The Sixth Annual West Windsor-Plainsboro Mathematics Tournament Saturday October 27th, 2018 Grade 7 Test RULES The test consists of 25 multiple choice problems and 5 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 9 5, then 1 4 18 5 (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).
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 SCORED. 1. What is (9 + 99 + 999)/9? A. 99 B. 100 C. 101 D. 111 E. 123 Solution: 123 Split up the fraction into 9 9 + 99 9 + 999 9 = 1 + 11 + 111 = 123 2. Today is Saturday, October 27th, 2018. What day of the week is Halloween (Oct. 31st)? A. Monday B. Tuesday C. Wednesday D. Friday E. Sunday Solution: W ednesday The 28th is Sunday, the 29th is Monday, the 30th is Tuesday, the 31st is Wednesday. 3. David and his dad were born on the same day of the year! 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. 17 B. 18 C. 34 D. 35 E. 51 Solution: 34 David s dad is 3 17 = 51 years old. When David was born 17 years ago, his dad was 51 17 = 34. 4. Bessie the Cow escapes from the farm. She heads 5 miles north and 12 miles east. Farmer Joe then goes in a straight line from his farm to Bessie s current location. How much more distance did Bessie travel than farmer Joe? A. 3 miles B. 4 miles C. 5 miles D. 6 miles E. 7 miles Solution: 4 This creates a right triangle with legs 5 and 12. Using the Pythagorean theorem, Farmer Joe travels 5 2 + 12 2 = 169 = 13 miles. Bessie traveled 5 + 12 = 17 miles. Bessie traveled 17-13=4 more miles than Farmer Joe. 5. Which of the following is the average of the remaining 4 answer choices? A. 4 B. 10 C. 14 D. 16 E. 36 Solution: 16 If the average of 4 numbers is x, then adding a 5th number equal to x still keeps the average of x. Then the answer is the average of all 5 answer choices, which is 4+10+14+16+36 5 = 16.
6. Bessie the Cow loves eating lemons and oranges. If eating a lemon increases her fullness by 6 and eating an orange increases her fullness by 5, then what is the largest amount of fullness she can never achieve just by eating oranges and lemons? A. 4 B. 7 C. 13 D. 17 E. 19 Solution: 19 Using the Chicken McNugget theorem, the answer is 6 5 6 5 = 19. 7. Bessie the Cow has invented a new operation, ℵ. For any two numbers, a ℵ b = (a + a b + b). What is 4 ℵ 9? A. 13 B. 49 C. 76 D. 81 E. 144 Solution: 49 4ℵ9 = 4 + 4 9 + 9 = 49. 8. 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 B. C is true C. B is true D. E is true E. None of the above Solution: ChoiceA If one of choice B or choice C is correct, then the other is correct. This means that both choice B and choice C are false, so choice A is true. 9. A cube of play-doh is broken apart and formed into three new cubes of side lengths 3, 4 and 5. What was the original side length of the cube? A. 5 B. 5 2 C. 6 D. 10 2 E. 8 Solution: 6 The volume of the three cubes are 3 3 = 27, 4 3 = 64, and 5 3 = 125. This adds to 27+64+125 = 216, which is 6 3, so the original side length is 6. 10. Jack and Jill are rounding numbers. They are given a number, 49.49, 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 B. 50 C. 100 D. 150 E. 200 Solution: 100 Jack rounds 49.49 to 49.5, then 50, then 100. Jill rounds it from 49.49 to 0. The difference between 0 and 100 is 100. 11. The numbers on the opposite faces of a die add up to 7. What is the sum of the numbers on the 4 faces touching the face with the number 1?
A. 12 B. 13 C. 14 D. 15 E. 16 Solution: to 14. 14 The 6 will be opposite the 1, so the numbers touching the 1 are 2, 3, 4, and 5, which add up 12. Which of the following has the largest area? A. A square with side length 3 B. A rectangle with dimensions of 2 4 C. An equilateral triangle with side length 3 D. A triangle with side lengths 3, 4 and 5 E. A circle with radius of 2 Solution: A circle with radius of 2 The square has an area of 9. The rectangle has an area of 8. The equilateral triangle can fit inside the square of side length 3, so it is smaller. The triangle with side lengths 3, 4, and 5 is a right triangle with an area of 6. The circle has an area of 4π 12.57, so it has the largest area. 13. Triangle ABC is a triangle where AB = 24, BC = AC = 13. What is the area of ABC? A. 48 B. 60 C. 120 D. 180 E. 240 Solution: 60 The triangle is an isosceles triangle. If you draw the altitude to side AB, it forms two right triangles, with hypotenuse 13 and one leg equal to 12. By the Pythagorean theorem, the altitude is 5. The area is then 1/2 24 5 = 60 14. Adam wants to travel to Canada, but he only has USD and Canada uses CAD. 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 = 5 GBP 3 GBP = 5 CAD 2 GBP = 5 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 900 USD, what is the greatest amount of CAD he can get? A. 2000 B. 2500 C. 3000 D. 3600 E. 5000 Solution: 5000 Adam will get the most CAD is he uses the following exchanges: USD GBP HKD CAD. Then, 900 USD 1500 GBP 3750 HKD 5000 CAD.
15. An ant is standing on a cube with a side length of 3. The ant has to get from one corner to the opposite corner, but can only travel along the surface of the cube. What is the shortest distance the ant can travel? A. 3 2 B. 3 3 C. 3 5 D. 6 E. 9 Solution: 3 5 Draw the net of the cube. The closest way to get to the opposite vertex of a cube is through the diagonal of a 6 x 3 rectangle. This length is 3( 2 2 + 1 2 ) = 3 5. 16. 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 is 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 B. 3 C. 4 D. 6 E. 9 Solution: 6 ( r + 6 ) 2 = 4 r r + 6 = 2r r = 6 17. I have a dysfunctional calculator that can only divide by 3 or subtract 1. What is the minimum amount of steps it takes to get from 2018 to 0? A. 14 B. 16 C. 18 D. 20 E. 22 Solution: 16 Whenever you can divide by 3, use that function since it decreases the value faster. Then our numbers will be: 2018, 2017, 2016, 672, 224, 223, 222, 74, 73, 72, 24, 8, 7, 6, 2, 1, 0, for a total of 16 steps. 18. Determine the number of digits that 10! = 10 9 8 7 6 5 4 3 2 1 has. A. 3 B. 4 C. 7 D. 10 E. 11 Solution: 7 10!=3628800, which has 7 digits. A second way to do it is by eliminating the other answers. 10! is less than 10 digits, since it is less than 10 9. It is also more than 4 digits, since 8!=40320 has more than 4 digits already. 19. Al is taking a test he did not study for. He can only correctly answer 10% of the problems, and will guess the answer for the rest. If the test is 120 multiple choice with four choices per answer, how many problems is he expected to get right? A. 12 B. 30 C. 36 D. 39 E. 42
Solution: 39 For 12 questions, he will answer them right. For the other 108, he will have a 1/4 chance to get them right. This means he has an expected value of 12 + 108 4 = 39 problems. 20. What is the area of the triangle with vertices A(5, 4, 3), B(5, 0, 0), C(17, 0, 0)? A. 30 B. 36 C. 45 D. 48 E. 60 Solution: 30 Graphing the points then dropping a perpendicular from A to the xy-plane and connecting point B to that perpendicular will reveal AB to be 5. BC is 12, and ABC is a 5-12-13 right triangle, meaning the area of this triangle is 30. 