was born? A C E Bessie the Cow escapes from the farm. She heads miles north and 12 miles east. Joe then goes in a straight line from his farm to Bessie s current location. How much more distance did Bessie travel than Joe? A. 3 miles 4 miles C. miles 6 miles E. 7 miles 4. Which of the following is the average of the remaining 4 answer choices? A C E. 36. What is the slope between the two points: (40, 6.3), (-23, 70.3)? A. C. E Which of the following is true? Only one answer choice is correct. The letters in each answer choice refer to the other answer choices. A. B & C are false C is true C. B is true E is true E. None of the above 7. A cube of play-doh is broken apart and formed into three new cubes of side lengths 3, 4 and. What was the original side length of the cube? A. 2 C E Jack and Jill are rounding numbers. They are given a number, 4.4, and decide to round it. Jack rounds it by first rounding to the hundredths place, then the tenths place, then the ones place, then the tens place, then the hundreds place. Jill rounds by rounding it to the nearest hundreds place. What is the difference between Jack s final number and Jill s final number? A. 0 0 C E. 200

3 . Which of the following ordered pairs (a, b, c) satisfy the following conditions? a + b + c = a 2 + b 2 + c 2 = a b c A. ( 2, 1, 2) ( 1, 1, 2) C. (1, 2, 2) ( 2, 1, 2) E. (0, 2, 3) 10. Which of the following has the largest area? A. A square with side length 3 A rectangle with dimensions of 2 4 C. An equilateral triangle with side length 3 A triangle with side lengths 3, 4 and E. A circle with radius of Bessie the Cow loves going for car rides. Bessie first travels at 28 km/hr for 30 minutes, then 3 km/hr for a distance of 7 km. What is her average speed the entire ride? A. 28 km/hr 30 km/hr C. 32 km/hr 3 km/hr E. 42 km/hr 12. Triangle ABC is a triangle where AB = 24, BC = AC = 13. What is the area of ABC? A C E Adam wants to travel to Canada, but he only has USD and Canada uses CA In addition, the places to exchange currency were all broken except for a couple strange options. The following are the exchange rates: 1 USD = 4 CAD 3 USD = GBP 3 GBP = CAD 2 GBP = HKD 3 HKD = 4 CAD (Note: These exchange rates are made up for this problem) Adam does not want too many exchanges so he will never exchange a CAD for another type of currency. If he starts with 00 USD, what is the most amount of CAD he can get? A C E Bessie the cow in on a cube with a side length of 3. She has to get from one corner to the opposite corner, but she can only travel along the surface. What is the shortest distance she can travel? A C. 3 6 E. 1. Allen wishes to make some changes to the design of the cylindrical Catfood containers. He forgot the radius of the cylinder, but he knows that the height of the cylinder was 2 units. When the height is quadrupled, the volume increases by the same amount as when the radius is increased by 6 units. What is the radius of the cylinder? A. 2 3 C. 4 6 E.

4 16. What is the area of the triangle with vertices A(, 4, 3), B(, 0, 0), C(17, 0, 0)? A C E Allen has 340 dollars. He wants to use this money to make the most people happy. Each candy costs 2 dollars and makes 1 person happy, and each decoration costs 24 dollars and makes 16 people happy, but for every decoration he buys, he must buy at least 3 candies. What is the largest number of people that Allen can make happy? A C E Anna and Elsa are tour guides. They want to split 8 tourists into 2 groups. How many ways can they split up the tourists if each group has at least 1 person? A C E In 184, January was on a Monday. What day will January, 201 be on? (Every year which was a multiple of 4 between those two dates is a leap year). A. Monday Tuesday C. Wednesday Friday E. Saturday 20. When a robot is asked to create a random number, it first checks the clock and takes note of the two digits in the minute s portion (Ex. if the time is 4:23, it notes down 2 and 3). If the numbers are noted as a and b, its random number will be a b + a + b. Which of the following answers cannot be its random number at any time? A C E Let n = Find the units digit of n 3. A. 1 3 C. 7 E. 22. Two ants start on adjacent corners of a square. Each minute, they both randomly walk to an adjacent corner of the square. What is the probability that the ants have not crossed paths after 4 minutes? A C E There are 3 freshmen and 6 sophomores in a class. How many ways are there to form groups of 3 such that none of them have more than one freshman? A C E Alice has 3 distinct gold coins and distinct silver coins. Bob has distinct gold coins and 3 distinct silver coins. Alice randomly picks two of her coins, and Bob randomly picks two of his coins. They simultaneously swap these pairs of coins. What is the probability that Alice will have more gold coins than Bob? A. 2 2 C E A triangle has two altitudes of length 6 and 1. How many possible whole number values can there be for the last altitude? A. 3 4 C. 6 E. 7

5 Short Answer Questions Answers to this section of the test should go on the short answer questions answer form given to you. NOTHING WRITTEN ON THIS TEST PAPER WILL BE SCORE 1. Compute the following: In Math Club, there are three types of people: Snakes, Cats, and Cows. 20% of Snakes attend every meeting, 30% of Cats attend every meeting, and 0% of Cows attend every meeting. Assuming that there is a whole number of each animal, what is the least amount of total members in Math Club? 3. How many digits does have? 4. What is the tens digit of! + 6! + 7! !? (The! is a factorial, and n! = n (n 1) (n 2) 2 1. For example, 4! = = 24). The cow number system only uses the digits 1. In addition, for a number to exist in this system, its digits must be strictly increasing from left to right. For example, 23 exists in the cow number system but 36 and 44 do not since is larger than 3 and 4 is the same as 4. How many numbers in the cow number system have 3 digits?