ΒΆ 1985 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1985 AHSME problems, please refer below:
Problem 1: If , then
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Problem 2: In an arcade game, the "monster" is the shaded sector of a circle of radius cm, as shown in the figure. The missing piece (the mouth) has central angle . What is the perimeter of the monster in cm?
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Problem 3: In right with legs and , arcs of circles are drawn, one with center and radius , the other with center and radius . They intersect the hypotenuse in and . Then has length
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Problem 4: A large bag of coins contains pennies, dimes and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is
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Problem 5: Which terms must be removed from the sum
if the sum of the remaining terms is to equal
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A. and
B. and
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E. and
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Problem 6: One student in a class of boys and girls is to be chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is
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Problem 7: In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from right to left. Thus, in such languages means the same as in ordinary algebraic notation. If is evaluated in such a language, the result in ordinary algebraic notation would be
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Problem 8: Let be real numbers with and nonzero. The solution to is less than the solution to if and only if
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Problem 9: The odd positive integers, , , are arranged in five columns continuing with the pattern shown below. Counting from the left, the column in which appears is the
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A. first
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E. fifth
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Problem 10: An arbitrary circle can intersect the graph of in
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A. at most points
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E. more than points
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Problem 11: How many distinguishable rearrangements of the letters in have both the vowels first? (For instance, is one such arrangement, but is not.)
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Problem 12: Let and be distinct prime numbers, where is not considered a prime. Which of the following is the smallest positive perfect cube having as a divisor?
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Problem 13: Pegs are put in a board unit apart both horizontally and vertically. A rubber band is stretched over pegs as shown in the figure, forming a quadrilateral. Its area in square units is
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Problem 14: Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon?
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Problem 15: If and are positive numbers such that and , then the value of is
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Problem 16: If and , then the value of is
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Problem 17: Diagonal of rectangle is divided into three segments of length by parallel lines and that pass through and and are perpendicular to . The area of , rounded to one decimal place, is
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Problem 18: Six bags of marbles contain , , , , and marbles, respectively. One bag contains chipped marbles only. The other bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?
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Problem 19: Consider the graphs of and , where is a positive constant and and are real variables. In how many points do the two graphs intersect?
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A. exactly
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C. at least , but the number varies for different positive values of
D. for at least one positive value of
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Problem 20: A wooden cube with edge length units (where is an integer ) is painted black all over. By slices parallel to its faces, the cube is cut into smaller cubes each of unit edge length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is
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Problem 21: How many integers satisfy the equation
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Problem 22: In a circle with center is a diameter, is a chord, and . Then the length of is
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Problem 23: If and , where , then which of the following is not correct?
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Problem 24: A non-zero digit is chosen in such a way that the probability of choosing digit is . The probability that the digit is chosen is exactly the probability that the digit chosen is in the set
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Problem 25: The volume of a certain rectangular solid is , its total surface area is , and its three dimensions are in geometric progression. The sum of the lengths in cm of all the edges of this solid is
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Problem 26: Find the least positive integer for which is a non-zero reducible fraction.
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Problem 27: Consider a sequence , defined by:
What is the smallest value of for which is an integer?
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Problem 28: In , we have and . What is
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Problem 29: In their base representations, the integer consists of a sequence of eights and the integer consists of a sequence of fives. What is the sum of the digits of the base representation of the integer
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Problem 30: Let be the greatest integer less than or equal to . Then the number of real solutions to is
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The problems on this page are the property of the MAA's American Mathematics Competitions