21. Anna and Elsa are tour guides. They want to split 8 tourists into two groups. How many ways can they split up the tourists if each group has at least 1 person? A. 84 B. 126 C. 196 D. 216 E. 254 Solution: 254 For each tourist, you can assign him/her to either Anna s or Elsa s tour group, so there are 2 ways for each person, which makes it 2 8 = 256 possibilities. However, you can t have all 8 tourists to either tour guide, so you subtract 2 to get 254. 22. When Adam is asked to create a random number, he first checks the clock and takes note of the two digits in the minute s portion(ex. if the time is 4:23, he notes down 2 and 3). If the numbers are noted as a and b, his random number will be a b + a + b. Which of the following answers cannot be his random number at any given time? A. 34 B. 40 C. 41 D. 44 E. 47 Solution: 40 a b + a + b = (a + 1) (b + 1) 1. If we add 1 to all the choices, we get 35, 41, 42, 45, and 48. 41 is a prime number, so you cannot get two numbers (other than 1 and 41) to multiply to that.then 41-1=40 is our answer. 23. Let S = 1 + 2 + 3... + 999. What is the sum of the digits of S? A. 10 B. 15 C. 27 D. 30 E. 40 Solution: 27 The sum is equal to 999*500, which is divisible by 9. Then the sum of the digits is also divisible by 9, so it has to be 27. 24. Let n = 2018 3 + 3 2018. Find the units digit of n 3. A. 1 B. 3 C. 5 D. 7 E. 9
Solution: 1 The units digit of 8 3 is 2. The units digits for the powers of 3 have a repeating pattern with 3, 9, 7, 1. Since 2018 has a remainder of 2 when dividing by 4, the units digit of 3 2018 is 9. Then 9+2=11, which has a units digit of 1. 25. Adam and Allen are two ants who start on adjacent corners of a square. After a minute passes, they both randomly walk to an adjacent corner of the square. If Adam or Allen happen to ever cross each other on their path to the next corner, they fall off the square, ending their journey. What is the probability that Adam and Allen will not encounter each other for 4 minutes? A. 1 2 B. 4 9 C. 9 16 D. 16 81 E. 81 256 81 Solution: Label the corners 1-4. Adam is on corner 1, Allen is on corner 2. After the first turn, 256 either the positions are (2,3), (4,3) or (4,1). The other option is (2,1), which is a fail case. All three of these positions are identical to the original position, so there is another 3 4 chance they will survive again. After this point, we can see that there is always a 3 4 chance that they will pass given any amount of turns if they start adjacent to each other. Therefore, the chance they won t fall off within 4 minutes is (3/4) 4 = 81/256
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 SCORED. 1. What is 37 7 + 3 37? Solution: 370 37 7 + 3 37 = 37 (7 + 3) = 37 10 = 370 2. A set of numbers is formed by taking the product of two integers which are 6 apart from each other. What is the minimum possible value of a number in the set? Solution: 9 If x is one number, then 6 + x is the other number. Then we want to minimize x(x + 6), which is obtained when x = 3, for a minimum of 9. 3. The number 201 can be written as the sum of two primes. What is the difference between these two primes? Solution: 197 201 is odd, so it must be the sum of an odd and an even number. 2 is the only even prime, so the other number must be 199. Then the answer is 199 2 = 197. 4. Bessie the Cow loves going for car rides. Bessie first travels at 28km/hr for 30 minutes, then 35km/hr for a distance of 7km. What is her average speed the entire ride? Solution: 30 In the first portion, Bessie travels 28 1 2 = 14 km. In the second portion, Bessie travels for 7 35 = 1 5 of an hour. Then in total, Bessie travels for 14+7=21 km in a time of 1 2 + 1 5 = 7 10 hours, so her average speed is 30 km/hour. 5. What is the tens digit of 5! + 6! + 7! +... + 999!? (The! is a factorial, and n! = n (n 1) (n 2) 2 1. For example, 4! = 4 3 2 1 = 24) Solution: 8 5! has a tens digit of 2. 6! has a tens digit of 2. 7! has a tens digit of 4. 8! has a tens digit of 2. 9! has a tens digit of 8. All factorials 10! and higher have a tens digit of 0, since there are at least two factors of 10. Then the final answer is 8.