2016 School Competition Sprint Round Problems 1 30

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1 Name 2016 School Competition Sprint Round Problems DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. 9 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. 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 Correct Scorer s Initials National Sponsors Raytheon Company Northrop Grumman Foundation U.S. Department of Defense National Society of Professional Engineers CNA Foundation Phillips 66 Texas Instruments Incorporated 3Mgives Art of Problem Solving NextThought Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation Copyright MATHCOUNTS, Inc All rights reserved.

2 1. C What is the temperature according to this thermometer that measures temperature in degrees Celsius? $ Cory has a summer job at the local grocery store. How much will Cory get paid to work 20 hours each week for 8 weeks if he earns $10 an hour? 3. What is the value of 21 divided by 7 8? T E 4. percent Teachers The graph shows the results when 100 teachers reported the number of years they have been teaching. What percent of these teachers have been teaching more than 5 years but no more than 15 years? Years 5. Let a b = 2a 3b. What is the value of (1 2) 7? 6. If 8 14 = 4, what is the value of d? d Copyright MATHCOUNTS, Inc All rights reserved School Sprint Round

3 7. dots The figure shows the first four stages of a dot pattern. If the pattern continues, how many dots will be in the figure for Stage 7? Stage 1 Stage 2 Stage 3 Stage 4 8. $ According to this order form for Tessa s art supplies, the total cost for the 20 items she ordered was $ Some of the details of Tessa s order are missing from the order form, but based on the information given, what was the unit price for each sketch pad Tessa ordered? ORDER FORM I Q U P T C Paint Brush $ 8.00 $ Acrylic Paint 10 $ $ Sketch Pad 4 $ $ 20 $ feet A length of 240 feet of fencing is to be supported by a total of 13 evenly spaced fence posts, including one at each end, as shown. How far apart must the fence posts be spaced? 10. units 2 This coordinate grid shows pentagon ABCDE with vertices A(3, 0), B(4, 0), C(4, 7), D(0, 7) and E(0, 5). What is its area? Express your answer as a decimal to the nearest tenth. D E y C A B x 11. cookies Maria baked a batch of cookies. She ate three cookies and then 4 5 were left. How many cookies did Maria bake? of the cookies 12. If + = + + and = 16 ( + ), what is the value of +? Copyright MATHCOUNTS, Inc All rights reserved School Sprint Round

4 13. pounds The weights of four students on a wrestling team are 100 pounds, 108 pounds, 113 pounds, and 119 pounds. What is the mean weight of the four students? 14. inches A 5-foot ribbon is cut into two pieces with lengths in the ratio 1:4. How many inches long is the shortest piece? 5 feet 15. gallons While jogging, Cher s heart pumped 450 gallons of blood in 1 hour, 15 minutes. On average, how many gallons of blood did her heart pump each minute? y (, ) What are the coordinates of the image of P, when this figure is reflected across the y-axis? Express your answer as an ordered pair P 2 4 x 17. boxes The glee club had a fund-raiser selling cookies to raise money for a spring concert. The club kept half of the money collected from selling cookies for $5 per box. How many boxes of cookies were sold if the glee club kept $1250? 18. cm The length of a rectangle is twice its width. The rectangle has an area of 128 cm 2. What is its perimeter? Copyright MATHCOUNTS, Inc All rights reserved School Sprint Round

5 19. integers How many two-digit positive integers less than 25 have the property that they are divisible by each of their digits? 20. The three-digit integer 5A4 is a multiple of six. What is the sum of all the possible values for the digit represented by A? 21. coins The total cost of Keegan s lunch was $4.56. He paid with a 5-dollar bill and received exact change. What is the fewest number of coins Keegan could have received? 22. The line passing through ( 3, 2) and (7, 1) intersects the x-axis at the point (a, 0). What is the value of a? Express your answer as a common fraction. 23. $ The prize money in a science fair is divided among the top three winners so that the amount awarded for first place is four times the amount for third place, and the amount for second place is three times the amount for third place. If the total amount of prize money is $2400, what is the amount awarded for second place? 24. degrees B C In right triangle ABC, shown here, m A = 40, and D is on side AC such that segment BD bisects CBA. What is the degree measure of BDC? 40 D A Copyright MATHCOUNTS, Inc All rights reserved School Sprint Round

6 Addie s one-quart fruit salad is strawberries and blueberries. Ashlyn s one-quart fruit salad is strawberries and blueberries. When Addie and 2 2 Ashlyn combine their fruit salads, what portion of the mixture is strawberries? Express your answer as a common fraction. 26. triangles How many triangles are in the figure shown here? 27. J.R.R. Tolkien, author of The Hobbit, stated in 1970 that on one of his birthdays the age he turned was the square root of the year. In what year was he born? 28. hours Twins Cara and Dara drove from their house to the beach in exactly 5 hours, not including rest stops. First, Cara drove at an average speed of 40 mi/h and stopped at a rest area located halfway between their house and the beach. Then Dara drove from that location to the beach at an average speed of 60 mi/h. For how many hours did Dara drive? 29. units 3 A D E B C F For the rectangular solid shown, AB = 12 units, BC = 8 units and AE = 6 units. What is the volume of the tetrahedron DHBC? H G 30. Bobby attends an event along with two friends. At the event, there is a drawing for three door prizes. Each person in attendance has an equal chance of winning a prize and no one can win more than one prize. If there are 20 people at the event, including Bobby and his two friends, what is the probability that Bobby and both of his friends each will win a prize? Express your answer as a common fraction. Copyright MATHCOUNTS, Inc All rights reserved School Sprint Round

7 2016 School Competition Target Round Problems 1 & 2 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of eight problems, which will be presented in pairs. Work on one pair of problems will be completed and answers will be collected before the next pair is distributed. The time limit for each pair of problems is six minutes. The first pair of problems is on the other side of this sheet. 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 left-hand column of the problem sheets. If you complete the problems before time is called, use the time remaining to check your answers. Problem 1 Problem 2 Scorer s Initials National Sponsors Raytheon Company Northrop Grumman Foundation U.S. Department of Defense National Society of Professional Engineers CNA Foundation Phillips 66 Texas Instruments Incorporated 3Mgives Art of Problem Solving NextThought Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation Copyright MATHCOUNTS, Inc All rights reserved.

8 1. dimes Kelly has $12.00 in dimes and quarters. She has six more quarters than dimes. How many dimes does Kelly have? 2. day The height of a bean plant is measured daily. At the end of the second day of observations, it measured 1 inch in height. Thereafter, its height doubled every two days. At the end of which day will the plant measure exactly 16 inches tall? Copyright MATHCOUNTS, Inc All rights reserved School Target Round

9 2016 School Competition Target Round Problems 3 & 4 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Problem 3 Problem 4 Scorer s Initials National Sponsors Raytheon Company Northrop Grumman Foundation U.S. Department of Defense National Society of Professional Engineers CNA Foundation Phillips 66 Texas Instruments Incorporated 3Mgives Art of Problem Solving NextThought Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation Copyright MATHCOUNTS, Inc All rights reserved.

10 3. students Koda teaches a total of 25 students in three different swimming classes. He teaches his first two classes on Friday and his third class on Saturday. There are seven students in Koda s Saturday class. What is the average number of students in Koda s Friday classes? 4. What is the least positive integer n such that the product of 2016 and n ends in the digits 00? Copyright MATHCOUNTS, Inc All rights reserved School Target Round

11 2016 School Competition Target Round Problems 5 & 6 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Problem 5 Problem 6 Scorer s Initials National Sponsors Raytheon Company Northrop Grumman Foundation U.S. Department of Defense National Society of Professional Engineers CNA Foundation Phillips 66 Texas Instruments Incorporated 3Mgives Art of Problem Solving NextThought Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation Copyright MATHCOUNTS, Inc All rights reserved.

12 5. inches A cubic yard of sand is spread uniformly to completely cover a rectangular path that is 3 feet wide and 24 feet long. How many inches deep is the sand that covers this path? Express your answer as a mixed number. 6. paths In the figure shown, how many paths are there that move only upward from A to B? B A Copyright MATHCOUNTS, Inc All rights reserved School Target Round

13 2016 School Competition Target Round Problems 7 & 8 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Problem 7 Problem 8 Scorer s Initials National Sponsors Raytheon Company Northrop Grumman Foundation U.S. Department of Defense National Society of Professional Engineers CNA Foundation Phillips 66 Texas Instruments Incorporated 3Mgives Art of Problem Solving NextThought Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation Copyright MATHCOUNTS, Inc All rights reserved.

14 7. sets With a $20 bill, Britt can buy 40 pencils, or 25 erasers, or 80 bookmarks, with no change left over. If Britt wants to use $20 to make sets consisting of one pencil, one eraser, and one bookmark each, how many complete sets can he buy? 8. ft 2 A piece of wire measuring 48 feet long is cut into two pieces of equal length. One piece is bent into an equilateral triangle and the other is bent into a circle. What is the combined area of the two shapes? Express your answer to the nearest whole number. Copyright MATHCOUNTS, Inc All rights reserved School Target Round

15 2016 School Competition Team Round Problems 1 10 Team Members, Captain DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of 10 problems which the team has 20 minutes to complete. Team members may work together in any way to solve the problems. Team members may talk to each other during this section of the competition. 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. The team captain must record the team s official answers on his/her own competition booklet, which is the only booklet that will be scored. If the team completes the problems before time is called, use the remaining time to check your answers. Total Correct Scorer s Initials National Sponsors Raytheon Company Northrop Grumman Foundation U.S. Department of Defense National Society of Professional Engineers CNA Foundation Phillips 66 Texas Instruments Incorporated 3Mgives Art of Problem Solving NextThought Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation Copyright MATHCOUNTS, Inc All rights reserved.

16 1. grams A particular coin that is composed only of silver and copper has a mass of 5 grams. If copper accounts for 9 of the coin s mass, what is the mass of the 10 silver in the coin? Express your answer as a decimal to the nearest tenth. 2. ordered pairs How many ordered pairs of integers (a, b) satisfy a b = 60? 3. $ The final bill for the Freeman s dinner, including a 20% tip, was $72. What was the total cost of the Freeman s dinner before the tip was added? 4. marbles Tucker has a jar containing 4 green and 5 blue marbles. What is the minimum number of marbles Tucker needs to remove from the jar, without replacement, to guarantee that he has at least four marbles of the same color among those removed? 5. in 2 A rectangular piece of paper measures 8 inches by 4 inches. The paper is folded along a crease from the upper left corner to a point located 3 inches from the lower left corner along the bottom edge, creating a right triangle as shown. What is the area of the shaded region? Copyright MATHCOUNTS, Inc All rights reserved School Team Round

17 6. The product of three consecutive prime numbers is 105. What is their sum? 7. hits Everett s batting average is after getting 120 hits in 400 times at bat. How many hits must he get in his next 100 times at bat to raise his average to 0.400? 8. The lines given by the equations y = x + 1 and y = ax + b are perpendicular and intersect at the point (2, 3). What is the value of a + b? 9. ft 3 The Farmington family has two different silos on their farm. The first silo is a circular cylinder of height 20 ft with parallel bases of diameter 14 ft. The second silo is a cylinder of height 22 ft topped by a cone of height 10 ft, both with bases of diameter 12 ft. What is the absolute difference between the volumes of the two silos? Express your answer in terms of π. 10. Regular octagon ABCDEFGH with side length 2 units is placed on a coordinate plane such that side AB lies on the positive y-axis and side CD lies on the positive x-axis. What is the absolute difference between the x- and y-coordinates of F? Express your answer in simplest radical form. Copyright MATHCOUNTS, Inc All rights reserved School Team Round

2019 State Competition Sprint Round Problems 1 30

2019 State Competition Sprint Round Problems 1 30 1 19 State Competition Sprint Round Problems 1 3 HONOR PLEDGE I pledge to uphold the highest principles of honesty and integrity as a Mathlete. I will neither give nor accept unauthorized assistance of