MATHCOUNTS ® 2006 Chapter Competition Sprint Round Problems 1–30 Name 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. 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 Sponsors National Society of Professional Engineers ADC Foundation Raytheon Company National Council of Teachers of Mathematics General Motors Foundation Shell Oil Company Lockheed Martin Texas Instruments Incorporated National Aeronautics and Space Administration 3M Foundation CNA Foundation Northrop Grumman Foundation Xerox Corporation Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 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 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? R R R 1 2 3 4 5 6 7 8 9 10 11 12 ounces 1. ________________ 2. ________________ 13 3. What is the greatest number of cubes, with edge length 1 inch, that can be placed into a rectangular box measuring 3 inches by 1 3 inches by 9 inches? 9 cubes 3. ________________ 3 3 4. What fraction is equal to six times one-seventh? Express your answer as a common fraction. 4. ________________ 5. 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? 5. ________________ 6. According to the table below, what is the median value of the 59 salaries paid to this company’s employees? $ 6. ________________ Position Title President Vice-President Director Associate Director Administrative Specialist # of Employees with this Title 1 5 10 6 37 Salary for Position $130,000 $90,000 $75,000 $50,000 $23,000 Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round 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 C A B H G E F sq cm 7. ________________ 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? a.m. 8. ________________ 9. 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 9. ________________ Type A 45 55 100 Male Female TOTAL Type B 60 40 100 Type O 80 100 180 Type AB 15 5 20 TOTAL 200 200 400 pounds 10. ________________ 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 1789? Express your answer as a common fraction. 11. ________________ # of States that Joined the Union 10. One type of cat food recommends that a cat have a daily serving of 13 ounce of dry cat food per pound of body weight. If a cat is fed 3 23 ounces of dry food a day according to these recommendations, how many pounds does the cat weigh? 12 10 8 6 4 2 9 178 0178 69 29 89 49 09 09 -18 20-18 40-18 60-18 80-18 00-19 0 0 18 18 18 18 18 19 9 195 0195 Decades Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round 12. In the sequence 0, 1, 1, 3, 6, 9, 27, ..., the ﬁrst term is 0. 12. ________________ 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 +1 ×1 +2 ×2 1 to the ﬁrst term; the third term is produced by multiplying the second term by 1; the fourth term 0, 1, 1, 3, 6, 9, 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? 13. Carolyn, Julie and Roberta share $77 in a ratio of 4:2:1, respectively. How much money did Carolyn receive? $ 13. ________________ 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, 3, 3, 9, 9, 27 and 27. What is the minimum number of these weights it would take to balance the total weight of A plus B? weights 14. ________________ A A 3 9 3 1 B 9 3 9 3 1 B ? 15. If a = 2, b = 3 and c = 4, what is the numerical value of the expression (b – c)2 + a (b + c) ? 15. ________________ 16. 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? ways 16. ________________ R 17. If one quart of paint is exactly enough for two coats of paint on a 9-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. quarts 17. ________________ Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round sq units 18. ________________ 19. 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.? miles per hour 19. ________________ Total Driving Distance Since 6 a.m. (miles) 18. Ten unit cubes are glued together as shown. What is the surface area of the resulting solid? 160 120 80 40 0 6 7 8 9 10 11 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? percent 20. ________________ 21. The numbers 1, 2 and 3 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? 21. ________________ 22. 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. 22. ________________ 23. 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? 23. ________________ Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round 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? 24. ________________ 25. Two consecutive even numbers are each squared. The difference of the squares is 60. What is the sum of the original two numbers? 25. ________________ 26. What is the area of the pentagon shown here with sides of length 15, 20, 27, 24 and 20 units? 15 20 square units 26. ________________ 27 20 24 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 27. ________________ 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 8th day of reading, she added the numbers on his list and thought Charlie had read 432 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 8th day? pages 28. ________________ COU 29. Suppose that each distinct letter in the + NT S equation MATH = COU + NTS is replaced MA T H by a different digit chosen from 1 through 9 in such a way that the resulting equation is true. If H = 4, what is the value of the greater of C and N? 29. ________________ 1 30. One gear turns 33 3 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? seconds 30. ________________ Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round MATHCOUNTS ® 2006 Chapter Competition Target Round Problems 1 and 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. Record only ﬁnal answers in the designated blanks on the problem sheet. All answers must be complete, legible and simpliﬁed to lowest terms. This round assumes the use of calculators, and calculations may also be done on scratch paper, but no other aids are allowed. If you complete the problems before time is called, use the time remaining to check your answers. Total Correct Scorer’s Initials Founding Sponsors National Sponsors National Society of Professional Engineers ADC Foundation Raytheon Company National Council of Teachers of Mathematics General Motors Foundation Shell Oil Company Lockheed Martin Texas Instruments Incorporated National Aeronautics and Space Administration 3M Foundation CNA Foundation Northrop Grumman Foundation Xerox Corporation Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 1. Brass is an alloy created using 80% copper and 20% zinc. If Henri’s brass trumpet contains 48 ounces of copper, how many ounces of zinc are in the trumpet? ounces 1. ________________ 2. In the maze below, a player may only move toward the right. At each junction the player chooses a path that includes an operation (+ or –) and then a number. The player keeps a running total of her score throughout her journey, starting with 7, and she may only choose among paths that keep her score between 2 and 14 points at all times. After performing four operations and ending at the 5, what is the lowest score with which she can ﬁnish? points 2. ________________ 3 2 Start with 7 4 5 3 2 4 3 Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Target Round MATHCOUNTS 2006 Chapter Competition Target Round Problems 3 and 4 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Total Correct Scorer’s Initials Founding Sponsors National Sponsors National Society of Professional Engineers ADC Foundation Raytheon Company National Council of Teachers of Mathematics General Motors Foundation Shell Oil Company Lockheed Martin Texas Instruments Incorporated National Aeronautics and Space Administration 3M Foundation CNA Foundation Northrop Grumman Foundation Xerox Corporation Copyright MATHCOUNTS, Inc. 2005. All rights reserved. ® 3. In the ﬁgure below, side AE of rectangle ABDE is parallel to the x-axis, and side BD contains the point C. The vertices of triangle ACE are A(1, 1), C(3, 3) and E(4, 1). What is the ratio of the area of triangle ACE to the area of rectangle ABDE? Express your answer as a common fraction. B A 3. ________________ C D E 4. The European equivalent of 8 12 ” by 11” paper is called A4 paper, and its dimensions are 0.21 meters by 0.297 meters. What is the greatest total area, in square meters, that can be covered by 21 sheets of rectangular A4 paper? Express your answer as a decimal to the nearest tenth. sq meters 4. ________________ Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Target Round MATHCOUNTS 2006 Chapter Competition Target Round Problems 5 and 6 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Total Correct Scorer’s Initials Founding Sponsors National Sponsors National Society of Professional Engineers ADC Foundation Raytheon Company National Council of Teachers of Mathematics General Motors Foundation Shell Oil Company Lockheed Martin Texas Instruments Incorporated National Aeronautics and Space Administration 3M Foundation CNA Foundation Northrop Grumman Foundation Xerox Corporation Copyright MATHCOUNTS, Inc. 2005. All rights reserved. ® 5. The arithmetic mean (or average) of A, B and C is 10. The value of A is six less than the value of B, and the value of C is three more than the value of B. What is the value of C? 5. ________________ 6. A value is assigned to each letter of the alphabet such that A = 1, B = 2, C = 3,..., Z = 26. A nine-digit code is then created for each letter using the prime factorization of its assigned value. The ﬁrst digit of a letter’s code is the number of times 2 is used as a factor, the second digit is the number of times 3 is used as a factor, the third digit is the number of times 5 is used as a factor, and so on. For example, since N is the 14th letter of the alphabet and N = 14 = 21 × 71, the code for the letter N is 100100000 with 1s in the ﬁrst and fourth positions because its prime factorization has one 2 (the ﬁrst prime number) and one 7 (the fourth prime number). What 6-letter word does the following sequence of six codes represent? The ﬁrst row is the code for the ﬁrst letter of the word, the second row is the code for the second letter of the word, and so on. 6. ________________ 0 0 0 0 2 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Target Round MATHCOUNTS 2006 Chapter Competition Target Round Problems 7 and 8 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Total Correct Scorer’s Initials Founding Sponsors National Sponsors National Society of Professional Engineers ADC Foundation Raytheon Company National Council of Teachers of Mathematics General Motors Foundation Shell Oil Company Lockheed Martin Texas Instruments Incorporated National Aeronautics and Space Administration 3M Foundation CNA Foundation Northrop Grumman Foundation Xerox Corporation Copyright MATHCOUNTS, Inc. 2005. All rights reserved. ® 7. On a particular quiz, there are 15 easy questions and 15 hard questions. The easy questions are worth 4 points each, and the hard questions are worth 10 points each. No partial credit is possible. Sam earns a total of 92 points on the quiz. What is the greatest number of hard questions that he could have answered correctly? hard questions 7. ________________ 8. Eli throws ﬁve darts at a circular target, and each one lands within one of the four regions. The point-value of a dart landing in each region is indicated. What is the least score greater than ﬁve points that is not possible when the pointvalues of the ﬁve darts are added together? points 8. ________________ 6 4 2 1 Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Target Round MATHCOUNTS ® 2006 Chapter Competition Team Round Problems 1–10 DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of ten 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 during this section of the competition. This round assumes the use of calculators, and calculations may also 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 problem sheet, which is the only sheet that will be scored. If the team completes the problems before time is called, use the remaining time to check your answers. Team Members , Captain Total Correct Scorer’s Initials Founding Sponsors National Sponsors National Society of Professional Engineers ADC Foundation Raytheon Company National Council of Teachers of Mathematics General Motors Foundation Shell Oil Company Lockheed Martin Texas Instruments Incorporated National Aeronautics and Space Administration 3M Foundation CNA Foundation Northrop Grumman Foundation Xerox Corporation Copyright MATHCOUNTS, Inc. 2005. All rights reserved. $ 1. ________________ 2. A ball bounces back up 23 of the height from which it falls. If the ball is dropped from a height of 243 cm, after how many bounces does the ball ﬁrst rise less than 30 cm? bounces 2. ________________ 450 3. The graph shows the maximum 400 speed and maximum height of the eight fastest roller coasters 350 in the world in 2003. In 2003, 300 the Dodonpa roller coaster had 250 a maximum speed of 200 106.9 miles per hour, making 150 it one of the eight fastest roller 80 90 100 110 120 coasters. In 2003, how many Speed (mph) roller coasters had a maximum speed that was faster than the Dodonpa roller coaster? roller coasters 3. ________________ 4. When Marika bought her house she paid 80% of the purchase price of the house with a loan. She paid the remaining $49,400 of the purchase price with her savings. What was the purchase price of her house? $ 4. ________________ 5. One interior angle of a convex polygon is 160 degrees. The rest of the interior angles of the polygon are each 112 degrees. How many sides does the polygon have? sides 5. ________________ 6. A 25-passenger bus rents for $110, and a 40-passenger bus rents for $170. What is the minimum cost for renting enough buses for a school trip with 475 passengers? $ 6. ________________ Height (feet) 1. In 1992, a scoop of gelato could be purchased in Italy for 1200 lire. The same gelato would have cost $1.50 in the U.S. At the equivalent exchange rate between the lire and the dollar, how many dollars would be equivalent to 1,000,000 lire? Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Team Round 7. Arpan and Tomika want to place straw in the ﬂower beds surrounding their house. In the ﬁgure below, every angle is a right angle, and the shaded 62’ regions represent the 20’ ﬂower beds. The 4’ 20’ dimensions shown 6’ 6’ 6’ are measurements for the house and do not include the ﬂower beds. The width of the ﬂower beds is 2 feet. One bale of straw, costing $2.75, will cover 9 square feet of ground. With 6% sales tax, how much will it cost them to cover the ﬂower beds with straw, if only whole bales of straw may be purchased? $ 7. ________________ 8. A “value meal” consists of exactly one selection from the entree menu, the drink menu and the dessert menu below. The price of any value meal is calculated by subtracting 20 cents from the sum of the prices of the three individual items. For example, the least expensive value meal costs $2.45. If a customer orders exactly one of each of the three most expensive value meal combinations, what is the total cost of the customer’s three value meals? $ 8. ________________ Entree Hot dog - $1.25 Hamburger - $1.65 Chicken - $1.80 Pizza - $2.25 Drink Coffee - $0.75 Soda - $1.05 Juice- $1.25 Dessert Cookie - $0.65 Pudding - $1.30 9. A 6 by 6 grid of 36 unit squares is completely covered in T-shaped pieces consisting of four unit squares. The T-shaped pieces do not overlap, but pieces may extend over the side of the 6 by 6 grid. What is the fewest number of T-shaped pieces required to cover the 6 by 6 grid? pieces 9. ________________ 10. Three friends share a full bag of jellybeans. Mike took 13 of the jellybeans in the full bag, Zac took 12 of the jellybeans in the full bag and Kary took what was left. Mike ate 12 of his jellybeans, Zac ate 13 of his jellybeans and Kary ate all of hers. If Mike and Zac now have a total of 45 jellybeans together, how many jellybeans did Kary eat? jellybeans 10. ________________ Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Team Round

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