ΒΆ 1986 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1986 AHSME problems, please refer below:
Problem 1:
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Problem 2: If the line in the -plane has half the slope and twice the -intercept of the line , then an equation for is
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Problem 3: In the figure, has a right angle at and . If is the bisector of , then
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Problem 4: Let be the statement
"If the sum of the digits of the whole number is divisible by , then is divisible by ."
A value of which shows to be false is
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Problem 5: Simplify .
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Problem 6: Using a table of a certain height, two identical blocks of wood are placed as shown in Figure . Length is found to be inches. After rearranging the blocks as in Figure , length is found to be inches. How high is the table?
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Problem 7: The sum of the greatest integer less than or equal to and the least integer greater than or equal to is . The solution set for is
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Problem 8: The population of the United States in was . The area of the country is square miles. There are square feet in one square mile. Which number below best approximates the average number of square feet per person?
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Problem 9: The product equals
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Problem 10: The permutations of are arranged in dictionary order, as if each were an ordinary five-letter word. The last letter of the word in this list is
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Problem 11: In , , and . Also, is the midpoint of side and is the foot of the altitude from to . The length of is
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Problem 12: John scores on this year's . Had the old scoring system still been in effect, he would score only for the same answers. How many questions does he leave unanswered?
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Problem 13: A parabola has vertex . If is on the parabola, then equals
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Problem 14: Suppose hops, skips and jumps are specific units of length. If hops equals skips, jumps equals hops, and jumps equals meters, then one meter equals how many skips?
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Problem 15: A student attempted to compute the average, , of , and by computing the average of and , and then computing the average of the result and . Whenever , the student's final result is
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Problem 16: In and side is extended, as shown in the figure, to a point so that is similar to . The length of is
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Problem 17: A drawer in a darkened room contains red socks, green socks, blue socks and black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.)
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Problem 18: A plane intersects a right circular cylinder of radius forming an ellipse. If the major axis of the ellipse is longer than the minor axis, the length of the major axis is
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Problem 19: A park is in the shape of a regular hexagon km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of km. How many kilometers is she from her starting point?
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Problem 20: Suppose and are inversely proportional and positive. If increases by , then decreases by
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Problem 21: In the configuration below, is measured in radians, is the center of the circle, and are line segments, and is tangent to the circle at .
A necessary and sufficient condition for the equality of the two shaded areas, given , is
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Problem 22: Six distinct integers are picked at random from . What is the probability that, among those selected, the second smallest is
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Problem 23: Let . How many positive integers are factors of
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Problem 24: Let , where and are integers. If is a factor of both and , what is
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Problem 25: If is the greatest integer less than or equal to , then
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Problem 26: It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the and axes and so that the medians to the midpoints of the legs lie on the lines and . The number of different constants for which such a triangle exists is
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Problem 27: In the adjoining figure, is a diameter of the circle, is a chord parallel to , and intersects at , with . The ratio of the area of to that of is
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Problem 28: is a regular pentagon. dropped from onto extended and extended, respectively. Let be the center of the pentagon. If , then equals
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Problem 29: Two of the altitudes of the scalene triangle have length and . If the length of the third altitude is also an integer, what is the biggest it can be?
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Problem 30: The number of real solutions of the simultaneous equations
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The problems on this page are the property of the MAA's American Mathematics Competitions