• Document: MATHCOUNTS Sample Problems
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MATHCOUNTS Sample Problems (1) 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? (2) A play has two different male roles, two different female roles and two different 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? (3) Zach has three bags and a bunch of pencils to be placed into the bags. He is told to place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same. What is the greatest number of pencils he could have left over? (4) If the ratio of 2x − y to x + y is 2 to 3, what is the ratio of x to y ? Express your answer as a common fraction. (5) Set R is a set of rectangles such that (1) only the grid points shown here are used as vertices, (2) all sides are vertical or horizontal and (3) no two rectangles in the set are congruent. If R contains the maximum possible number of rectangles given these conditions, what fraction of the rectangles in set R are squares? Express your answer as a common fraction. (6) For a list of five positive integers, none greater than 100, the mean is 1.5 times the mode. If 31, 58, 98, x and x are the five integers, what is the value of x ? (7) The members of a band are arranged in a rectangular formation. When they are arranged in 8 rows, there are 2 positions unoccupied in the formation. When they are arranged in 9 rows, there are 3 positions unoccupied. How many members are in the band if the membership is between 100 and 200? (8) On planet Larky, 7 ligs = 4 lags, and 9 lags = 20 lugs. How many ligs are equivalent to 80 lugs? √ (9) What is the positive value of the expression x 3 − 2y when x = 5 and y = 2? (10) The first term of a sequence of positive integers is any two-digit integer. Each subsequent term is the sum of the tens digit and the square of the ones digit of the previous term. One possible sequence is 14, 17, 50, 5, 25,... . If the first term of one such sequence is 97, what is the 2008th term of the sequence? (11) The point at (a, b) on a Cartesian plane is reflected over the y -axis to the point at (j, k). If a + j = 0 and b + k = 0, what is the value of b? (12) Two identical CDs regularly cost a total of $28. What is the cost in dollars of five of these CDs? (13) Grady rides his bike 60% faster than his little brother Noah. If Grady rides 12 miles further than Noah in two hours, how fast in miles per hour does Noah ride? (14) For a particular circle, a central angle of 75◦ will intercept an arc of length 10π feet. What is the radius of this circle? (15) Suppose that each distinct letter in the equation MATH = COU + NTS is replaced 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? C O U + N T S M A T H (16) Given that 6x + y = 15, the value of 3x can be written in terms of y as ay + b for some numbers a and b. What is the simplified value of a + b? (17) What is the least positive multiple of 72 that has exactly 16 positive factors? (18) Five termites are eating through a piece of wood, all beginning at the same edge and going in the same direction. Woody is 20 mm ahead of Muncher, Cruncher is 10 mm behind Woody, Muncher is 5 mm behind Nibbler, and Biter is 15 mm ahead of Cruncher. How many millimeters is the distance between the two termites that are the farthest apart? (19) 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 lis

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