MOCK MATHCOUNTS Chapter Competition Sprint Round Problems Signature Date. Printed Name School

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1 0 MOCK MATHCOUNTS 018 Chapter Competition Sprint Round Problems 1 30 HONOR PLEDGE I pledge to uphold the highest principles of honesty and integrity. 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 pledge is subject to disqualification. Signature Date Printed Name School Team 1 or Team or Team 3 (circle one) 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 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 the time is called, use the remaining time to check your answers. In each written round of the competition, the required unit for the answer in 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 Produced by - Junwon Lee - Ted Lin - Curtis Liu - George Harpole - Jocelyn Hsu - Alvin Kim - Nathan Kim - Edward Xu - Eric Xu - Daniel Yang Copyright South Bay MOCK MATHCOUNTS 017. All rights reserved.

2 1. $ If computers are worth 500 dollars and 3 monitors are worth 90 dollars, what is the cost of 1 computer and monitors in dollars? 0.1. What is the value of? Express your answer as a common fraction in simplest form The sum of two prime numbers is 01. What is the value of the greater prime number? 4. cups A 5 pound sack of sugar was donated to a birdwatcher s club for their members to mix hummingbird nectar. The nectar recipe calls for a hot water to sugar volume ratio of 4 to 1. One cup of sugar weighs 0.4 pounds. How many cups of hot water are needed to mix the entire sack of sugar? 5. years Six years ago, Bill s dad was 5 times as old as Bill. If his dad is 41 now, how old is Bill? Copyright South Bay MOCK MATHCOUNTS 017. All rights reserved. 018 Sprint Round

3 6. Devin has twelve homework papers for every textbook he owns. If he has a sum total of 91 homework papers and textbooks, how many homework papers does Devin have? 7. How many positive integers are factors of 36 or 63, but not both? 8. Alvin has taken four tests so far in his math class, getting a 86, 88, 88, 89 on his tests. What is the lowest score that Alvin can get on his next test in order to raise his average test score to at least a 90? 9. The shortest possible route from Earth to Mars is 34 million miles. If a spacecraft, traveling this route, leaves Earth at 1:00 AM, Monday, maintaining an average speed of 34,000 miles per hour, on what day of the week would the craft reach Mars? 10. A standard 6-sided die is rolled. If the number on the die is odd, it is rolled again. If the number is even, nothing happens. What is the probability that the first or second roll is a 6? Express your answer as a common fraction in simplest form. Copyright South Bay MOCK MATHCOUNTS 017. All rights reserved. 018 Sprint Round

4 11. What is the measure of angle A, in degrees, given that lines x and y are parallel? x 30 A y Candle A is 4 inches long and candle B is 1 inches long. Candle A melts down 8 inches per hour when lit, while candle B melts down 5 inches per hour when lit. Candle A is lit on 3:00 PM. At what time must candle B be lit so that Candle A and B completely melt simultaneously? x, if x 5 x, if 18 x If f( x), what is the value of x 9, if 0 x 18 x, if x 0 f( f( f( f( f( f( f(19)))))))? 14. Circles A and B overlap each other and share an area K. K is of circle A 1 5 and of circle B. What is the ratio of the area of circle A to the area of 8 circle B? Express your answer to the simplest fraction. 15. Consider the following two sequences: an arithmetic sequence, a, in which a 1 = and a = 4, and a geometric sequence, b, in which b 1 = and b = 4,. Compute a 10 + b 10. Copyright South Bay MOCK MATHCOUNTS 017. All rights reserved. 018 Sprint Round

5 16. Six people are evenly divided into three groups. How many different ways can the three groups be formed? 17. An equilateral triangle is circumscribed by a circle. If the ratio of the triangle s perimeter to the circle s circumference is a, and the ratio of the triangle s area to the circle s area is b, then what is the value of a/b? 18. What is the remainder when is divided by 7? 19. A square with side length 6 is inscribed in a square with side length 8. A vertex of the smaller square divides a side of the larger square into two segments of lengths a and b. What is ab? a b What is the value of 3 3? Copyright South Bay MOCK MATHCOUNTS 017. All rights reserved. 018 Sprint Round

6 1. Find the smallest positive integer x such that (x!)(16!) is divisible by What is the smallest positive integer n such that 6n is a perfect square and 8n is a perfect cube? 3. In triangle ABC, BC = 4, CA = 5, and AB = 6. Draw three circles of radius two, each centered on a different vertex of the triangle. What is the sum of the area of the overlapping regions of the triangle and each of the three circles? What is the value of x if x 3? x x 5. If 1009 f ( x ) f ( x ) 5, then what is the value of f (018)? x Copyright South Bay MOCK MATHCOUNTS 017. All rights reserved. 018 Sprint Round

7 6. Let f (n) = (n 7) (n + 3) +7(n + 3) + n + n. f (109) is the product of two large prime numbers. Find the larger prime number. 7. Jack plays a game where if he flips heads he moves 1 foot to the west, while if he flips tails he moves 1 foot to the east. If Jack flips six coins, what is the probability that Jack ends up in his starting position? 8. Evaluate the following expression as a common fraction in lowest terms Quadrilateral ABCD is inscribed in a circle of radius 6. If BDA = 40 and AD = 6, what is the measure of BAD in degrees? D A B C What is the value of? Express your answer as a common fraction in lowest terms. Copyright South Bay MOCK MATHCOUNTS 017. All rights reserved. 018 Sprint Round

8 Forms of Answers All answers must be expressed in simplest form. A common fraction means a fraction in the form. a..., b where a and b are natural numbers and GCF(a,b) = 1. Examples: Problem: Express 8 divided by 1 as a common fraction. Answer: 4 3 Unacceptable: 6 Problem: Express 1 divided by 8 as a common fraction. Answer: 3 Unacceptable: 1 8, 1 1, 1.5 Ratios shall be expressed as simplified common fractions unless otherwise specified. Radicals must be simplified. A simplified radical must satisfy the following: 1) Radicands cannot have a factor with the root of the index (e.g., no 4 or 9 factors inside a square root). ) No radicands contain fractions. 1 3) No radicals appear in the denominator of a fraction (e.g., is not allowed; is allowed). Example: Problem: Evaluate Answer: 5 3 Unacceptable: 75 Dollar amount answers shall be expressed in the form ($) a.bc, where a is an integer and b and c are digits. The only exceptions to this rule are when a is zero, it may be omitted, or when b or c are both zero, they may be omitted. Examples: Acceptable:.35, 0.38,.38, 5.00, 5 Unacceptable: 4.9, 8.0 Units of measure are not required in answers, but they must be correct if given. When a problem asks for an answer expressed in a specific unit of measure, or when a unit of measure is provided in the answer blank, equivalent answers in other units are not acceptable. For example, if a problem asks for the number of ounces and 36 oz is the correct answer, lbs 4 oz will not be accepted. If a problem asks for the number of cents and 5 cents is the correct answer, $0.5 will not be accepted. Do not make approximations for numbers (e.g., π,, ) in the data given or the answer unless the problem says to do so, such as when it says to round to a specific decimal place. Do not do any intermediate rounding (other than what the calculator performs) when calculating solutions. Any rounding should be done at the end of the calculation process, and only when the problem requests it. An answer expressed to a greater or lesser degree of accuracy than called for in the problem will not be accepted. Whole number answers should be expressed in their whole number form. Thus, 5.0 will not be accepted for 5, and 5 will not be accepted for 5.0. The plural form of the units will always be provided in the answer blank, even if the answer appears to require the singular form of the units. Answers must be written on the line after the problem number. Any other blank space may be used as needed to solve the problems, but will be ignored for grading. South Bay MOCK MATHCOUNTS is not affiliated with the MATHCOUNTS organization. Still, the South Bay Invitational MOCK MATHCOUNTS Competition is practice in preparation for the Chapter level MATHCOUNTS Competition, so the same rules are used.

2018 Sprint Solutions

2018 Sprint Solutions 08 Sprint s. $ 30 If computers are worth 00 dollars and 3 monitors are worth 90 dollars, what is the cost of computer and monitors in dollars? One computer is worth 00/ = 0 dollars and monitors are worth