Individual Problems and Solutions
For problems and detailed solutions to each of the 1951 AHSME problems, please refer below:
Problem 1: The percent that is greater than , is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 2: A rectangular field is half as wide as it is long and is completely enclosed by yards of fencing. The area in terms of is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 3: If the length of a diagonal of a square is , then the area of the square is:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 4: A barn with a flat roof is rectangular in shape, 10 yd. wide, 13 yd. long and 5 yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 5: Mr. A owns a home worth . He sells it to Mr. B at a profit. Mr. B sells the house back to Mr. A at a loss. Then:
Answer Choices:
A. A comes out even
B. A makes on the deal
C. A makes on the deal
D. A loses on the deal
E. A loses on the deal
Solution:
Problem 6: The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to:
Answer Choices:
A. The volume of the box
B. The square root of the volume
C. Twice the volume
D. The square of the volume
E. The cube of the volume
Solution:
Problem 7: An error of is made in the measurement of a line long, while an error of only is made in a measurement of a line long. In comparison with the relative error of the first measurement, the relative error of the second measurement is:
Answer Choices:
A. greater by
B. the same
C. less
D. times as great
E. correctly described by both (a) and (d)
Solution:
Problem 8: The price of an article is cut . To restore it to its former value, the new price must be increased by:
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
Problem 9: An equilateral triangle is drawn with a side of length . A new equilateral triangle is formed by joining the mid-points of the sides of the first one. Then a third equilateral triangle is formed by joining the mid-points of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the triangles thus drawn is:
Answer Choices:
A. Infinite
B.
C.
D.
E.
Solution:
Problem 10: Of the following statements, the one that is incorrect is:
Answer Choices:
A. Doubling the base of a given rectangle doubles the area.
B. Doubling the altitude of a triangle doubles the area.
C. Doubling the radius of a given circle doubles the area.
D. Doubling the divisor of a fraction and dividing its numerator by changes the quotient.
E. Doubling a given quantity may make it less than it originally was.
Solution:
Problem 11: The limit of the sum of an infinite number of terms in a geometric progression is where denotes the first term and denotes the common ratio. The limit of the sum of their squares is:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 12: At oclock, the hour and minute dials of a clock form an angle of:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 13: A can do a piece of work in days. B is more efficient than . The number of days it takes to do the same piece of work is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
Problem 14: In connection with proof in geometry, indicate which one of the following statements is incorrect:
Answer Choices:
A. Some statements are accepted without being proved.
B. In some instances there is more than one correct order in proving certain propositions.
C. Every term used in a proof must have been defined previously.
D. It is impossible to have a correct conclusion if in the beginning an incorrect proposition is introduced followed by correct reasoning.
E. Indirect proof can be used whenever there are two or more contrary propositions.
Solution:
Problem 15: The largest number by which the expression is divisible for all possible positive values of , is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 16: If in applying the quadratic formula to a quadratic equation , it happens that , then the graph of will certainly:
Answer Choices:
A. have a maximum
B. have a minimum
C. be tangent to the -axis
D. be tangent to the -axis
E. lie in one quadrant only
Solution:
Problem 17: Indicate in which one of the following equations is neither directly nor inversely proportional to :
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 18: is to be factored into prime binomial factors and without a numerical monomial factor. This can be done if the value ascribed to is:
Answer Choices:
A. any odd number
B. some odd number
C. any even number
D. some even number
E. zero
Solution:
Problem 19: A six place number is formed by repeating a three place number; for example, or etc. Any number of this form is always exactly divisible by:
Answer Choices:
A. only
B. only
C. only
D.
E.
Solution:
Problem 20: When simplified and expressed with negative exponents, the expression is equal to:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 21: Given: and . The inequality which is not always correct is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 22: The values of in the equation: is:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 23: The radius of a cylindrical box is 8 inches and the height is 3 inches. The number of inches that may be added to either the radius or the height to give the same increase in volume is:
Answer Choices:
A.
B.
