Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Gauss School and Gauss Math Circle 2016 Gauss Math Tournament Grade 5-6 (Sprint Round 50 minutes) 1. What is (1003)(997)-(1008)(992)? 2. Solve the system of equations to get (x, y) a. 3x+7y=38 b. x+10y=44 3. a/b=2, b/c=3. Find (a+c)/b. 4. Bob has a picture of dimensions 6 inch by 10 inch to which he has a ½ inch border made. What is the ratio between the area of the border and the area of the picture? 5. A plane makes a round trip from the East Pole to the West Pole. However, the whole time, there is a constant strong wind going from east to west, making the trip from east to west take 10 hours, and the trip west to east take 12 hours. If the West Pole is 21,000 miles away from the East Pole and the plane’s speed is constant, find the speed of the plane in mph. 6. What is x/y if (9x-5y)/(3x+11y) = ⅓ 7. The sum of the squares of two numbers is 205 and their product is 78. Find the larger of the 2 numbers. 8. Jane starts with 720 cookies and then gives ⅕ of them to Tiffany. Jane then gives ⅓ of the remaining cookies to Lily and splits the rest with her sister. How many cookies does Jane end up with? 9. In the school of West Pageborough, there are 253 students, each of which takes orchestra, ba(n)d, both or none. If there are 163 orchestra students and 111 band students, and 15 take none, how many students take both orchestra and band? 10. In circle O with radius 5, minor arc EF measures 2pi, what is the angle measure of <EOF? 11. Points E and F are the midpoints of AB and CD in rectangle ABCD, respectively. DAEF is similar to ABCD. If AB =16, what is the area of ABCD? www.gaussmath.org 12. What is the side length of a cube whose volume is numerically 3 times its surface area? 13. What is the sum of the external angles of a regular pentagon? 14. A rubber band is wrapped tightly around the outside of three tangent circles with radius 2. What is the length of the rubber band? 15. In triangle ABC, AB=BC=1, and angle B is a right angle. Isosceles right triangles are constructed on the hypotenuse of the previous triangle. Shown are the first 3 triangles. How many times larger is the area of the 10th triangle than the 1st triangle. 16. An icecream cone of radius 6 cm and height 20 cm is filled with liquefied strawberries until the depth of the liquefied strawberries is ½ of the height of the cone. What is the volume of the liquefied strawberries? 17. How many ways are there to rearrange the letters of the word GAUSS? 18. David is on the coordinate plane, and can only move 1 unit at a time in the positive x or y directions. How many ways are there for David to get from (0, 0) to (3, 5)? 19. How many triangles are in the following diagram? 20. What is the chance of drawing two red marbles from a bag of 8 marbles with 5 red marbles? www.gaussmath.org 21. How many 3 digit numbers satisfy the condition that the tens digit is the largest, and the hundreds digit is smaller than the ones digit? 22. License plates are made by 3 digits (0 through 9) followed by 2 letters, but the letters cannot both be consonants (letters other than “a, e, i, o, or u”). How many different license plates can be made? 23. A circle with is inscribed inside an equilateral triangle with side length 6in. What is the probability that a random point chosen inside the triangle is within the circle? 24. What is the probability that when you roll 2 dice, the sum of the numbers you roll is 8? 25. From a gumball machine, there is a 20% chance you will roll a green gumball. If you roll the gumball machine 5 times, what is the chance you will get exactly 2 green gumballs? 