¶ 1956 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1956 AHSME problems, please refer below:
Problem 1: The value of when is:
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Problem 2: Mr. Jones sold two pipes at each. Based on the cost his profit on one was and his loss on the other was . On the sale of the pipes, he
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A. broke even
B. lost cents
C. gained cents
D. lost cents
E. gained cents
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Problem 3: The distance light travels in one year is approximately miles. The distance light travels in 100 years is:
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A. miles
B. miles
C. miles
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Problem 4: A man has to invest. He invests at and at . In order to have a yearly income of , he must invest the remainder at:
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Problem 5: A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is:
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Problem 6: In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. The number of cows was:
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Problem 7: The roots of the equation will be reciprocal if:
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Problem 8: If , then, when
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Problem 9: Simplify:
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Problem 10: A circle of radius 10 inches has its center at the vertex of an equilateral triangle and passes through the other two vertices. The side extended through intersects the circle at . The number of degrees of angle is:
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Problem 11: The expression equals
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Problem 12: If is divided by the quotient is:
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Problem 13: The percent that is less than is
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Problem 14: The points , and are on a circle 0 . The tangent line at and the secant intersect at . If and , then equals
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Problem 15: The roots of are
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A. -5 and 3
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C. only 2
D. -3 and 5
E. only 3
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Problem 16: The sum of three numbers is 98. The ratio of the first to the second is , and the ratio of the second to the third is . The second number is:
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Problem 17: The fraction was obtained by adding the two fractions and . The values of and must be:
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A. ,
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D. ,
E. ,
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Problem 18: If , then equals:
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Problem 19: Two candles of the same height are lighted at the same time. The first is consumed in 4 hours and the second in 3 hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second?
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Problem 20: If and , then the value of to the nearest tenth is:
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Problem 21: If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is:
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B. or
C. or
D. or
E. or
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Problem 22: Jones covered a distance of miles on his first trip. On a later trip he traveled miles while going three times as fast. His new time compared with the old time was:
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A. three times as long
B. twice as long
C. the same
D. half as long
E. a third as long
Solution:
Problem 23: For the equation we have . A description of the roots is that they are:
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A. equal and rational
B. unequal and rational
C. equal and irrational
D. unequal and irrational
E. equal and imaginary
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Problem 24: In the figure , angle , and .
Then equals:
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Problem 25: The sum of all numbers of the form , where takes on integral values from 1 to is:
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Problem 26: Which one of the following combinations of given parts does not determine the indicated triangle?
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A. base angle and vertex angle; isosceles triangle
B. vertex angle and the base; isosceles triangle
C. the radius of the circumscribed circle; equilateral triangle
D. one arm and the radius of the inscribed circle; right triangle
E. two angles and a side opposite one of them; scalene triangle
Solution:
Problem 27: If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by:
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Problem 28: Mr. J left his entire estate to his wife, his daughter, his son, and the cook. His daughter and son got half the estate, sharing in the ratio of 4 to 3. His wife got twice as much as the son. If the cook received a bequest of , then the entire estate was:
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Problem 29: The points of intersection of and are joined. The resulting figure is:
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A. a straight line
B. an equilateral triangle
C. a parallelogram
D. a rectangle
E. a square
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Problem 30: If the altitude of an equilateral triangle is , then the area is:
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Problem 31: In our number system the base is ten. If the base were changed to four you would count as follows: The twentieth number would be:
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Problem 32: George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time?
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Problem 33: The number is equal to:
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A. a rational fraction
B. a finite decimal
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D. an infinite repeating decimal
E. an infinite non-repeating decimal
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Problem 34: If is any whole number, is always divisible by:
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Problem 35: A rhombus is formed by two radii and two chords of a circle whose radius is 15 feet. The area of the rhombus in square feet is:
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Problem 36: If the sum is a perfect square and if is less than 100 , then the possible values for are:
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A. only 1
B. 1 and 8
C. only 8
D. 8 and 49
E. 1,8 , and 49
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Problem 37: On a map whose scale is 400 miles to an inch and a half, a certain estate is represented by a rhombus having a angle. The diagonal opposite is . The area of the estate in square miles is:
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Problem 38: In a right triangle with sides and , and hypotenuse , the altitude drawn on the hypotenuse is . Then
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Problem 39: The hypotenuse and one arm of a right triangle are consecutive integers. The square of the second arm is:
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Problem 40: If and then equals
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Problem 41: The equation where is satisfied by:
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A. no value of
B. all values of
C. only
D. all integral values of only
E. all rational values of only
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Problem 42: The equation has:
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A. no root
B. one real root
C. one real root and one imaginary root
D. two imaginary roots
E. two real roots
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Problem 43: The number of scalene triangles having all sides of integral lengths, and perimeter less than 13 is:
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Problem 44: If means that and are numbers such that is less than and is less than zero, then
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Problem 45: A wheel with a rubber tire has an outside diameter of 25 in . When the radius has been decreased a quarter of an inch, the number of revolutions of the wheel in one mile will:
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A. be increased about
B. be increased about
C. be increased about 20%
D. be increased by
E. remain the same
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Problem 46: In the equation where is positive, can have:
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A. any positive value less than 1
B. any value less than 1
C. the value zero only
D. any non-negative value
E. any value
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Problem 47: An engineer said he could finish a highway section in 3 days with his present supply of a certain type of machine. However, with 3 more of these machines the jab could be done in 2 days. If the machines all work at the same rate, how many days would it take to do the job with one machine?
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Problem 48: If is a positive integer, then can be a positive integer, if and only if is:
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A. at least 3
B. at least 3 and no more than 35
C. no more than 35
D. equal to 35
E. equal to 3 or 35
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Problem 49: Triangle is formed by three tangents to circle . Angle , then angle equals:
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Problem 50: In triangle . On square is constructed away from the triangle. If is the number of degrees in angle , then
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A. depends upon triangle
B. is independent of the triangle
C. may equal angle
D. can never equal angle
E. is greater than but less than
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions