MATHCOUNTS Chapter Competition Sprint Round Problems 1 30 DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.

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1 MATHCOUNTS 2006 Chapter Competition Sprint Round Problems 1 0 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of 0 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 right-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 Founding Sponsors National Society of Professional Engineers National Council of Teachers of Mathematics CNA Foundation National Sponsors ADC Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Northrop Grumman Foundation Raytheon Company Shell Oil Company Texas Instruments Incorporated M Foundation Xerox Corporation Copyright MATHCOUNTS, Inc All rights reserved.

2 1. Two identical blue boxes together weigh the same as three identical red boxes together. Each red box weighs 10 ounces. How much does one blue box weigh? B B R R R 1. ounces 2. Three black chips have been placed on the game board shown. What is the number in the square where a fourth chip should be placed so that it shares neither a row nor a column with any of the existing chips? What is the greatest number of cubes, with edge length 1 inch, that can be placed into a rectangular box measuring inches by inches by inches? 1. cubes 4. What fraction is equal to six times one-seventh? Express your answer as a common fraction Krista put 1 cent into her new bank on a Sunday morning. On Monday she put 2 cents into her bank. On Tuesday she put 4 cents into her bank, and she continued to double the amount of money she put into her bank each day for two weeks. On what day of the week did the total amount of money in her bank first exceed $2? According to the table below, what is the median value of the 5 salaries paid to this company s employees? 6. $ Position Title # of Employees with this Title Salary for Position President 1 $10,000 Vice-President 5 $0,000 Director 10 $75,000 Associate Director 6 $50,000 Administrative Specialist 7 $2,000

3 7. Each side of square ABCD is doubled in length to form square EFGH. The perimeter of square EFGH is 40 cm. What is the area of square ABCD? D A C B H G 7. sq cm E F 8. New York and Denver are in different time zones. When it is noon in New York, it is 10 a.m. in Denver. A train leaves New York at 2 p.m. (New York time) and arrives in Denver 45 hours later. What time is it in Denver when the train arrives? 8. a.m.. A survey of 400 patients at a hospital classified the patients by gender and blood type, as shown in the table below. What percent of the patients with type AB blood are male?. percent Type A Type B Type O Type AB TOTAL Male Female TOTAL One type of cat food recommends that a cat have a daily serving of 1 ounce of dry cat food per pound of body weight. 2 If a cat is fed ounces of dry food a day according to these recommendations, how many pounds does the cat weigh? 10. pounds 11. Roger has exactly one of each of the first 22 states new U.S. quarters. The quarters were released in the same order that the states joined the union. The graph below shows the number of states that joined the union in each decade. What fraction of Roger s 22 coins represents states that joined the union during the decade 1780 through 178? Express your answer as a common fraction. 11. # of States that Joined the Union Decades

4 12. In the sequence 0, 1, 1,, 6,, 27,..., the first term is Subsequent terms are produced by alternately adding and multiplying by each successive integer beginning with 1. For instance, the second term is produced by adding to the first term; the third term is produced by multiplying the second term by 1; the fourth term 0, 1, 1,, 6,, 27,... is produced by adding 2 to the third term; and so on. What is the value of the first term that is greater than 125? 1. Carolyn, Julie and Roberta share $77 in a ratio of 4:2:1, respectively. How much money did Carolyn receive? 1. $ 14. To weigh things on a balance scale, one or more objects are placed on one pan and weights are placed onto the other pan until the two pans are balanced. We see A is balanced by the 4 weights shown, and B is balanced by the 5 weights shown. We have the following weights: 1, 1,,,,, 27 and 27. What is the minimum number of these weights it would take to balance the total weight of A plus B? A 14. weights A 1 B 1 B? 15. If a = 2, b = and c = 4, what is the numerical value of the expression (b c) 2 + a (b + c)? The hexagon with the R is colored red. Each hexagon is colored either red, yellow or green, such that no two hexagons with a common side are colored the same color. In how many different ways can the figure be colored? R 16. ways 17. If one quart of paint is exactly enough for two coats of paint on a -foot by 10-foot wall, how many quarts of paint are needed to apply one coat of paint to a 10-foot by 12-foot wall? Express your answer as a common fraction. 17. quarts

5 18. Ten unit cubes are glued together as shown. What is the surface area of the resulting solid? 18. sq units 1. The graph shows the total distance Sam drove from 6 a.m to 11 a.m. How many miles per hour is the car s average speed for the period from 6 a.m. to 11 a.m.? Total Driving Distance Since 6 a.m. (miles) miles per hour Time of Day (a.m.) 20. A stock loses 10% of its value on Monday. On Tuesday it loses 20% of the value it had at the end of the day on Monday. What is the overall percent loss in value from the beginning of Monday to the end of Tuesday? 20. percent 21. The numbers 1, 2 and are written in these nine unit squares. Each of the numbers appears three times, and there is only one number placed in each of the nine unit squares. Each number is in a unit square horizontally or vertically adjacent to a unit square with the same number. The sum of the numbers in the leftmost column and the sum of the numbers in the top row are each 7. What is the sum of the numbers in the four shaded squares? Five balls are numbered 1 through 5 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the two numbers will be even and greater than 10? Express your answer as a common fraction If a certain negative number is multiplied by six, the result is the same as 20 less than the original number. What is the value of the original number? 2.

6 24. If 2004 is split after the third digit into a three-digit integer and a one-digit integer, then the two integers, 200 and 4, have a common factor greater than one. The years 2005 and 2006 each have this same property, too. What is the first odd-numbered year after 2006 that has this property? Two consecutive even numbers are each squared. The difference of the squares is 60. What is the sum of the original two numbers? What is the area of the pentagon shown here with sides of length 15, 20, 27, 24 and 20 units? square units 27. At the beginning of my bike ride I feel good, so I can travel 20 miles per hour. Later, I get tired and travel only 12 miles per hour. If I travel a total of 122 miles in a total time of 8 hours, for how many hours did I feel good? Express your answer as a common fraction hours 28. When reading a book, Charlie made a list by writing down the page number of the last page he finished reading at the end of each day. (He always finished reading a page that he started.) His mom thought his list indicated the amount of pages he had read on each day. At the end of the 8 th day of reading, she added the numbers on his list and thought Charlie had read 42 pages. If Charlie started reading the book on page one, and he read the same amount of pages each day of this eightday period, how many pages did he actually read by the end of the 8 th day? 28. pages 2. Suppose that each distinct letter in the C O U equation MATH = COU + NTS is replaced + N T S by a different digit chosen from 1 through M A T H in such a way that the resulting equation is true. If H = 4, what is the value of the greater of C and N? One gear turns 1 times in a minute. Another gear turns 45 times in a minute. Initially, a mark on each gear is pointing due north. After how many seconds will the two gears next have both their marks pointing due north? 0. seconds