26. In Matthew’s wardrobe, there are 5 different T shirts, 3 different pairs of pants and 4 different pairs of socks, and 2 types of Gauss makeup. How many outfits consisting of one T shirt, two pairs of pants and one pair of socks, and one application of makeup can Matthew make? 27. Two points C and D are selected from a line segment AB with C between A and D, and D between C and B. What is the probability that AC+DB>CD. 28. March 6th, 2019 is a Wednesday. What day of the week will it be 192 days from then? 29. Find the number of divisors of 2016. 30. How many integers between 1 and 50 inclusive are divisible by 2 or 3 (but not both)? 31. How many two-digit numbers with 6 divisors are there? 32. How many different ordered pairs (A, B) are there such that A and B are digits and make 12A73B divisible by 11? 33. What is the last digit of 13332016 ? 34. A number with 34 factors is squared. How many factors does this new number have? www.gaussmath.org 35. A man who loves chocolate buys 100lbs of it. Every day, he eats 2/7 of the chocolate. However, when there is less than 3 pounds of chocolate remaining, he will eat all of it. How many days will it take him to finish the chocolate? 36. A ball is dropped vertically from 40 feet. When the ball bounces, it only bounces up to ½ the height of the previous bounce. How many feet would the ball have traveled after hitting the ground for the 3rd time? 37. 4 lines and a circle are drawn on a plane. What is the maximum number of intersections possible? 38. in triangle ABC with B=90. AD=4, DC=9, the altitude from B to AC intersects AC at D. What is length of BD? 39. Let r and s be the roots of the equation 40. If x and y are integers such that solutions are there for x? Sprint Round Ends www.gaussmath.org . Find r+s? , how many distinct Gauss School and Gauss Math Circle 2016 Gauss Math Tournament Grade 5-6 (Target Round 20 minutes) 1. In triangle ABC with B=90. The altitude from B to AC intersects AC at D AD=4, DC=9. What is length of BD? 2. David can eat 6 hot dogs in 3 minutes. Lawrence can eat 2 hot dogs in 4 minutes. Rachel can eat 4 hot dogs in 6 minutes. Finally, William can eat 1 hot dog in 9 minutes. Working together, how many minutes will it take for the four people to eat 20 hot dogs? Express your answer to the nearest tenth. 3. A classroom has 10 boys and 12 girls. How many ways are there to pick a group of 4 students such that not all of them are the same gender? 4. How many ways are there to put a red ball, a blue ball, a green ball, and a yellow ball into 2 indistinguishable boxes? 5. Three faces of a rectangular prism have areas 16, 20, and 45. What is the volume of this prism? 6. Circles O and P are congruent circles with radius 3 and intersect at points A and B. If AB = 3, what is the area of the intersection? Express your answer to the nearest hundredth. 7. How many 8 digit integers are not palindromes? A palindrome is a number that is the same when reading forwards to backwards as when it is read backwards to forwards. For example, 14541, 108801, 5, and 104929401 are all palindromes. 8. A sev-peating number is an integer consisting of only the digit 7. What is the smallest integer n such that a sev-peating number with n digits is divisible by 133? Target Round Ends www.gaussmath.org Name: ______________________________ Grade:___________________________________ Sprint Round Answers: 1 21 2 22 3 23 4 24 5 25 6 26 7 27 8 28 9 29 10 30 11 31 12 32 13 33 14 34 15 35 16 36 17 37 18 38 19 39 20 40 Target Round Answers: 1 5 2 6 3 7 4 8 www.gaussmath.org G5-6 Answer Keys: Sprint Round: 1. 55 2. (10/3, 4) 3. 7/3 4. 1/3 5. 1925 6. 13/12 7. 13 8. 192 9. 36 10. 72 11. 256 root(2) 12. 18 13. 360 14. 12+4pi 15. 1024 16. 30pi 17. 60 18. 56 19. 16 20. 5/14 21. 84 22. 23500 23. pi/3root(3) 24. 5/36 25. 128/536 26. 120 27. ¾ 28. Saturday 29. 24 30. 33 31. 16 32. 5 33. 1 34. 99 35. 12 36. 100 37. 14 38. 6 39. -3 www.gaussmath.org 40. 8 Target Keys: 1. 2. 3. 4. 5. 6. 7. 8. 6 5.3 6610 8 120 1.63 89991000 18 www.gaussmath.org