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(1) Rico can walk 3 miles in the same amount of time that Donna can walk 2 miles. Rico walks a rate 2 miles per hour faster than Donna. At that rate, what is the number of miles that Rico walks in 2 hours and 10 minutes? (2) The first 102 positive integers are separated into successive groups of three consecutive integers. These groups are arranged in an array of three columns as shown 1 2 3 6 5 4 7 8 9 below. What is the sum of the numbers in the first column? 12 11 10 13 14 15 .. .. .. . . . 102 101 100 (3) (4) The cost of holding a concert is the sum of the fixed cost, which is the same no matter how many people attend, and the variable cost, which depends on the number of people attending. If the total cost for a concert attended by 1000 people is $75, 000 and the total cost of a concert attended by 1200 people is $85, 000, what is the number of dollars in the fixed cost when holding a concert? 84 is 40% of what number? (5) The sum of two numbers is 15. One number is doubled and the other is tripled. The sum of the two new numbers is 39. What is the positive difference between the original numbers? (6) Susie’s parents paid x dollars for their house and Brian’s parents paid $30,000 less for their house than Susie’s parents paid. How many dollars did Brian’s parents pay for their house? Express your answer as an algebraic expression in terms of x . (7) The formula d = 16t 2 is used to calculate the distance, d, in feet, a free falling object, starting from rest, will travel in t seconds. How many seconds will it take for a ball, starting from rest, to free fall from a height of 64 feet to the ground? (8) Tidy Painters will paint Taylor’s house for a charge of $30 per hour. It takes 40 hours for their crew to complete the job. How many dollars does the paint job cost? (9) Given the proportions your answer as a common fraction. a b = 3 7 and c b = 9 14 , what is the value of ca ? Express (10) John computes the sum of the elements of each of the 15 two-element subsets of {1, 2, 3, 4, 5, 6}. What is the sum of these 15 sums? (11) Juan, Carlos and Manu take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Manu will win? Express your answer as a common fraction. (12) A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to the number of units in its perimeter. What is the number of units in the perimeter of this rectangle? (13) Of the eggs produced by salmon, 80% hatch, and of those, 25% survive to migrate to the ocean. How many eggs are needed to produce 100 salmon that migrate to the ocean? (14) Of the 2400 jellybeans in a jar, 1650 are purple. Lucia then removes 150 purple jellybeans from the jar. What fraction of the jellybeans left in the jar are purple? Express your answer as a common fraction. (15) The lengths of the sides of isoceles △ABC are 3x + 62, 7x + 30, and 5x + 50 feet. What is the least possible number of feet in the perimeter of △ABC? (16) (17) Express the reciprocal of 2.3 as a common fraction. The rectangular region bounded by the lines with equations x = 1.2, x = 2.6, y = −0.2 and y = d has area 14 square units. What is the greatest possible value of d? Express your answer as a decimal to the nearest tenth. (18) Think of a number n. Double the number. Subtract 160. Divide the result by 4. Add 60. Subtract half the original number. Now square what you have. What is your answer? (19) Mathman rides speed in miles per hour? (20) Lee can make 18 cookies with two cups of flour. How many cookies can he make with three cups of flour? (21) The Benton Hotel has six floors and the same number of stairs between floors. It takes Peter 30 seconds to climb the stairs from the first floor to the third floor. At this rate, how many seconds will it take Peter to climb the stairs from the first floor to the sixth floor? 3 5 of a mile in 1 21 minutes on his bicycle. What is his average (22) What is the number of units in the distance between (2, 5) and (−6, −1)? (23) What fraction of a foot is equal to nine inches? Express your answer as a common fraction. x π ≤ 10? (24) What is the sum of all integers x that satisfy −5 ≤ (25) A number x is 3 larger than its reciprocal. What is the value of x − (26) Solve for x : 1 6 + 1 6 + 1 6 + 1 6 = 1 4 ? x x 24 . 1000 1001 , B) 1001 1000 , C) (27) Which of the following has a value closest to 1: A) 101 2 1 − 2−10 , or D) 100 ? (28) The operation ∗ is defined for non-zero integers as follows: a ∗ b = 1a + b1 . If a + b = 9 and a × b = 20, what is the value of a ∗ b? Express your answer as a common fraction. What is the greatest possible value of x + y such that x 2 + y 2 = 90 and (29) x y = 27. (30) The sum of the first twenty-one terms of an arithmetic series is 273. The fifth term is 7. What is the 49th term? Copyright MATHCOUNTS Inc. All rights reserved Answer Sheet Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Answer 13 1751 25,000 dollars 210 3 x − 30, 000 or −30, 000 + x 2 seconds 1200 dollars 3/2 105 1/7 18 units 500 eggs 2 3 232 10/23 9.8 400 24 27 cookies 75 seconds 10 3/4 376 81 16 C 9/20 12 51 Problem ID 2A001 A1A4 11D4 05B41 B355 CBC2 C1A2 C4A2 5B13 AC13 D1B4 4443 D1D2 D432 22C31 405C A5D3 C1B5 B3A2 CBC1 10D4 C1A5 C303 DCA4 4C14 2AC3 0455 3003 AB001 25B5 Copyright MATHCOUNTS Inc. All rights reserved Solutions (1) 13 ID: [2A001] Let the rate (in mph) at which Rico walks be r and the rate (in mph) at which Donna walks be d. We have the two equations 2 3 = r d r =d +2 The first equation comes from the fact that the time each of Rico and Donna traveled is equal to the distance traveled divided by the rate of travel. Substituting the second equation into the first to eliminate r , we have 2 3 = d +2 d ⇒ 3d = 2d + 4 ⇒d =4 Plugging the value of d into the second equation to find r , we get r = 4 + 2 = 6. Thus, the number of miles Rico traveled in 2 hours and 10 minutes is 6 · (2 + 10/60) = 13 . (2) 1751 ID: [A1A4] We split the numbers in the first column into the two arithmetic sequences 1, 7, . . . , 97, and 6, 12, . . . , 102. Each arithmetic sequence contains 102/6 = 17 terms. The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum of the terms in the first series is (1 + 97)/2 · 17 = 833, and the sum of the terms in the second series is (6 + 102)/2 · 17 = 918. Therefore, the total sum is 833 + 918 = 1751 . (3) 25,000 dollars ID: [11D4] No solution is available at this time. (4) 210 ID: [05B41] No solution is available at this time. (5) 3 ID: [B355] No solution is available at this time. (6) x − 30, 000 or −30, 000 + x ID: [CBC2] No solution is available at this time. (7) 2 seconds ID: [C1A2] No solution is available at this time. (8) 1200 dollars ID: [C4A2] No solution is available at this time. (9) 3/2 ID: [5B13] No solution is available at this time. (10) 105 ID: [AC13] Among the two-element subsets of {1, 2, 3, 4, 5, 6}, each element in {1, 2, 3, 4, 5, 6} appears 5 times, one time in the same subset with each other element. Thus, the desired 6·7 sum is 5(1 + 2 + 3 + 4 + 5 + 6) = 5 2 = 105 . (11) 1/7 ID: [D1B4] For Manu to win on his first turn, the sequence of flips would have to be TTH, which has 1 3 probability 2 . For Manu to win on his second turn, the sequence of flips would have to 6 be TTTTTH, which has probability 21 . Continuing, we find that the probability that 3n Manu wins on his nth turn is 12 . The probability that Manu wins is the sum of these probabilities, which is 1 1 1 1 23 = 1/7 , + + +··· = 23 26 29 1 − 213 where we have used the formula a/(1 − r ) for the sum of an infinite geometric series whose first term is a and whose common ratio is r . (12) 18 units ID: [4443] Let the two sides of the rectangle be a and b. The problem is now telling us ab = 2a + 2b. Putting everything on one side of the equation, we have, ab − 2a − 2b = 0 This looks tricky. However, we can add a number to both sides of the equation to make it factor nicely. 4 works here: ab − 2a − 2b + 4 = 4 ⇒ (a − 2)(b − 2) = 4 Since we don’t have a square, a and b must be different. It doesn’t matter which one is which, so we can just say a = 6 and b = 3. The perimeter is then 2(6 + 3) = 18 (13) 500 eggs ID: [D1D2] No solution is available at this time. (14) 2 3 ID: [D432] No solution is available at this time. (15) 232 ID: [22C31] No solution is available at this time. (16) 10/23 ID: [405C] No solution is available at this time. (17) 9.8 ID: [A5D3] No solution is available at this time. (18) 400 ID: [C1B5] No solution is available at this time. (19) 24 ID: [B3A2] No solution is available at this time. (20) 27 cookies ID: [CBC1] Let x be the number of cookies that Lee can make with three cups of flour. We can set up x the proportion 18 2 = 3 . Solving for x , we find that x = 27 . (21) 75 seconds ID: [10D4] No solution is available at this time. (22) 10 ID: [C1A5] p √ We use the distance formula: (−6 − 2)2 + (−1 − 5)2 = 64 + 36 = 10 . - OR We note that the points (2, 5), (−6, −1), and (2, −1) form a right triangle with legs of length 6 and 8. This is a Pythagorean triple, so the length of the hypotenuse must be 10 . (23) 3/4 ID: [C303] No solution is available at this time. (24) 376 ID: [DCA4] Multiply by π to rewrite the inequality as −5π ≤ x ≤ 10π. Since π ≈ 3.14, we have −5π ≈ −15.7 and 10π ≈ 31.4. Therefore, the integers satisfying the inequality are those between −15 and 31 inclusive. The sum of these integers is −15 − 14 − 13 − · · · + 29 + 30 + 31 = (−15 − 14 − · · · + 14 + 15) + 16 + · · · + 31 = (0) + 16 + · · · + 31 (16 + 31)(16) 2 = 376 , = where we have used the formula (first term + last term)(number of terms) 2 for the sum of an arithmetic series. (25) 81 ID: [4C14] The sentence is telling us, in algebra, x =3+ 1 x A more useful form for us is 1 =3 x From there, we can bring both sides to the fourth power: 4 1 = 81 x− x x− (26) 16 ID: [2AC3] To start, multiply both sides of this equation by 24, to give: 1 1 1 1 24 =x + + + 6 6 6 6 Distributing, we find: 24 6 + 24 6 + 24 6 + 24 6 = x , so 4 + 4 + 4 + 4 = x , which means x = 16 . (27) C ID: [0455] No solution is available at this time. (28) 9/20 ID: [3003] Note that a ∗ b = 1a + 1 b = a+b ab . We are given that a + b = 9 and ab = 20. If we substitute 9 these values into a+b . ab , we can see that a ∗ b = 20 (29) 12 ID: [AB001] (x + y )2 = x 2 + y 2 + 2x y = 90 + 2 · 27 = 144, so x + y = 12 or x + y = −12. We want the larger value, or x + y = 12 . (30) 51 ID: [25B5] No solution is available at this time. Copyright MATHCOUNTS Inc. All rights reserved