C. any number
D. non-existent
E. none of these
Solution:
Problem 24: when simplified is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 25: The apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be:
Answer Choices:
A. equal to the second
B. times the second
C. times the second
D. times the second
E. indeterminately related to the second
Solution:
Problem 26: In the equation the roots are equal when
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 27: Through a point inside a triangle, three lines are drawn from the vertices to the opposite sides forming six triangular sections. Then:
Answer Choices:
A. The triangles are similar by opposite pairs.
B. The triangles are congruent by opposite pairs.
C. The triangles are equal in area by opposite pairs.
D. Three similar quadrilaterals are formed.
E. None of the above relations are true.
Solution:
Problem 28: The pressure ( ) of wind on a sail varies jointly as the area ( ) of the sail and the square of the velocity of the wind. The pressure on a square foot is 1 pound when the velocity is 16 miles per hour. The velocity of the wind when the pressure on a square yard is 36 pounds is:
Answer Choices:
A.
B. mph
C. mph
D.
E. mph
Solution:
Problem 29: Of the following sets of data the only one that does not determine the shape of a triangle is:
Answer Choices:
A. the ratio of two sides and the included angle
B. the ratios of the three altitudes
C. the ratios of the three medians
D. the ratio of the altitude to the corresponding base
E. two angles
Solution:
Problem 30: If two poles and high are apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 31: A total of 28 handshakes were exchanged at the conclusion of a party. Assuming that each participant was equally polite toward all the others, the number of people present was:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 32: If is inscribed in a semicircle whose diameter is , then must be:
(Hint: is read " is equal to or greater than .")
Answer Choices:
A. equal to
B. equal to
C.
D.
E.
Solution:
Problem 33: The roots of the equation can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 34: The value of is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 35: If and , then:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 36: Which of the following methods of proving a geometric figure a locus is not correct?
Answer Choices:
A. Every point on the locus satisfies the conditions and every point not on the locus does not satisfy the conditions.
B. Every point not satisfying the conditions is not on the locus and every point on the locus does satisfy the conditions.
C. Every point satisfying the conditions is on the locus and every point on the locus satisfies the conditions.
D. Every point not on the locus does not satisfy the conditions and every point not satisfying the conditions is not on the locus.
E. Every point satisfying the conditions is on the locus and every point not satisfying the conditions is not on the locus.
Solution:
Problem 37: The number which when divided by 10 leaves a remainder of 9 , when divided by 9 leaves a remainder of 8 , by 8 leaves a remainder of 7 , etc., down to where, when divided by 2 , it leaves a remainder of 1 , is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
Problem 38: A rise of 600 feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from to is approximately:
Answer Choices:
A. feet
B. feet
C. feet
D. feet
E. none of these
Solution:
Problem 39: A stone is dropped into a well and the report of the stone striking the bottom is heard 7.7 seconds after it is dropped. Assume that the stone falls feet in seconds and that the velocity of sound is 1,120 feet per second. The depth of the well is:
Answer Choices:
A. ft.
B. ft.
C. ft.
D. ft.
E. none of these
Solution:
Problem 40: equals:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 41: The formula expressing the relationship between and in the table is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 42: If , then:
Answer Choices:
A.
B.
C.
D. is infinite
E. but finite
Solution:
Problem 43: Of the following statements, the only one that is incorrect is:
Answer Choices:
A. An inequality will remain true after each side is increased, decreased, multiplied or divided (zero excluded) by the same positive quantity.
B. The arithmetic mean of two unequal positive quantities is greater than their geometric mean.
C. If the sum of two positive quantities is given, their product is largest when they are equal.
D. If and are positive and unequal, is greater than .
E. If the product of two positive quantities is given, their sum is greatest when they are equal.
Solution:
Problem 44: If and , where , and are other than zero, then equals:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 45: If you are given and , then the only logarithm that cannot be found without the use of tables is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 46: is a fixed diameter of a circle whose center is 0 . From , any point on the circle, a chord is drawn perpendicular to . Then, as moves over a semicircle, the bisector of angle cuts the circle in a point that always:
Answer Choices:
A. bisects the arc
B. trisects the arc
C. varies
D. is as far from as from
E. is equidistant from and .
Solution:
Problem 47: If and are the roots of the equation , the value of is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
Problem 48: The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 49: The medians of a right triangle which are drawn from the vertices of the acute angles are 5 and respectively. The value of the hypotenuse is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
Problem 50: Tom, Dick and Harry started out on a 100 -mile journey. Tom and Harry went by automobile at the rate of 25 mph , while Dick walked at the rate of 5 mph . After a certain distance, Tom let Harry off, who walked on at 5 mph , while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was:
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions