Individual Problems and Solutions
For problems and detailed solutions to each of the 1950 AHSME problems, please refer below:
Problem 1: If is divided into three parts proportional to and , the smallest part is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers.
Solution:
Problem 2: Let . When . When is equal to:
Answer Choices:
A.
B.
C.
D.
E. none of these answers.
Solution:
Problem 3: The sum of the roots of the equation is equal to:
Answer Choices:
A.
B.
C.
D.
E. none of these answers.
Solution:
Problem 4: Reduced to lowest terms. is equal to:
Answer Choices:
A.
B.
C.
D.
E. none of these answers.
Solution:
Problem 5: If five geometric means are inserted between and , the fifth term in the geometric series is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers.
Solution:
Problem 6: The values of which will satisfy the equations
Answer Choices:
A.
B.
C.
D.
E. none of the above equations
Solution:
Problem 7: If the digit is placed after a two digit number whose tens' digit is , and units' digit is , the new number is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers.
Solution:
Problem 8: If the radius of a circle is increased , the area is increased:
Answer Choices:
A.
B.
C.
D.
E. by none of these
Solution:
Problem 9: The area of the largest triangle that can be inscribed in a semicircle whose radius is is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 10: After rationalizing the numerator of , the denominator in simplest form is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers.
Solution:
Problem 11: If in the formula is increased while and are kept constant, then :
Answer Choices:
A. decreases
B. increases
C. remains constant
D. increases and then decreases
E. decreases and then increases
Solution:
Problem 12: As the number of sides of a polygon increases from to , the sum of the exterior angles formed by extending each side in succession:
Answer Choices:
A. increases
B. decreases
C. remains constant
D. cannot be predicted
E. becomes straight angles.
Solution:
Problem 13: The roots of are
Answer Choices:
A.
B. and
C. and
D. and
E. and
Solution:
Problem 14: In the simultaneous equations
Answer Choices:
A.
B.
C.
D. there is no solution
E. there are an infinite number of solutions
Solution:
Problem 15: The real factors of are:
Answer Choices:
A.
B.
C.
D.
E. non-existent
Solution:
Problem 16: The number of terms in the expansion of when simplified is:
Answer Choices:
A.
B.
C.
D.
E. 8
Solution:
Problem 17: The formula which expresses the relationship between and as shown in the accompanying table is:
Answer Choices:
A.
B.
C.
D.
E. none of the above formulae.
Solution:
Problem 18: Of the following
(1)
(2)
(3)
(4)
(5)
Answer Choices:
A. only and are true
B. only and are true
C. only and are true
D. only and are true
E. only is true
Solution:
Problem 19: If men can do a job in days, then men can do the job in:
Answer Choices:
A. days
B. days
C. days
D. days
E. none of the above
Solution:
Problem 20: When is divided by , the remainder is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
Problem 21: The volume of a rectangular solid each of whose side, front, and bottom faces are sq. in., sq. in., and sq. in. respectively is:
Answer Choices:
A. .
B. .
C. .
D. .
E. none of the above answers
Solution:
Problem 22: Successive discounts of and are equivalent to a single discount of:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 23: A man buys a house for and rents it. He puts of each month's rent aside for repairs and upkeep; pays a year taxes and realizes on his investment. The monthly rent is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 24: The equation has:
Answer Choices:
A. real roots
B. real and imaginary root
C. imaginary roots
D. no roots
E. real root and extraneous root
Solution:
Problem 25: The value of is equal to
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
Problem 26: If , then
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 27: A car travels miles from to at miles per hour but returns the same distance at miles per hour. The average speed for the round trip is closest to:
Answer Choices:
A. mph
B. mph
C. mph
D. mph
E. mph
Solution:
Problem 28: Two boys and start at the same time to ride from Port Jervis to Poughkeepsie, miles away. travels miles an hour slower than . reaches Poughkeepsie and at once turns back meeting miles from Poughkeepsie. The rate of was:
Answer Choices:
A. mph
B. mph
C. mph
D. mph
E. mph
Solution:
Problem 29: A manufacturer built a machine which will address envelopes in minutes. He wishes to build another machine so that when both are operating together they will address envelopes in minutes. The equations used to find how many minutes () it would require the second machine to address envelopes alone is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
Problem 30: From a group of boys and girls, girls leave. There are then left two boys for each girl. After this boys leave. There are then girls for each boy. The number of girls in the beginning was:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 31: John ordered pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by . The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 32: A foot ladder is placed against a vertical wall of a building. The foot of the ladder is feet from the base of the building. If the top of the ladder slips feet, then the foot of the ladder will slide:
Answer Choices:
