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Warm-Up 1 settings George is planning a dinner meeting for 150 people. Each table will be set for nine 1. ________ people. If there are enough tables to accommodate everyone, what is the fewest possible number of unused table settings? $ 2. ________ Sami decided to park her car at Park&Go for her four-day trip. The cost is usually $8 per day or partial day. However, Sami has a 20% discount coupon for her entire bill. How much did Sami‛s parking cost? 3. ________ If x = 3, what is the value of 2x + 3? Newborn Infant Weights (lbs, oz) lbs oz What is the median weight of the newborn 4. ________ infant weights represented in the stem-andleaf plot shown to the right? (2 4 represents 2 lbs 4 oz.) 5 3 6 7 10 6 0 1 3 4 7 8 11 12 14 7 0 4 10 10 10 12 14 8 2 3 4 7 8 10 9 2 4 6 5. ________ Haley has enlarged a 3” by 5” picture so that both the length and width are tripled. By what number must the original area of the picture be multiplied to get the enlarged area? bananas A shake from Jake‛s Shakes uses 2 tsp vanilla, 2 scoops of ice cream, 4 oz fresh 6. ________ banana and ½ cup milk. A banana (without the peel) weighs 6 oz on average. How many whole bananas must Jake order so that he can make 500 shakes? % The data in the graph below show the number of jobs in this week‛s classified ads 7. ________ that require various levels of math knowledge. Each job was placed in exactly one of the four categories. What percent of the jobs needs either practical “shop” math or arithmetic? Express your answer to the nearest whole number. Number of Jobs Mathematics Required for Different Jobs 25 25 20 15 22 16 20 10 5 0 Theoretical Applied Math Math “Shop” Math Arithmetic weeks Game birds such as ducks and pheasants that are raised on a farm need a brooder 8. ________ house temperature of 95°F to start. The temperature is then decreased five degrees per week until the outside temperature is reached. How many weeks would it take for the brooder house temperature to reach 60°F? 9. ________ What is the value of 5 (3 · 2) − 3 (2 · 5) ? % Tara rolls three standard dice once. What is the probability that the sum of the 10. _______ numbers rolled will be three or more? Express your answer as a percent. MATHCOUNTS 2005-2006 17 Warm-Up 2 16 cm cm A regular octagon has the same perimeter as the regular hexagon shown 1. ________ here with side length 16 cm. How long is each side of the octagon? 2. ________ What is the smallest prime factor of 1821? marbles Three bags each contain the same number of marbles. (The weight of an empty bag 3. ________ would not affect the weight on the scale.) If three bags of marbles plus two more marbles balance 14 marbles, as shown, how many marbles are in each bag? factors A certain number n has factors of 15 and 10. What is the fewest total number of 4. ________ positive factors that n can have? 5. ________ Jacque and Emily work in a shoe store. Emily was trying to carry too many boxes and dropped all of them, spilling four different pairs of shoes into a pile on the floor. Jacque picked up one shoe from the pile, and then Emily picked up a shoe from the pile. What is the probability that the shoe Emily‛s holding does not belong in the same box as the shoe Jacque‛s holding? Express your answer as a common fraction. faces How many total faces does a trapezoid-based prism have? 6. ________ % The odds of an event happening successfully are expressed as a :b , where a :b is the 7. ________ ratio of the probability of a successful event to the probability of an unsuccessful event. The odds of rolling a 5 on an unfair die are 1:3. What is the probability of rolling a 5 with this die? Express your answer as a percent. 8. ________ In this maze you may move only in a downward direction. Starting with 0.4 at the top, you must continue to add each new number you land on to the sum of the previous numbers you‛ve been on. Your 0.4 trail is complete when you land on and add the last value, which is 0.4 at the 0.1 0.7 0.3 bottom. What is the least possible sum for a complete trail through the maze? Express your answer as a decimal to the 0.8 0.9 0.7 0.6 0.5 0.6 0.3 nearest tenth. 0.6 0.3 0.7 0.4 meters2 The area of a square garden plot is 64 square meters. When the length is doubled 9. ________ and the width is increased by three meters, how much area is gained? feet During a severe thunderstorm, Tim saw a flash of lightning and then six seconds 10. _______ later heard a clap of thunder. The speed of sound is 1088 ft/sec. How far is Tim from the lightning? Express your answer to the nearest thousand feet. MATHCOUNTS 2005-2006 19 Warm-Up 3 $ 1. ________ Last month Sue made $200 by baby-sitting for 12 hours and tutoring for 20 hours. This month she worked the same number of hours at each job as she did last month, but increased her baby-sitting fee by $2 per hour and her tutoring fee by $1 per hour. How much did she earn this month? $ 3. ________ This graph shows the raise schedule for Mr. Bright‛s employees. If the linear pattern continues, what is the amount of the raise for six years of service? Amount of Raise lines How many lines of symmetry does the symbol Ω have? 2. ________ $4000 $3500 $3000 $2500 $2000 $1500 $1000 $500 $0 1 2 3 4 5 6 Years of Service 4. ________ A British television program shows homes that are for sale. One home is listed as costing 500,000 pounds, or $850,000. The ratio of pounds to dollars used in this TV program can be written in the form “1 pound : x dollars.” What is the value of x ? Express your answer as a decimal to the nearest tenth. 5. ________ What is the sum of the first 10 terms of the arithmetic sequence 1, −1, −3, … ? mm Kat has a collection of porcelain dolls that have heights, in millimeters, of 24, 32, 24, 6. ________ 27, 40, 36 and 34. What is the average height of the dolls? 7. ________ At an awards ceremony 280 personalized certificates were given out in the categories and quantities shown here. Julie earned certificates in both fine arts and mathematics. The principal grabbed a certificate out of the stack at random after all of the science and social studies certificates were given out. What is the probability that he pulled out one of Julie‛s certificates? Express your answer as a common fraction. Type of Award Number of Awards Mathematics 35 Science 38 Social Studies 29 Business 40 Physical Education 23 English 25 Fine Arts 30 Foreign Language 60 % A restaurant mixes two gallons of milk containing 1% fat and three gallons of milk 8. ________ containing 2% fat. What is the percent of fat in the mixture, expressed to the nearest tenth? turns In a particular dice game a player rolls two dice. The player can choose either the 9. ________ sum or the product of the numbers rolled for her score for that turn. To win the game, a player must get a total score of exactly 102 points from the sum of her scores. What is the least possible number of turns needed to win the game? diags How many diagonals does a regular octagon have? 10. _______ MATHCOUNTS 2005-2006 23 Warm-Up 4 units Using the grid below, follow the clues to find the location of the treasure. 1. ________ y Start at (−1, −1). Follow the line y = x until x = 4. Reflect this new point over the x-axis. From this new point add seven to the y-coordinate and subtract five from the x-coordinate. This is the location of the treasure. y=x x How far is the treasure point from the starting point? $ 2. ________ Jane bought six items from a basket of 25-cent and 50-cent items. Kyle bought six items from a basket of 50-cent and 75-cent items. What is the difference between the lowest possible cost to Jane and the highest possible cost to Kyle? 4c m cm The two triangles shown are similar. What is 4. ________ the perimeter of the larger triangle? 5 cm 8c m 3. ________ Roman numerals are letters that represent numbers. Some of the most common ones are I = 1, V = 5, X = 10, L = 50 and C = 100. The Roman system represents the number 3 as III and 30 as XXX. If a symbol with a lower value is placed in front of a symbol with a greater value, the lesser number is subtracted from the greater number. For example, IV represents 5 – 1 = 4. What is the three-digit value of the following subtraction problem: CCCXLVII – CLXVIII ? 10 c m 7 cm pairings Chandra has four bowls. Each one is a different color (red, blue, yellow, green). She 5. ________ also has exactly one glass the same color as each bowl. If she chooses a bowl and a glass from the cupboard, how many pairings are possible? One such pairing is a blue bowl and a yellow glass. 1 inch 6. ________ One color printer produces 300 dots per linear inch while another can produce 600 dots per linear inch. If the two printers both print the same 8-inch by 10-inch photograph, what is the ratio of the total number of dots produced 1 inch by the first printer to the total number of dots produced by the second printer? This figure shows what a printer using four dots per linear inch would produce. Express your answer as a common fraction. degrees If the measure of angle C is double the measure of angle B, 7. ________ what is the measure of angle A in triangle ABC? C A 21 B zeros How many zeros are there in the product of 24 and 55? 8. ________ 9. ________ What is the greatest possible value of ( ) x ( ) + ( ) if the parentheses are filled with 2, 4 and 12, and each of the three integers is used exactly once? 10. _______ What is the least positive integer value of x for which the inequality 3x > 2x + 1 is true? MATHCOUNTS 2005-2006 25 Warm-Up 5 1. ________ The number 169 can be written as the sum of three positive perfect squares. What is the greatest of these perfect squares? Greatest Depth Attained by Submersibles Number of Feet vehicles The world‛s tallest mountain, 2. ________ measured from its base, is Mauna Loa. Only 13,680 feet of this 33,356-foot mountain are visible above the water; the rest of the mountain is under the Pacific Ocean. Of the seven submersible vehicles in the graph, how many could get to the bottom of Mauna Loa on the ocean floor? 35790 36000 30000 24000 19700 19700 18000 13000 12000 6000 9775 3000 3300 e -1 na vin ow on ste er s a Al ph SRV Cy p T - Ja Trie s e y D e D rgo th Ba A 3. ________ If x = −2, what is the value of 2x − 7? girls The ratio of boys to girls in a class of 35 students is 3:4. How many more girls than 4. ________ boys are in the class? (x, y) 5. ________ On a coordinate grid, the triangle is translated from position A to position B. What is the value of x + y ? B (2, 3) (9, 4) A (0, 0) (4, 0) meters The diagonals of a rhombus measure 10 meters and 24 meters and are perpendicular 6. ________ bisectors of each other. What is the perimeter of the rhombus? 7. ________ The Fibonacci sequence is 1, 1, 2, 3, 5, 8, … . Each number after the first two numbers is the sum of the preceding two numbers. What is the first perfect square greater than 1 to occur in this sequence? tpks How many toothpicks would you need in order to 8. ________ build the sixth figure in the pattern? The first three figures of the pattern are shown here. The first figure is made from four toothpicks. 9. ________ When four standard six-sided dice are rolled, what is the most likely sum of the four numbers? 10. _______ A rectangular frame measuring 10” by 15” has five square holes for photos. Four of the holes are each 2” by 2”, and one hole is 3” by 3”. What portion of the frame‛s area was not cut away for the photo holes? Express your answer as a common fraction. MATHCOUNTS 2005-2006 29 Warm-Up 6 1. ________ The set {2, 3, 4, 5, 9} contains five of the factors of which one of the following four integers? 480 1260 1890 3888 $ 2. ________ The table below shows the average amount paid to winners of the Kentucky Derby races from 1976 through 2004. For example, there were four years that 15 horses started the race, and the first-place winners for those years averaged $527,175. For the 29 races shown, what is the average winnings amount associated with the median number of starters? No. of Starters Frequency First Prize No. of Starters Frequency First Prize 9 1 $165,200 16 3 $688,467 10 1 $228,650 17 3 $680,600 11 1 $186,900 18 2 $1,299,900 13 3 $452,450 19 6 $777,750 14 1 $628,800 20 3 $605,967 15 4 $527,175 21 1 $317,200 3. ________ What is the 11th term in the geometric sequence 3, answer as a common fraction. −3 3 −3 , , , 2 4 8 … ? Express your 4. ________ Goran has a standard deck of 52 cards. He considers an ace to have a value of 1, a jack has a value of 11, a queen has a value of 12, a king has a value of 13, and all other cards are their face value. What is the probability that a randomly selected card will have an even value? Express your answer as a common fraction. 5. sq ft ________ Jen and Jerry need to carpet their L-shaped room, having dimensions 10’ as shown. The carpet they chose is a solid color and is sold in rolls 4’ that are 12 feet wide. How many square feet of wasted carpet will they have if the minimum length from the roll is used and only one 10’ seam is permitted? 12’ 6. ________ What is the 100th term of the sequence 2, 3, 5, 6, 7, 10, 11, … which consists of all of the positive integers that are neither perfect squares nor perfect cubes? 7. ________ What is the integer value of x that minimizes the absolute value of (6x 2 – 2x )? ways Cheryl, Don, Oleg, Shandra and Hans each have exactly one pet. Their five pets are 8. ________ a ferret, parrot, dog, cat and fish, but not necessarily in that order. In how many different ways can the five friends be paired with the five animals? 9. ________ The mean of a collection of five positive integers is 7. If the collection has a unique mode, what is the greatest possible member of the collection? 10. Figure _______ Maria sees these first four figures of a sequence in her textbook. The first figure is made from 6 toothpicks. Maria has 41 toothpicks. What is the largest figure of this sequence Maria can create if the sequence were to continue forever? ... Figure 1 MATHCOUNTS 2005-2006 Figure 2 Figure 3 Figure 4 31 Warm-Up 7 cu ft Annie estimated that her household recycles five cubic feet of materials per 1. ________ collection day. There are 26 collection days per year in her town of 16,000 households. If every household is similar to Annie‛s, how much recycled material is collected in one year in Annie‛s town? sec The distance from the earth to the sun is 93,000,000 miles, and light travels at 2. ________ 186,000 miles per second. How many seconds does light from the sun take to reach the earth? $ 3. ________ Jorge bought a TV at a 15% discount. Later he found that there should have been a 20% discount, so the store gave him the $17 he was owed. What was the price of the TV before any discount? 4. ________ The whole numbers are arranged in an array as shown. If the pattern continues going down one column and up the next, what is the number at the bottom of the 19th column? 1 8 9 16 ... 2 7 10 15 3 6 11 14 4 5 12 13 patterns A bracelet is made by stringing together 5. ________ four beads. Each bead is either red or R green. How many different color patterns are possible for the bracelet, where patterns are considered the same if turning one will produce the other, as shown here? ... R R G G G R G 4 6. ________ A rectangular 7-inch by 12-inch picture is framed with a mat so that four inches of the mat are seen on the top and on each of the sides. Five inches of the mat are seen on the bottom. What fraction of the finished product‛s area (picture and mat) is the picture? Express your answer as a common fraction. 4 4 7 12 5 7. ________ 1 3 6 10 The numbers 1, 3, 6, 10, … are called triangular numbers, as shown geometrically here. What is the 20th triangular number? 8. ________ If a # b = a 2 + b and a @ b = b − a , what is the value of ((1 # 3) @ 2)? 9. ________ If y = 2x + 1, which of the following equations is true? A) x = 2y + 1 B) x = (½)y + 1 C) x = (½)y − ½ D) x = (½)y + ½ yel cds A special deck of cards consists of green cards and yellow cards. The odds of 10. _______ randomly selecting a green card from the deck are 3:5. If there are 40 cards in the deck, how many yellow cards are in the deck? MATHCOUNTS 2005-2006 35 Warm-Up 8 $ 1. ________ A resort hotel charges a regular rate of $100 per night, but Wednesday nights are $150, and Saturday nights and Sunday nights are each $200. What is the price difference between the most expensive and cheapest possible stays of four consecutive nights? % 2. ________ There are three times the number of orange fish as blue fish in a tank at the pet store, and there are no other fish. If Kaya randomly pulls out one fish from the tank, what is the probability that it is orange? Express your answer as a percent. sq cm A large square region is divided into four congruent, non3. ________ overlapping square regions. If the perimeter of one of these small square regions is 60 cm, what is the area of the large square? 4. ________ Mary will pick a positive integer less than 80 that is a multiple of 7. Susan will pick a positive integer less than 80 that is a multiple of 9. What is the probability that they both will pick the same number? Express your answer as a common fraction. hours A certain amount of work can be done by seven identical machines in 30 hours. How 5. ________ long will it take 10 of these same machines to do the same amount of work? cubes How many 1” cubes will pack into a rectangular box with dimensions 8” by 6” by 6.5”? 6. ________ : a.m. Changee is scheduling 7. ________ use of the pool for the 2008 Summer Olympic Games. He has decided to allow 15 minutes between events, and he will use the historic average time for each event rounded up to the next whole minute for the length of each event. He needs to schedule these 10 events, in order, beginning at 8 a.m. At what time will the men‛s relay begin, according to his schedule? Event Avge Time of Event (min:sec) Women‛s Indiv Freestyle 2:23 Women‛s Indiv Butterfly 3:20 Women‛s Indiv Backstroke 3:10 Women‛s Indiv Breaststroke 2:56 Women‛s Relay 9:27 Men‛s Indiv Freestyle 2:10 Men‛s Indiv Butterfly 3:01 Men‛s Indiv Backstroke 2:54 Men‛s Indiv Breaststroke 2:45 Men‛s Relay 8:15 °F Each degree in the Celsius temperature system is 9 of a degree in the Fahrenheit 8. ________ 5 system, and 32°F = 0°C. What is the Fahrenheit temperature when the Celsius temperature is 45°? 9. ________ What number should be added to both the numerator and the denominator of get a fraction equivalent to 54 ? 1 5 to cm A rectangle has a perimeter of 38 cm. Three-fourths of its width is equal to one10. _______ fifth of its length. What is the measure of the length? MATHCOUNTS 2005-2006 37 Warm-Up 9 inches The perimeter of a rectangle is 24 inches. The length is 8½ inches. What is the 1. ________ width? Express your answer as a mixed number. $ 2. ________ Mrs. Goodteacher is ordering supplies for her 28 students. The chart shows what she needs and how these items can be purchased, only in the quantities listed, through her available supplier. A gross is 12 dozen, and a pound is 16 ounces. What is the minimum total cost for her supplies? Item Quantity Needed Purchase Info A Each student needs 2 pieces 1 gross for $2.38 B Every 2 students share 1 piece 1 dozen for $2.27 C Each student needs 1 ounce 1 pound for $2 D Every 4 students share 1 piece $10 each ways Four people are sitting around a table. A name tag is made for each person. In how 3. ________ many ways can these name tags be passed out so that every person gets the wrong name tag? flowers Every other time that Cheri Linn saw a friend today, she gave away half of 4. ________ her flowers. At the end of the day, she had five flowers left. How many flowers did she start the day with if she saw eight friends? pgs A recent autobiography used 2808 digits to number the pages of the book. How 5. ________ many numbered pages are in the book? 6. ________ A positive two-digit integer less than 50 with two different digits, both of which are odd, is chosen at random. What is the probability that the number is prime? Express your answer as a common fraction. inches Points A, B, E and F are the corners of an 8½-inch by 11-inch sheet of paper. 7. ________ Corners A and B are folded as shown below so that they meet at point D. The paper is then unfolded. In the same manner, corners E and F meet at point G. What is the length of segment GD? Express your answer as a mixed number. B E 11” A B C F A B D E G A D F squares What is the least number of additional squares that must be drawn to 8. ________ make this figure have a vertical line of symmetry and a horizontal line of symmetry? The figure may not be rotated. solns How many positive integer solutions does the inequality 2x + 9 > 4x + 1 have? 9. ________ 10. _______ A sequence of 20 terms is formed by using the rule f (x ) = 4x − 1 for x = 1, 2, 3, …, 20. What is the sum of the last eight terms? MATHCOUNTS 2005-2006 41 Warm-Up 10 coasters The box-and-whisker plot below shows the speed of the 25 fastest roller coasters in 1. ________ the world. What is the largest possible number of roller coasters that could have a speed greater than 80 mph? Speeds (mph) of the Fastest Roller Coasters in the World 74 75.5 80 94 120 sq cm Each curve in the logo is a semicircle with a 2. ________ radius of 8 cm. What is the area of the logo? Express your answer in terms of π. liters A 10% potassium chloride solution is needed for use in a lab. How much water must 3. ________ be added to dilute 300 liters of a 50% potassium chloride solution for the lab‛s use? 4. ________ The sum of the digits of a positive two-digit integer is 45 less than the integer. What is the tens digit of the integer? triangles How many non-congruent triangles can be drawn if each one must have one side of 5. ________ length four inches, one side of length five inches and one 90º angle? diagonals A square has two diagonals, and a convex pentagon has five diagonals. How many 6. ________ diagonals does a convex decagon have? $ 7. ________ Connie has 53 the amount of money as Joe. If Joe gives Connie $3, then Connie would have three times the money Joe has. How many dollars did Connie have originally? 20 points Vanessa takes a test worth 85 points. If the minimum percent required to earn an 8. ________ A is 90%, and no rounding can be done, what is the least number of whole points she can earn on the test to receive an A? students A college class has exactly enough students to form eight equal 9. ________ rows. On Monday a student is absent, and the professor is able to seat the students into five equal rows. On Tuesday two students are absent, and the professor can seat the students into nine equal rows. What is the least possible number of students in the class? 10. _______ Bag A contains four yellow marbles, two green marbles and three blue marbles. Bag B contains four blue marbles and four green marbles. If you choose one marble from each bag, what is the probability that you end up with one blue marble and one yellow marble? Express your answer as a common fraction. MATHCOUNTS 2005-2006 43 Warm-Up 11 1. ________ A flight from San Francisco to New York takes five hours, but the return flight takes six hours. When it is 4 p.m. in New York, it is only 1 p.m. in San Francisco. The flight leaves San Francisco at noon and arrives in New York at x o‛clock, and the return flight the next day leaves New York at noon and arrives in San Francisco at y o‛clock. What is the positive difference of x and y ? pieces A 6” by 6” grid made from 1” pieces of wire glued together at their 2. ________ ends is shown here. How many pieces of wire would be required to create a similar 10” by 10” grid? 3. ________ What is the least positive integer that has each of the first eight positive integers as factors? $ 4. ________ Jackson and Lisa divide some money in a ratio of 3:2, respectively. If Lisa gives Jackson $6, then Jackson will have twice as much money as Lisa will have. How much money did Lisa have after the original division of money? sq units The length of a rectangle is 2x + 5 units, the width is 4x + 15 units and the 5. ________ perimeter of the rectangle is 100 units. What is the area of the rectangle? ° 6. ________ The consecutive angles of a particular trapezoid form an arithmetic sequence. If the largest angle measures 120°, what is the measure of the smallest angle? voters Sandy originally beat Terry by 82 votes in a school election. In a 7. ________ re-vote, all of Terry‛s supporters voted for her again, but some of Sandy‛s supporters switched to Terry. If Terry beat Sandy by four votes in the re-vote, how many voters switched? VOTES 8. ________ Which of the following graphs represents the solution to x 2 > −4x − 3? A. 4 2 0 2 4 B. 4 2 0 2 4 4 2 0 2 4 D. C. 4 2 0 2 4 4 2 0 2 4 E. integers How many positive integers less than or equal to 250 are relatively prime with 250? 9. ________ 10. _______ The positive integers 2 through 2006 have median 1004. If the odd integers are all increased by 1 and the even integers are all decreased by 1, what is the new median? MATHCOUNTS 2005-2006 47 Warm-Up 12 1. ________ A particular positive three-digit integer is divisible by 5. The integer also is divisible by 11. The sum of the three digits of the integer is 13. What is the integer? 2. ________ A European train compartment has six seats. Four of the seats are broken. Wilhelm needs to fill out a form like the one here to indicate that there are broken seats. If he randomly checks off four of the seats in the diagram, what is the probability that he marked the correct seats? Express your answer as a common fraction. Maintenance Report Door $ 3. ________ At the fish market, a whole fish costs $4.20. If the body costs $1 more than the tail, and the body costs $2 more than the head, how much does the head cost? 4. ________ What is the greatest integer value of n such that 635,040 is divisible by 2n ? cu in A ream of paper consists of 500 sheets. If a standard box of 8½” by 11” paper holds 5. ________ exactly ten reams with no extra space, and a sheet of paper is .004 inches thick, what is the box‛s volume? 6. ________ Four positive four-digit integers with a sum of 23,750 each contain the digit 1 in a different position. If the 1s are all removed, the list of integers becomes 982, 829, 982, 298. What was the last four-digit number in the original list? 7. ________ Scott and Deric are playing the Shade Game. Two standard dice are rolled, and the sum of the numbers shown is the number of blank squares that must now be shaded in. It is Deric‛s turn, and his grid is shown here. What is the probability that Deric will win on this turn by rolling the exact sum needed to shade the rest of the grid? Express your answer as a common fraction. 8. ________ Given the function f (a , b , c , d ) = d ÷ a − a b c − b c , what is the value of f (1, 1, 1, −1)? 9. ________ A positive integer is 24 less than half its square. What is the integer? 10. _______ A polygonal region is bounded by the lines x = 0, x = 4, y = −1 and y = 4. The region is then rotated about the y -axis to form a solid. Which of these nets is the approximate net of the solid formed? (The nets are not drawn to scale.) MATHCOUNTS 2005-2006 A. B. C. D. 49 Warm-Up 13 1. ________ What is the maximum product of two integers whose sum is 18? regions When two lines are drawn in a plane, they can create four non-overlapping regions. 2. ________ If five straight lines are drawn in the plane, what is the largest number of nonoverlapping regions they can create? floor One elevator leaves the 1st floor of a building at exactly 9 a.m. It takes 3. ________ five seconds to travel between floors, stays 10 seconds at each floor and stops at each floor on its way to the top. Another elevator leaves the 12th floor (the top floor) at exactly 9 a.m., takes five seconds to travel between floors, stays 15 seconds at each floor and stops at each floor on its way to the bottom. What is the first floor on which both elevators will be available at the same time? sq cm 4. ________ 26 cm A right square pyramid with a base measuring 20 cm by 20 cm is unfolded into this net shown. Each side of the net not connected to the square measures 26 cm. What is the area of the net of the square pyramid? 5. ________ A set of five different positive integers has mean 8 and median 8. What is the greatest possible value one of the integers could have? coins Sam and Sally each have a pocketful of U.S. coins. Sam has 6. ________ 49 cents while Sally has 43 cents. Neither has a sufficient number of any coin to exchange for a larger value coin. For example, neither has five pennies to exchange for one nickel. What is the least number of coins they could have if they combine their coins? 7. ________ Russell rolls a pair of dice with his eyes closed. His sister tells him that the sum of the dice is greater than or equal to nine. Knowing this information, Russell calculates the probability that he rolled doubles. What is that probability expressed as a common fraction? 8. ________ The numbers 3, a , b , c , d , 38, … form an arithmetic sequence. What is the value of the sum b + c ? sq units Each student in a class is asked to draw a rectangle with integral side lengths and a 9. ________ perimeter of 30 units. Each student then calculates the area of the rectangle s/he drew. What is the greatest possible difference of the areas of two of the students‛ rectangles? grams Four cashews and three walnuts weigh 10 grams; two walnuts and six pecans weigh 10. _______ 13 grams; and one cashew, one walnut and two pecans weigh 6 grams. Assuming each cashew has the same weight, each walnut has the same weight and each pecan has the same weight, how much does one walnut weigh? MATHCOUNTS 2005-2006 53 Warm-Up 14 items Pens cost $2, and mechanical pencils cost $3. If Dominic spent exactly $100 on 1. ________ these items, what is the least number of items he could have bought? sq cm The boundary of the figure to the right is comprised of eight 2. ________ semicircular arcs, each with a diameter of 7 cm. What is the area of the enclosed region? min Each cartoon on a certain DVD lasts eight minutes, and there is a ten-second break 3. ________ between every two consecutive cartoons. Including the breaks, how long will it take to watch all ten cartoons? Express your answer as a mixed number. runs The Monroe Monkeys scored in each of the odd-numbered innings of a 9-inning 4. ________ baseball game. The Freeport Ferrets scored in each of the even-numbered innings of the game. The two teams never scored in the same inning. The Monkeys scored consecutive even numbers of runs starting with two runs in the first inning. The Ferrets scored consecutive odd numbers of runs starting with one run in the second inning. By how many runs did the winning team win? buses An Ausdal bus from Austin to Dallas leaves every hour on the hour. An Ausdal bus 5. ________ from Dallas to Austin leaves every hour on the half-hour. The trip between the two cities takes three hours. How many Ausdal buses going in the opposite direction does an Ausdal bus from Austin to Dallas pass? $ 6. ________ The chart here shows the prices for different seattypes at a stadium for professional soccer. If every seat in the stadium is sold for a game, what is the average price per seat? Section Price No. of Seats Avail. West, center $10 1000 West, end $8 1500 East, center $8 800 East, end $6 1700 weeks Paitlyn estimates she will need $750 for the computer she wants. She baby-sits for 7. ________ the Knox family every Saturday for $5.50 per hour. On average, she baby-sits 4.5 hours each Saturday. What is the fewest number of weeks she must work in order to save enough money for the computer? 8. ________ Three digits are randomly selected without replacement from two 2s, three 6s and one 9. What is the probability that the sum of the three digits is a multiple of 2? Express your answer as a common fraction. 9. ________ If the reciprocal of (x − 2) is (x + 2), what is the greatest possible value of x ? Express your answer in simplest radical form. sq cm In this figure of three non-overlapping regular polygons, the 10. _______ area of the equilateral triangle is 43 square centimeters. What is the area of the hexagon? Express your answer as a common fraction in simplest radical form. MATHCOUNTS 2005-2006 55 Warm-Up 15 min Alex can bike one lap of a racecourse in 30 minutes at 25 miles per hour How many 1. ________ minutes does it take Lance to bike the same lap at 30 miles per hour? 2. ________ A point (a , b ) is located in the third quadrant of the Cartesian plane. The coordinates are changed according to the functions given: f (a ) = a 2 + 2 and g (b ) = b − 4. In which quadrant is (f (a ), g (b )) located? 3. ________ The mean of a collection of five positive integers is 6. What is the greatest possible median of the collection? Start 4. ________ In the maze to the right, you may move only vertically or horizontally. Each box you move to must have a fraction that is a value less than the fraction of the box you are currently in. If you start at the indicated box, what fraction is in the last box of your path on the bottom row? 34 8 20 5 21 10 4 2 32 8 25 5 5 2 9 3 24 5 22 4 12 5 7 3 23 18 9 6 2 4 11 18 20 3 5 5 24 20 33 8 6 9 8 34 15 2 9 8 5. ________ If 2x > 8000, what is the least possible integer value of x ? 13 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 ••• 6. ________ The whole numbers are listed in order and placed in this arrangement. Each row has two more entries in it than the row above it. What is the first number in the row containing the integer 500? sq units Quadrilateral ABCD is a rectangle, and its length is twice 7. ________ its width. The perimeter of the rectangle is 60 units. Points P, Q, R and S are midpoints of each side. What is the area of quadrilateral PQRS? A P S B Q D R C cuts A deck of 52 playing cards is printed on a 4 by 13 rectangular grid. If stacking the 8. ________ pieces is allowed after each cut, but no folding is allowed, what is the least number of cuts required to separate all the cards? (Cuts can be any length, but may go in only one direction per cut.) units In triangle ABC, m ∠ A = 30° and m ∠ C = 105°. If AC = 8 units, what is the length of 9. ________ segment BC? Express your answer in simplest radical form. 10. _______ A bag contains six white marbles, five red marbles and four blue marbles. Norka will reach into the bag and pull out three marbles at the same time. What is the probability that all three marbles will be the same color? Express your answer as a common fraction. MATHCOUNTS 2005-2006 59 Warm-Up 16 ways In how many ways can this figure be colored if the five regions 1. ________ must be colored with either red, white or blue and no two bordering regions can be the same color? 2. ________ The lines y = ax + 1 and y = 1x + a , with a ≠ 1, intersect in exactly one point. What is the x -coordinate of that point? ways A band has four trumpets, three trombones, three saxophones and two drums. If 3. ________ each section has its own set of seats, in how many ways can the 12 band members be seated? trumpets trombones drums saxophones 4. ________ Miss James has a license plate with six digits. If commas were 05 IDAHO 06 ______ placed after the second and fourth digits, it would look like the number is a list of three consecutive even numbers in increasing order. The first of those numbers has three different prime factors. What is the least six-digit number that can be on Miss James‛ license plate? 5. ________ When the least common multiple of 8 and 20 is multiplied by the greatest common factor of 8 and 20, what is the result? fractions Think about all of the positive common (reduced) fractions with a denominator of 11 6. ________ 7 or less. How many of these common fractions have a value less than 11 ? 7. ________ If 12x + 57 = 29, what is the value of 12 x + 92 ? min Mr. Harter drives from Chicago to Milwaukee on Sunday. On Monday when he makes 8. ________ the return trip, he figures that if he increases his speed by one-fifth, he can make the trip in 80 minutes. How many minutes did the first trip take? sq cm The sphere has radius 10 cm, and the smaller circle‛s center 9. ________ is 6 cm from the center of the sphere, as shown in the figure. What is the area of the smaller circle in terms of π ? 6 10 ways In how many ways can two slashes be inserted between the letters of 10. _______ MATHCOUNTS to create three groups of letters? Each “group” of letters must contain at least one letter. MATHCOUNTS 2005-2006 61 Warm-Up 17 Columns 1 2 3 4 Row 1 1 Row 2 2 3 Row 3 4 5 6 7 Row 4 8 9 10 11 5 6 ... 12 13 14 15 ... 1. ________ The positive integers are written in this pattern where the number of entries in each successive row is twice the number of entries in the previous row. In which column is the number 2006? 2. ________ The sum of the first N positive integers is 2006. Wait, that‛s impossible... one of those integers must not have been included! If in fact exactly one of the first N positive integers was not included in the sum, which integer must that have been? ways Each of the six identical faces of a cube is to be painted either gray or white. How 3. ________ many distinct cubes are possible? (Notice that the two cubes shown below are not distinct. They are different orientations of the same cube.) 4. ________ The sum of the numerator and denominator of a fraction is 160. The fraction is equivalent to 73 . What is the original fraction? 5. ________ Scott and Deric are playing the Shade Game. Three standard dice are rolled, and the sum of the numbers rolled is the number of blank squares that must then be shaded. It is Deric‛s turn, and his grid is shown here. What is the probability that Deric will roll the exact sum needed to shade the rest of the grid on this turn? Express your answer as a common fraction. 6. ________ Ben owns five pairs of socks, and each pair is a different color. If he loses three socks, what is the probability that he has only two complete pairs left? Express your answer as a common fraction. 7. ________ Parallelogram ABCD has diagonals that intersect at point E. Suppose that AB = 6x + 5, BC = 2x , AE = 4x − 3 and EC = 3x + 4. Is angle ABC an acute, right or obtuse angle? 8. ________ The three integers e , f and g are distinct primes such that e = f 2 + g 2. What is the greatest possible value of e less than 100? 9. ________ Nine points are to be evenly spaced on a standard number line. They are labeled A through I, from left to right. The coordinates A I of points A and I are 1 and 10, respectively. What is the coordinate of point B? Express 10 1 your answer as a mixed number. 10. _______ Define the operation ◊ for any two real numbers a and b as a ◊b = 2a (a + b ). What is the value of p such that 3◊(4◊p ) = 1122? Problem #2 submitted by 2005 national competitor Noah Arbesfeld, Lexington, Mass. MATHCOUNTS 2005-2006 65 Warm-Up 18 1. ________ In the game “Dicey Numbers,” two dice are rolled, and the two numbers are used to form a two-digit number. For example 6 and 1 form either the number 61 or the number 16. The number is then changed as described below. If your number is greater than 36, take 80% of the number and round to the nearest whole number. If your number is less than 15, take 120% of the number and round to the nearest whole number. If the original number was prime, add 5 to its current value. If Sherry rolls a 1 and 4, what is the difference between the greatest final value and least final value that could result? tickets At a carnival game, Katya chooses a number from 1 through 6 and rolls two ordinary 2. ________ dice. If her number comes up on one die, she wins three tickets. If her number comes up on both dice, she wins six tickets. If she plays all morning, how many tickets would she expect to win, on average, per turn? 3. ________ The points (x, y) represented in this table lie on a straight line. When the equation of this line is written in the form y = Ax + B, what is the value of A + B? x y 2 t−2 t 7 v v+6 4. ________ A non-zero fraction‛s denominator is increased by 1 and its numerator is decreased by 50%. If the resulting fraction is 31 of the original fraction, what is the original fraction‛s denominator? ° 5. ________ When the triangular piece of paper ABC is folded along the dotted line AF, point B lands on side AC. We know AB is 1 unit and AC is 2 units. If the measure of angle B is 90°, what is the measure of angle AFC? B F C A 6. ________ Pat and Grady were both born on May 15. Pat was born in 1958, and Grady was born in 1966. In what year did Pat‛s age become three times Grady‛s age? % 7. ________ Carbohydrates provide four calories per gram, and fats provide nine calories per gram. If 36% of the calories Curtis ate at lunch came from fats and 64% came from carbohydrates, what percent of the grams eaten came from carbohydrates? cards Jill gives half her cards to Bill; Bill then gives half of his cards to 8. ________ Jill. Jill and Bill repeat this two-step process twice more. If Jill now has 37 cards and Bill has 19 cards, with how many cards did Bill start? ways The local sandwich shop has three cash registers. In how many possible ways can 9. ________ four customers line up at the registers? cu cm A pyramid has a rectangular base 10 cm by 25 cm and four triangular faces; two 10. _______ opposing faces both with altitude 13 cm and the other two faces with altitudes of 20 cm and 15 cm. What is the volume of the pyramid? MATHCOUNTS 2005-2006 67