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(1) The median of the set of numbers {12, 38, 45, x , 14} is five less than the mean. If x is a negative integer, what is the value of x ? (2) The Asian elephant has an average gestation period of 609 days. How many weeks is this gestation period? What is the sum of the reciprocals of the positive integer factors of 28? (3) (4) The circumference of a particular circle is 18 cm. In square centimeters, what is the area of the circle? Express your answer as a common fraction in terms of π. (5) For what value of x is 23 × 3x = 72? (6) Based on this graph, what percent of viewers watched one hour or more of television? More than 2 hours less than 1 hour 1-2 hours (7) (8) Lee can make 18 cookies with two cups of flour. How many cookies can he make with three cups of flour? √ The hypotenuse and a leg of a particular right triangle are 97 inches and 4 inches, respectively. The area of this triangle is what common fraction of a square foot? (9) The measures of the interior angles of a particular triangle are in a 5:6:7 ratio. What is the measure, in degrees, of the smallest interior angle? (10) The ratio of girls to boys participating in intramural volleyball at Ashland Middle School is 7 to 4. There are 42 girls in the program. What is the total number of participants? (11) What is the median of all values defined by the expression 2x − 1, where x is a prime number between 0 and 20? (12) What is the median of the prime numbers between 20 and 50? (13) Before taking his last test in a class, the arithmetic mean of Brian’s test scores is 91. He has determined that if he scores 98 on his last test, the arithmetic mean of all his test scores will be exactly 92. How many tests, including the last test, does Brian take for this class? (14) A particular triangle has sides of length 14 cm, 8 cm and 9 cm. In centimeters, what is the perimeter of the triangle? (15) One of the following four-digit numbers is not divisible by 4: 3544, 3554, 3564, 3572, 3576. What is the product of the units digit and the tens digit of that number? (16) (17) (18) What is the mean of 1 2 and 87 ? Express your answer as a common fraction. The set {5, 8, 10, 18, 19, 28, 30, x } has eight members. The mean of the set’s members is 4.5 less than x . What is the value of x ? Calculate: 9 − 8 + 7 × 6 + 5 − 4 × 3 + 2 − 1 (19) There are eight furlongs in a mile. There are two weeks in a fortnight. The British cavalry traveled 2800 furlongs in a fortnight. How many miles per day did the cavalry average? (20) An environmental agency needs to hire a number of new employees so that 85 of the new employees will be able to monitor water pollution, 73 of the new employees will be able to monitor air pollution, and 27 of the new employees will be able to monitor both. (These 27 are included in the 85 and 73 mentioned above.) What is the minimum number of employees that need to be hired? (21) Simplify: √ 2.52 −0.72 2.7−2.5 . (22) Think of a number. Double the number. Add 200. Divide the answer by 4. Subtract one-half the original number. What is the value of the result? (23) The mean of three test scores is 74. What must a fourth test score be to increase the mean to 78? (24) What is the greatest common factor of 154 and 252? (25) How many positive integer values of x are solutions to the inequality 10 < −x + 13? (26) Raquel has collected $3.80 in nickels and dimes. She has exactly 48 nickels. How many dimes does she have? (27) The gauge of an oil tank indicated that the tank was 17 full. After 240 gallons of oil were added to the tank, the gauge indicated that the tank was 47 full. How many gallons of oil will the tank hold, assuming the gauge is accurate? (28) A figure skater is facing north when she begins to spin to her right. She spins 2250 degrees. Which direction (north, south, east or west) is she facing when she finishes her spin? (29) When Cedric walked into a party, two-thirds of those invited had already arrived. Six more people arrived just after Cedric, bringing the number at the party to 56 of those invited. What was the total number of invited guests? (30) Bekah has three brass house number digits: 2, 3 and 5. How many distinct numbers can she form using one or more of the digits? 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 -14 87 weeks 2 81 π square centimeters 2 75 percent 27 cookies 1/8 50 66 participants 1087 37 7 tests 31 centimeters 20 11/16 22 37 25 miles per day 131 12 50 90 14 2 14 560 gallons east 42 guests 15 numbers Problem ID A34C 014C 1003 1CC1 0CC1 BC31 CBC1 2CC1 414C B1D2 A2AC 50B3 04A2 5C31 3222 B113 3102 22D2 B041 1BD3 D5D3 514C 00D4 0203 A14C BC543 CAA2 2322 22A4 2443 Copyright MATHCOUNTS Inc. All rights reserved Solutions (1) -14 ID: [A34C] Since x is negative, the median of the set is 14. Therefore, the mean of the set is 14 + 5 = 19, and its sum is 19 · 5 = 95. Since 12, 38, 45, and 14 sum to 109, the remaining integer x must be 95 − 109 = −14 . (2) 87 weeks ID: [014C] There are 7 days in one week, so 609 days equal 609/7 = 87 weeks. (3) 2 ID: [1003] The prime factorization of 28 is 22 · 7. Therefore, the positive integer divisors are 1, 2, 22 (= 4), 7, 2 · 7 (= 14), and 28 itself. The sum of their reciprocals is 1 1 1 1 1 1 1 + 2 + 4 + 7 + 14 + 28 . We can put this over a common denominator of 28, so that it simplifies to 28 · 1 14 · 1 7 · 1 4 · 1 2·1 1·1 + + + + + 28 · 1 14 · 2 7 · 4 4 · 7 2 · 14 1 · 28 28 + 14 + 7 + 4 + 2 + 1 56 = = 2. 28 28 (Notice that the original number, 28, can be used as a common denominator, and that when the reciprocals of the divisors are put over this common denominator, the new numerators are just the divisors of 28 listed in reverse order! Can you figure out why this happens?) = (4) 81 π square centimeters ID: [1CC1] If r is the radius of the circle, then the circumference is2πr .2 Setting 2πr equal to 18 cm, 9 81 square centimeters. we find r = 9/π cm. The area of the circle is πr 2 = π = π π (5) 2 ID: [0CC1] Since the prime factorization of 72 is 72 = 23 · 32 , we have x = 2 . (6) 75 percent ID: [BC31] The sector corresponding to “less than 1 hour” has a central angle of 90 degrees. 90 = 14 = 25% of viewers. It follows that 100% − 25% = 75% of Therefore, it represents 360 viewers watch one hour of television or more. (7) 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 . (8) 1/8 ID: [2CC1] q √ 2 The measure of the missing leg is 97 − 42 = 9 inches, by the Pythagorean theorem. Converting 9 inches and feet, we find that the area of the triangle is 4 inches to 1 1 1 3 1 (base)(height) = ft. ft. = square feet. 2 2 3 4 8 (9) 50 ID: [414C] Choose k so that the smallest angle measures 5k degrees. Then the measures of the other two angles are 6k degrees and 7k degrees. Since the measures of the angles in a triangle sum to 180 degrees, we have 5k + 6k + 7k = 180 =⇒ 18k = 180 =⇒ k = 10. The smallest angle measures 5k = 5(10) = 50 degrees. (10) 66 participants ID: [B1D2] We have girls : boys = 7 : 4. Multiplying both parts of the ratio by 6 gives girls : boys = 42 : 24, so there are 24 boys to go with the 42 girls. This gives a total of 42 + 24 = 66 participants. (11) 1087 ID: [A2AC] No solution is available at this time. (12) 37 ID: [50B3] The easiest way to attack this problem is to simply write out a list of the prime numbers between 20 and 50, as follows: 23, 29, 31, 37, 41, 43, 47. The median is the middle number, which is 37 . (13) 7 tests ID: [04A2] Let S be the sum of all of Brian’s test scores up to this point, and let n be the number of tests Brian has taken up to this point. Thus the arithmetic mean of his scores now is Sn and the arithmetic mean of his scores after getting a 98 on the last test will be S+98 n+1 . This gives the system of equations: S + 98 S = 91 = 92 n n+1 From the first equation we have S = 91n. Substituting this into the second equation gives: S + 98 = 92 n+1 S + 98 = 92(n + 1) 91n + 98 = 92n + 92 92n − 91n = 98 − 92 n=6 So Brian has to take n + 1 = 7 tests. (14) 31 centimeters ID: [5C31] The perimeter of a polygon is defined to be the sum of the measures of the sides of the polygon. Therefore, the perimeter of a triangle whose sides measure 14 cm, 8 cm, and 9 cm is 31 centimeters. (15) 20 ID: [3222] A number is divisible by 4 if its last two digits are divisible by 4. The only given number that is not divisible by 4 is 3554 because 54 is not divisible by 4. The product of the units digit and the tens digit of 3554 is 5 · 4 = 20 . (16) 11/16 ID: [B113] The sum of the two fractions is 1 2 + 7 8 = 11 8 . So their mean is 1 2 11 8 (17) 22 ID: [3102] Writing the information presented as an equation, 5 + 8 + 10 + 18 + 19 + 28 + 30 + x = x − 4.5. 