A. ft.
B. ft.
C. ft.
D. ft.
E. ft.
Solution:
Problem 33: The number of circular pipes with an inside diameter of inch which will carry the same amount of water as a pipe with an inside diameter of inches is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 34: When the circumference of a toy balloon is increased from inches to inches, the radius is increased by:
Answer Choices:
A. in.
B. in.
C. in.
D. in.
E. in.
Solution:
Problem 35: In , and . The radius of the inscribed circle is:
Answer Choices:
A. in.
B. in.
C. in.
D. in.
E. none of these answers
Solution:
Problem 36: A merchant buys goods at off the list price. He desires to mark the goods so that he can give a discount of on the marked price and still clear a profit of on the selling price. What per cent of the list price must he mark the goods?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 37: If , and , which of the following statements is incorrect?
Answer Choices:
A. If
B. If
C. If is imaginary (complex)
D. If is always less than 0 and decreases without limit as approaches zero
E. only some of the above statements are correct.
Solution:
Problem 38: If the expression
has the value for all values of and , then the equation
Answer Choices:
A. is satisfied for only value of .
B. is satisfied for values of .
C. is satisfied for no values of .
D. is satisfied for an infinite number of values of .
E. is satisfied for none of these answers.
Solution:
Problem 39: In the sequence .
(1) the sum increases without limit.
(2) the sum decreases without limit.
(3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small.
(4) The difference between the sum and can be made less than any positive quantity no matter how small.
(5) the sum approaches a limit.
Answer Choices:
A. only (3) and (4) are correct statements
B. only (5) is a correct statement
C. only (2) and (4) are correct statements
D. only (2), (3) and (4) are correct statements
E. statements (a), (b), (c) and (d) are all incorrect.
Solution:
Problem 40: The limit of as approaches as a limit is:
Answer Choices:
A.
B. indeterminate
C.
D.
E.
Solution:
Problem 41: The least value of the function is:
Answer Choices:
A.
B.
C.
D.
E. none of the above.
Solution:
Problem 42: If , then is equal to.
Answer Choices:
A. infinity
B.
C.
D.
E. none of these answers.
Solution:
Problem 43: The sum to infinity of is:
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
Problem 44: The graph of
Answer Choices:
A. cuts the -axis
B. cuts all lines the - axis
C. cuts the - axis
D. cuts neither axis
E. cuts all circles whose center is at the origin
Solution:
Problem 45: The number of diagonals that can be drawn in a polygon of sides is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 46: In triangle , and . If sides and are doubled while remains the same, then:
Answer Choices:
A. the area is doubled
B. the altitude is doubled
C. the area is four times the original area.
D. the median is unchanged
E. the area of the triangle is .
Solution:
Problem 47: A rectangle inscribed in a triangle has its base coinciding with the base ( ) of the triangle. If the altitude of the triangle is , and the altitude ( ) of the rectangle is half the base of the rectangle, then:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 48: A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is:
Answer Choices:
A. least when the point is at the center of gravity of the .
B. greater than the altitude of the triangle.
C. equal to the altitude of the triangle.
D. one-half the sum of the sides of the triangle.
E. the sum is greatest when the point is the center of gravity.
Solution:
Problem 49: A triangle has a fixed base that is inches long. The median from to side is inches long and can have any position emanating from . The locus of the vertex of the triangle is:
Answer Choices:
A. a straight line in. from .
B. a circle with as center and radius in.
C. a circle with as center and radius in.
D. a circle with radius in. and center in. from along .
E. an ellipse with as a focus.
Solution:
Problem 50: A privateer discovers a merchantman miles to leeward at a.m. and there being a good breeze bears down upon her at mph while the merchantman can only make mph in her attempt to escape. After a two hour chase, the top sail of the privateer being carried away, she can make only miles while the merchantman makes . The privateer will overtake the merchantman at:
Answer Choices:
A. p.m.
B. p.m.
C. p.m.
D. p.m.
E. p.m.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions