MATHCOUNTS 2015 ██ Mock Chapter Competition ██ Sprint Round Problems 130 Name __________________________________________________________________ State ___________________________________________________________________ DO NOT BEGIN UNTIL YOU HAVE SET YOUR TIMER TO FORTY MINUTES. This section of the competition consists of 30 problems. You will have 40 minutes to complete all the 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 in the blanks in the left-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers. In each written round of the 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. Total Score Scorer’s Initials 1. dollars A banana costs 50 cents and an apple costs 75 cents. How much would it cost to buy 2 bananas and 3 apples? Round your answer to the nearest cent. 2. outfits Derek has 3 shirts, 4 pairs of pants, and 9 hats. How many different outfits can Derek make if each outfit must consist of a shirt, pair of pants, and a hat? 3. The midpoints of an equilateral triangle are connected to form a smaller triangle. What is the ratio of the area of the smaller triangle to the larger triangle? Express your answer as a common fraction. 4. What is the positive difference between the sum of the first 3 prime numbers and the sum of the first 3 composite numbers? 5. diagonals A diagonal is a segment that connects two vertices of a polygon that do not share a common side. How many diagonals does a regular hexagon have? 6. ways I have a collection of 10 books, and I want to choose 3 books to take with me to school. How many ways can I do this? 7. What is the average of the values of x that satisfy the equation x2 + 10x + 21 = 0? 8. The radius of circle Q is doubled to form a new circle R . What is the ratio of the circumference of circle Q to circle R ? Express your answer as a common fraction. 9. people 10 people at my school are in the art club, and 18 people are in either the art club or the math club, or both. There are also 3 people in both clubs. How many people are in the math club? 10. When 2015 is divided by a positive integer, the quotient is 201 and the remainder is 5. What is the positive integer? 11. mph Kenny drives a car for 125 minutes and travels a total of 50 miles. What is his speed in miles per hour? 12. Larry writes down all the positive divisors of 2015. What is the sum of all the numbers he wrote? 13. ways How many ways can I choose 2 positive integers such that the sum of these two numbers is less than 7? 14. A square has an area of 36 units and is inscribed in a circle. What is the area of the circle? Express your answer in terms of π. 15. What is the least common multiple of 20 and 15? 16. bananas You can trade 4 bananas for 9 oranges and 8 kiwis for 10 oranges. How many bananas can you trade for 72 kiwis? 17. What is the positive difference between the mean and the median of the set of the first 2015 positive integers? 18. I multiply all nonprime positive integers less than 10. What is the remainder when I divide this product by 10? 19. The bases of an isosceles trapezoid measure 20 inches and 15 inches. If the area of the trapezoid is 105 square inches, what is the perimeter? 20. An abundant number is a positive integer that when all its proper divisors (divisors of the number except for itself) are added up, the sum is greater than the number itself. How many 1digit positive integers are abundant numbers? 2 21. cm A rectangle has a perimeter of 90 centimeters. If all the side lengths are multiples of 3, what is the maximum possible area? 22. How many positive integers less than 1000 have 3 divisors? 23. How many 4digit positive integers are divisible by 13? 24. units A hexagonal prism has a height of 6 units and a regular hexagonal base with a side length of 4 units. What is the length of the longest segment that can fit inside this prism? 25. I originally have 10000 marbles. I give away 50% of what I have on Monday, 60% on Tuesday, 70% on Wednesday, 80% on Thursday, and 90% on Friday. How many marbles will I have after Friday? 26. A jar contains 2 red, 3 white, and 4 blue chips. I randomly take 3 chips out of the jar. What is the probability that I end up with 3 chips of different colors? Express your answer as a common fraction. 27. George graphs the equation x2 − 6x + y2 − 10y − 18 = 16. What is the area of this region? Express your answer in terms of π. 28. I toss a fair coin 5 times. What is the probability that there are more heads than tails? Express your answer as a common fraction. x 29. Define a function t(x) to be the units digit of 2 . What is t(1) + t(2) + t(3) + ...t(2014) + t(2015)? 3 30. units A triangle with points located at (20, 0), (20, 15), and (25, 0) is rotated about the yaxis to create a 3dimensional figure. What is the volume of this figure? Express your answer in terms of π. MATHCOUNTS 2015 ██ Mock Chapter Competition ██ Target Round Problems 18 Name __________________________________________________________________ State ___________________________________________________________________ DO NOT BEGIN UNTIL YOU HAVE SET YOUR TIMER TO SIX MINUTES. This section of the competition consists of eight problems, which should be worked in pairs. The time limit for each pair of problems is six minutes. The first pair of problems is on the next page. When told to do so, turn the page over and begin working. This round assumes the use of calculators, and calculations also may be done on scratch paper, but no other aids are allowed. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the lefthand column of the problem sheets. If you complete the problems before time is called, use the time remaining to check your answers. Total Score Scorer’s Initials 1. dollars A cell phone company charges a flat fee each month as well as 10 cents per a minute. In January, I payed $25 for using a total of 50 minutes. In February, I payed $40 for using a total of 200 minutes. I used a total of 300 minutes in March and April. How much money did I pay for March and April combined? Round your answer to the nearest cent. 2. The base of a prism is a regular 2015gon. What is the sum of the number of vertices, faces, and edges of this figure? 3. How many ordered pairs of nonnegative integers (a, b) have the property that 4. Express 0.2015201520152015… as a common fraction in simplest form. 5. Elli writes down all the digits of the positive integers less than 2015. Then she randomly selects a digit. What is the expected value of this digit? Express your answer as a common fraction. 6. How many 3digit palindromes have the property that the sum of their digits is equal to the product of their digits? 7. ways I wish to walk from (0, 0) to (5, 4) without passing through (2, 2) or (4, 3) on the Cartesian Plane. I may only move one unit up or one unit to the right at a time. How many ways can I do this? 8. The side lengths of a rectangular prism are a , a + 1 , and a + 2 , where a is a positive integer. What is a if the length of a space diagonal is a + 3?
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