ΒΆ 1998 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1998 AHSME problems, please refer below:
Problem 1:
Each of the sides of five congruent rectangles is labeled with an integer, as shown above. These five rectangles are placed, without rotating or reflecting, in positions through so that the labels on coincident sides are equal.
Which of the rectangles is in position
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Problem 2: Letters , and represent four different digits selected from . If is an integer that is as large as possible, what is the value of
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Problem 3: If , and are digits for which
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Problem 4: Define to mean , where . What is the value of
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Problem 5: If , what is the value of
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Problem 6: If is written as a product of two positive integers whose difference is as small as possible, then the difference is
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Problem 7: If , then
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Problem 8: A square with sides of length is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find , the length of the longer parallel side of each trapezoid.
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Problem 9: A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average number of minutes of the talk heard by members of the audience?
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Problem 10: A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimeter of each of the congruent rectangles is . What is the area of the large square?
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Problem 11: Let be a rectangle. How many circles in the plane of have a diameter both of whose endpoints are vertices of
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Problem 12: How many different prime numbers are factors of if
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Problem 13: Walter rolls four standard six-sided dice and finds that the product of the numbers on the upper faces is . Which of the following could not be the sum of the upper four faces?
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Problem 14: A parabola has vertex at and has two -intercepts, one positive and one negative. If this parabola is the graph of , which of , and must be positive?
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Problem 15: A regular hexagon and an equilateral triangle have equal areas. What is the ratio of the length of a side of the triangle to the length of a side of the hexagon?
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Problem 16: The figure shown is the union of a circle and two semicircles of diameters and , all of whose centers are collinear. The ratio of the area of the shaded region to that of the unshaded region is
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Problem 17: Let be a function with the two properties:
for any two real numbers and , and
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What is the value of
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Problem 18: A right circular cone of volume , a right circular cylinder of volume , and a sphere of volume all have the same radius, and the common height of the cone and the cylinder is equal to the diameter of the sphere. Then
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Problem 19: How many triangles have area and vertices at , and for some angle
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Problem 20: Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that
the numbers are all different,
they sum to , and
they are in increasing order, left to right.
First, Casey looks at the number on the leftmost card and says, "I don't have enough information to determine the other two numbers." Then Tracy looks at the number on the rightmost card and says, "I don't have enough information to determine the other two numbers." Finally, Stacy looks at the number on the middle card and says, "I don't have enough information to determine the other two numbers." Assume that each person knows that the other two reason perfectly and hears their comments. What number is on the middle card?
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Problem 21: In an -meter race, Sunny is exactly meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finishes the second race?
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Problem 22: What is the value of the expression
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Problem 23: The graphs of and intersect when satisfies , and for no other values of . Find .
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Problem 24: Call a -digit telephone number memorable if the prefix sequence is exactly the same as either of the sequences or (possibly both). Assuming that each can be any of the ten decimal digits , the number of different memorable telephone numbers is
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Problem 25: A piece of graph paper is folded once so that is matched with , and is matched with . Find .
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Problem 26: In quadrilateral , it is given that , angles and are right angles, , and . Then
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Problem 27: A cube is composed of twenty-seven cubes. The big cube is 'tunneled' as follows: First, the six cubes which make up the center of each face as well as the center cube are removed as shown. Second, each of the twenty remaining cubes is diminished in the same way. That is, the center facial unit cubes as well as each center cube are removed.
The surface area of the final figure is
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Problem 28: In triangle , angle is a right angle and . Point is located on so that angle is twice angle . If , then , where and are relatively prime positive integers. Find .
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Problem 29: A point in the plane is called a lattice point if both and are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to
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Problem 30: For each positive integer , let
Let denote the smallest positive integer for which the rightmost nonzero digit of is odd. The rightmost nonzero digit of is
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The problems on this page are the property of the MAA's American Mathematics Competitions