November 8th, 2018 Sprint Round Problems 1-30

Transcription

1 November 8th, 2018 Sprint Round Problems 1-30 HONOR PLEDGE I pledge to uphold the highest principles of honest and integrity as a mathlete. I will neither give nor accept unauthorized assistance of any kind. I will not copy another s work and submit it as my own. I understand that any competitor found to be in violation of this honor pledge is subject to disqualification. Signature Print Name School DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of 30 problems. You will have 40 minutes to complete all problems. You are not allowed to use calculators, books or other aids during this round. If you are wearing a calculator wrist watch, please give it to your proctor now. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers on your answer sheet. If you complete the problems before time is called, use the remaining time to check your answers. In each written round of competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted.

2 1. There are 72 people in the Blue Valley West Math Club, and 38 are girls. What is the probability of choosing a girl when picking a random member of the club? Express your answer as a common fraction in simplest form. 2. If Andy can bench 300 pounds successfully with a probability of 10% each time, what is the probability that he fails to lift 300 pounds? Express your answer as a percent. 3. If = 17 and = 4, what is the value of the expression 3 x 4 x? 4. If Rohit is paid 0.01 cents per second of work, how much is he paid for an hour of work in cents? 5. The mean of four distinct positive integers is 10. If one of the numbers is 22, what is the sum of the other three numbers? 6. If Shivam flips a coin 2 times, what is the probability of getting at least one heads? Express your answer as a common fraction.

3 7. What is the slope of the line containing points (-1, -1) and (2, 3) on a standard coordinate graph? Express your answer as a common fraction. 8. At Pratik's school, which has 1000 students, 30% of the students are sick. 50% of the students who are sick have the flu. How many students do not have the flu? 9. Ben is on a road trip driving from Kansas City to Houston. He travels a total of 650 miles in 13 hours. What is his average speed in miles per hour for the entire trip? 10. The sum of the positive integers from 1 to n is equal to 91. What is the value of n? 11. Ricky went to piano class on a Thursday. If his next piano class is 11 days after his class on Thursday, on what day of the week will his next class be? 12. What is the sum of the factors of 12? 13. A triangles vertices lie on the points (1,3), (2,5), and (-1,4). What is the area of the triangle? Express your answer as a common fraction.

4 14. How many perfect cubes are factors of 1 4 x 2 4 x 3 4 x 4 4 x 5 4 x 6 4? 15. What value of b makes the equation 62b (base b) = 267 (base 7) true? [NOTE: This question has no correct answer and was therefore thrown out during the actual competition] 16. What is the sum of the roots of the equation y = 2x 2 + 4x? 17. What is the area of the largest rectangle that can be inscribed in a triangle with sides 5, 12, and 13? 18. Rohit is 3 times older than Ricky, who is 2 years younger than Abi. If Rohit s age will be double Abi s age in 12 years, how old is Ricky right now? 19. What is the last digit of 3 400?

5 20. How many ways are there to give 6 pieces of indistinguishable candy to 4 different children if every child receives at least one piece of candy and all pieces of candy must be given out to the children? 21. How many positive integers that are factors of 6 12 are also factors of 12 6? 22. Abi randomly selects two real numbers, x and y, between 0 and 10. He then calculates the probability that the quotient x/y < A, where A is some constant that satisfies the condition 0 < A < 1. If he can express this probability as ca, where c is some constant, what is the value of c? Express your answer as a common fraction. 23. Rohit selects two integers, x and y, such that (x + 5) 2 + (y - 12) 2 = 169. What is the greatest possible value of x 2 + y 2? 24. What positive, real value of x satisfies the equation: x x x = 0? 25. If f(x) = x 2 a 2, where a is a positive integer less than 20, find the value of a that would result in f(20) having exactly 20 factors.

6 26. A circle is inscribed in an equilateral triangle. The area of the region inside the triangle and outside the circle is 6 feet squared. What is the length of one side of the equilateral triangle? Round your answer to the nearest integer. 27. Alan has 14 feet of wood. He is trying to make triangles with all 14 feet of the wood where all 3 sides of the triangles he makes have positive integer values. How many distinct, non-congruent triangles with distinct side lengths can he make? 28. If AB = 30, BD = 36, AD = 50, and C is the midpoint of BD, then what is the length of segment AC? Express your answer in simplest radical form. 29. Ria has 10 cards, of which 5 are distinguishable green ones and 5 are distinguishable blue ones. Ria randomly chooses 5 cards and lays them side by side in a random order. She will be mad if more than two of the same color card are adjacent to each other. For example, she will be mad if the card order is BBBGB, but she will not be mad if the order is BBGBB. What is the probability that Ria will not be mad? Express your answer as a common fraction. 30. What is the sum of all values of x that satisfy?