2005 Chapter Competition Sprint Round Problems 1–30 Name School DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This round of the competition consists of 30 problems. You will have 40 minutes to complete the problems. You are not allowed to use calculators, books or any 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 right-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers. Total Correct Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics Scorer’s Initials National Sponsors ADC Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Texas Instruments Incorporated 3M Foundation ©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314 1. The chart below gives the air distance in miles between selected world cities. What is the distance from Honolulu to Cape Town? Bangkok Bangkok Cape Town Honolulu London 6300 6609 5944 Cape Town 6300 11,535 5989 2. The pie chart shown represents a survey of Canadians who do not use the Internet. What is the percent of Canadian non-users for whom cost is not the primary barrier? Honolulu 6609 11,535 7240 1. _________________ miles London 5944 5989 7240 Primary Barrier to Internet Access % 2. _________________ Canadian Non-Users 3. It is now 12:00:00 midnight, as read on a12-hour digital clock. In 122 hours, 39 minutes and 44 seconds the time will be A:B:C. What is the value of A + B + C? 3. _________________ 4. Shooting hoops for 30 minutes burns 150 calories. How many calories would Kendra burn shooting hoops if she shot hoops 30 minutes every day for one week? calories 4. _________________ 5. Olga purchases a rectangular mirror (the shaded region) that ﬁts exactly inside a frame. The outer perimeter of the frame measures 60 cm by 80 cm. The width of each side of the frame is 10 cm. What is the area of the mirror? sq cm 5. _________________ 6. The product of two positive whole numbers is 2005. If neither number is 1, what is the sum of the two numbers? 6. _________________ ©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round 7. Three friends each ordered a large cheese pizza. Shauntee ate 23 26 of her pizza, Carlos ate 89 of his pizza and Rocco ate 27 of his pizza. If the remaining portions from the three pizzas are put together, what fraction of a large pizza do they make? Express your answer as a common fraction. 7. _________________ 8. It takes 24 minutes for Jana to walk one mile. At that rate, how far will she walk in 10 minutes? Express your answer as a decimal to the nearest tenth. miles 8. _________________ 9. Roslyn has ten boxes. Five of the boxes contain pencils, four of the boxes contain pens, and two of the boxes contain both pens and pencils. How many boxes contain neither pens nor pencils? 9. _________________ boxes 10. How many combinations of pennies, nickels and/or dimes are there with a total value of 25¢? combinations 10. _________________ 11. What is the value of the following expression: 1 − 3 + 5 − 7 + 9 − ….. − 43 + 45 − 47 + 49 ? 11. _________________ 12. A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet? tiles 12. _________________ 13. When plotted in the standard rectangular coordinate system, trapezoid ABCD has vertices A(1, −2), B(1, 1), C(5, 7) and D(5, 1). What is the area of trapezoid ABCD? square units 13. _________________ ©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round 14. Trey receives a 5% commission on every sale he makes. On the sale of a $60 coat (before any discounts), how many more cents will he receive if his commission is based on the original price of the coat rather than the price of the coat after a 20% discount? cents 14. _________________ 15. Five distinct points A, B, C, D and E lie on a line, but not necessarily in that order. Use the information below to determine the number of units in the length of segment DC. • E is the midpoint of segment AB. • D is the midpoint of segment AE. • Both C and E are the same distance from B. • The distance from D to B is 9 units. units 15. _________________ 16. In the 19th century, Britain used a money system which included pence, farthings, shillings and pounds. The following conversions were used: 4 farthings = 1 pence 12 pence = 1 shilling 20 shillings = 1 pound How many total farthings were equivalent to 1 pound and 5 pence? farthings 16. _________________ 17. What is the sum of all the distinct positive two-digit factors of 144? 17. _________________ 18. The points B(1, 1), I(2, 4) and G(5, 1) are plotted in the standard rectangular coordinate system to form triangle BIG. Triangle BIG is translated ﬁve units to the left and two units upward to triangle B’I’G’, in such a way that B’ is the image of B, I’ is the image of I, and G’ is the image of G. What is the midpoint of segment B’G’? Express your answer as an ordered pair. ( , ) 18. _________________ 19. The positive difference of the cube of an integer and the square of the same integer is 100. What is the integer? 19. _________________ ©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round 20. A rectangular sheet of paper is folded twice and then cut, as shown below. All fold lines are dashed, and the portion that is to be cut away is shaded. 20. _________________ Which of the following drawings (A, B, C, D, E or F) shows what the paper looks like when it is unfolded after the cuts? 21. Henry took ﬁve tests, and his average score was 57 points. He scored at least 50 points on each test. There were 100 points possible on each test. What is the highest score that Henry could have earned on any of the ﬁve tests? points 21. _________________ 22. How many ordered pairs (x, y) satisfy BOTH conditions below? Condition I: x = 1 or y = 0 or y = 2 Condition II: x = 0 or x = 2 or y = 1 ordered pairs 22. _________________ 23. Zan has created this rule for generating sequences of whole numbers. If a number is 25 or less, double the number. If a number is more than 25, subtract 12 from it. For example, if Zan starts with 10, she gets the sequence 10, 20, 40, 28, 16, … . If the third number in Zan’s sequence is 36, what is the sum of the four distinct numbers that could have been the ﬁrst number in her sequence? 23. _________________ 24. Reverse the two digits of my age, divide by three, add 20, and the result is my age. How many years old am I? years 24. _________________ ©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round 25. The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of N? 25. _________________ 26. To be able to walk to the center C of a circular fountain, a repair crew places a 16-foot plank from A to B and then a 10-foot plank from D to C, where D is the midpoint of AB . What is the area of the circular base of the fountain? Express your answer in terms of π. square feet 26. _________________ 27. The function f is deﬁned by f (n) = f (n − 1) + f (n − 2). It is also true that f (1) = 3 and f (3) = 10. What is the value of f (6)? 27. _________________ c 2 ad a 3 b 8 = , = and = , what is the value of 2 ? Express b 4 c 9 d 3 b your answer as a common fraction. 28. If 28. _________________ 29. A play has two male roles, two female roles and two roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If ﬁve men and six women audition, in how many ways can the six roles be assigned? ways 29. _________________ 30. What is the arithmetic mean of all of the positive two-digit integers with the property that the integer is equal to the sum of its ﬁrst digit plus its second digit plus the product of its two digits? 30. _________________ ©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round 2005 Chapter Competition Target Round Problems 1 and 2 Name School DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This round 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 your ﬁnal answer in the designated space 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. Total Correct Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics Scorer’s Initials National Sponsors ADC Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Texas Instruments Incorporated 3M Foundation ©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314 1. Francisco starts with the number 5, doubles it, adds 1, doubles the result, adds 1, doubles the result, adds 1, and continues this pattern of two alternating calculations. Phong, meanwhile, starts with 5, adds 1, doubles the result, adds 1, doubles the result, adds 1, doubles the result, and continues this pattern of two alternating calculations. They each do eight total calculations. What is the positive difference of their ﬁnal results? 1. __________________ 2. All vertices of this cube will be colored such that no two vertices on the same edge of the cube are the same color. What is the minimum number of colors that will be needed to color the vertices of this cube? colors 2. __________________ ©MATHCOUNTS Foundation: 2005 Chapter Target Round 2005 Chapter Competition Target Round Problems 3 and 4 Name School DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Total Correct Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics Scorer’s Initials National Sponsors ADC Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Texas Instruments Incorporated 3M Foundation ©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314 3. A store purchases televisions from a factory for $87.89 each. The store normally sells one of these televisions for 225% of the factory cost, but a store coupon gives 25% off this selling price. Ignoring tax, how much does a customer with this coupon pay for the television? Express your answer in dollars to the nearest hundredth. 3. __________________ $ 4. The symbols represent four different integers from 1 to 9. Using the equations below, what is the value of ? 4. __________________ ©MATHCOUNTS Foundation: 2005 Chapter Target Round 2005 Chapter Competition Target Round Problems 5 and 6 Name School DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Total Correct Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics Scorer’s Initials National Sponsors ADC Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Texas Instruments Incorporated 3M Foundation ©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314 5. What is the greatest whole number that MUST be a factor of the sum of any four consecutive positive odd numbers? 5. __________________ 6. In the ﬁgure below, the smaller circle has a radius of two feet and the larger circle has a radius of four feet. What is the total area of the four shaded regions? Express your answer as a decimal to the nearest hundredth. sq feet 6. __________________ ©MATHCOUNTS Foundation: 2005 Chapter Target Round 2005 Chapter Competition Target Round Problems 7 and 8 Name School DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Total Correct Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics Scorer’s Initials National Sponsors ADC Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Texas Instruments Incorporated 3M Foundation ©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314 7. Jamie has a jar of coins containing the same number of nickels, dimes and quarters. The total value of the coins in the jar is $13.20. How many nickels does Jamie have? nickels 7. __________________ 8. John, Mike and Chantel will divide a pile of pennies amongst themselves using the following process: pennies 8. __________________ The number of pennies in the pile is counted. • If the number of pennies in the pile is even, Mike will get half of the pile. • If the number of pennies in the pile is odd, one penny will be given to Chantel, and John will get half the pennies remaining in the pile. This process is then repeated until the pile is empty. How many pennies will Mike have at the end if the original pile contains 2005 pennies? ©MATHCOUNTS Foundation: 2005 Chapter Target Round 2005 Chapter Competition Team Round Problems 1–10 School Team Members , Captain DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This round of the competition consists of ten problems which the team has 20 minutes to complete. Team members may work together 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. The team captain must record the answers on his/her problem sheet, and all answers must be complete and legible. Only the team captain’s problem sheet will be scored. Total Correct Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics Scorer’s Initials National Sponsors ADC Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Texas Instruments Incorporated 3M Foundation ©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314 1. A competition problem requires one hour to fully develop (write, proofread, edit, and typeset). This problem is then given to 30,000 students, each working an average of 24 seconds to solve the problem. What is the ratio of a problem’s development time to the total time spent by the students to solve the problem? Express your answer as a common fraction. 1. _________________ 2. Select any three-digit multiple of 3. Calculate the sum of the cubes of the digits of that number. This is now your new number. Now calculate the sum of the cubes of the digits of this new number. Continue this procedure of adding the cubes of the digits of the resulting number until you arrive at a number that is equal to the sum of the cubes of its digits. What is this number? 2. _________________ 3. Emma and Ed walk into a room containing 30 assembled Tworks and enough pieces to assemble 100 more Tworks. A Twork takes eight minutes to assemble and 10 minutes to disassemble. If Emma starts assembling Tworks as Ed begins disassembling the ones that were already made, and they both continue to work until there are exactly 35 assembled Tworks in the room (and no partially assembled or disassembled Tworks), how many minutes will Emma have worked? minutes 3. _________________ 4. How many triangles are in the ﬁgure below? triangles 4. _________________ 5. How many integers are solutions to the equation integers 5. _________________ (x − 2 )( 25 − x 2 ) =1 ? ©MATHCOUNTS Foundation: 2005 Chapter Team Round 6. The minute hand of a clock measures 10 cm from its tip to the center of the clock face, and the hour hand from its tip to the center of the clock face is 5 cm. What is the sum of the distances, in meters, traveled by the tips of both hands in one 24-hour period? Express your answer to the nearest thousandth of a meter. 6. _________________ meters 7. These are two shoelace patterns for two identical shoes with fourteen holes each. Assume that the holes form a rectangular grid and that each hole is 1 cm from its nearest horizontal and vertical neighbor-holes. Calculate the ratio of the total length of the shoelace shown in Pattern #1 to the total length of the shoelace shown in Pattern #2. Express your answer as a decimal to the nearest hundredth. 7. _________________ Pattern #1 Pattern #2 8. A four-digit perfect square number is created by placing two positive two-digit perfect square numbers next to each other. What is the four-digit square number? 8. _________________ 9. If Ella rolls a standard six-sided die until she rolls the same number on consecutive rolls, what is the probability that her 10th roll is her last roll? Express your answer as a decimal to the nearest thousandth. 9. _________________ 10. A standard deck of playing cards with 26 red cards and 26 black cards is split into two piles, each having at least one card. In pile A there are six times as many black cards as red cards. In pile B, the number of red cards is a multiple of the number of black cards. How many red cards are in pile B? red cards 10. _________________ ©MATHCOUNTS Foundation: 2005 Chapter Team Round

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