8 Solving, x = 22 . = 11 . 16 (18) 37 ID: [22D2] By the order of operations, we perform the multiplications before the additions and subtractions: 9 − 8 + 7 × 6 + 5 − 4 × 3 + 2 − 1 = 9 − 8 + 42 + 5 − 12 + 2 − 1 = 1 + 42 + 5 − 12 + 2 − 1 = 48 − 12 + 2 − 1 = 36 + 1 = 37 . (19) 25 miles per day We have ID: [B041] 14 days = 1 fortnight and 8 furlongs = 1 mile, and we are asked to convert a quantity whose units are furlongs per fortnight to miles per day. We divide the first equation by 14 days to obtain a quantity which is equal to 1 and has units of fortnight in the numerator. 1 fortnight 1= . 14 days Similarly, 1 mile . 1= 8 furlongs Since the right-hand sides of both of these equations are equal to 1, we may multiply them by 2800 furlongs per fortnight to change the units without changing the value of the expression: 1 mile furlongs miles 1 fortnight = 25 2800 · . fortnight 14 days 8 furlongs day (20) 131 ID: [1BD3] There are 85 + 73 = 158 jobs to be done. 27 people do two of the jobs, so that leaves 158 − 27 · 2 = 158 − 54 = 104 jobs remaining. The remaining workers do one job each, so we need 27 + 104 = 131 workers. We also might construct the Venn Diagram below. We start in the middle of the diagram, with the 27 workers who do both: Water Air 85 − 27 73 − 27 27 This gives us 27 + (73 − 27) + (85 − 27) = 131 workers total. (21) 12 ID: [D5D3] We have p √ √ √ 576/100 2.52 − 0.72 6.25 − 0.49 5.76 = = = 2.7 − 2.5 2.7 − 2.5 0.2 0.2 √ √ 576/ 100 24/10 2.4 = = = = 12 . 0.2 0.2 0.2 (22) 50 ID: [514C] Let x be the number we think of. We double x to obtain 2x , add 200 to find 2x + 200, divide by 4 to find 2x 200 x 2x + 200 = + = + 50. 4 4 4 2 After subtracting one-half of the original number, we are left with 50 . (23) 90 ID: [00D4] No solution is available at this time. (24) 14 ID: [0203] The prime factorizations of these integers are 154 = 2 · 7 · 11 and 252 = 22 · 32 · 7. The prime factorization of their greatest common divisor (GCD) must include all of the primes that their factorizations have in common, taken as many times as both factorizations allow. Thus, the greatest common divisor is 2 · 7 = 14 . (25) 2 ID: [A14C] We first solve the inequality: 10 < −x + 13 −3 < −x 3 > x. The only positive integers less than 3 are 1 and 2, for a total of 2 solutions. (26) 14 ID: [BC543] The 48 nickels are worth 48 · $0.05 = $2.40. The remaining dimes are worth $3.80 − $2.40 = $1.40. Because each dime is worth $0.10, there are $1.40 ÷ $0.10 = 14 dimes. (27) 560 gallons ID: [CAA2] No solution is available at this time. (28) east ID: [2322] Each full circle is 360 degrees. Dividing 360 into 2250 gives a quotient of 6 with a remainder of 90. So, she spins 90 degrees to her right past north, which leaves her facing east . (29) 42 guests ID: [22A4] Let P be the number of people invited to the party. Before Cedric arrived, there were 32 P people at the party. After Cedric and six others arrived, there are 23 P + 7 people at the party. Since this is the same as 56 P , we solve 23 P + 7 = 65 P to find that P = 42 . (30) 15 numbers ID: [2443] If Bekah uses only one digit, she can form three numbers. If she uses two digits, she has three choices for the tens place and two for the units, so she can form six numbers. Finally, if Bekah uses all three digits, she has three choices for the hundreds place, two for the tens, and one for the units, so she can form six numbers. Thus, Bekah can form 3 + 6 + 6 = 15 distinct numbers. Copyright MATHCOUNTS Inc. All rights reserved