1971 AHSME Problems
1971 AHSC (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 Problem 31
- 32 Problem 32
- 33 Problem 33
- 34 Problem 34
- 35 Problem 35
- 36 See also
Problem 1
The number of digits in the number is
Problem 2
If men take days to lay bricks, then the number of days it will take men working at the same rate to lay bricks, is
Problem 3
If the point lies on the straight line joining the points and in the -plane, then is equal to
Problem 4
After simple interest for two months at % per annum was credited, a Boy Scout Troop had a total of in the Council Treasury. The interest credited was a number of dollars plus the following number of cents
Problem 5
Points , and lie on the circle shown and the measures of arcs and are and respectively. The sum of the measures of angles and is
Problem 6
Let be the symbol denoting the binary operation on the set of all non-zero real numbers as follows: For any two numbers and of , . Then the one of the following statements which is not true, is
Problem 7
is equal to
Problem 8
The solution set of is the set of all values of such that
Problem 9
An uncrossed belt is fitted without slack around two circular pulleys with radii of inches and inches. If the distance between the points of contact of the belt with the pulleys is inches, then the distance between the centers of the pulleys in inches is
Problem 10
Each of a group of girls is blonde or brunette and is blue eyed of brown eyed. If are blue-eyed blondes, are brunettes, and are brown-eyed, then the number of brown-eyed brunettes is
Problem 11
The numeral in base a represents the same number as in base . Assuming that both bases are positive integers, the least possible value of written as a Roman numeral, is
Problem 12
For each integer , there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by N. If , and are congruent in one such system, then in that same system, 8 is congruent to
Problem 13
If is evaluated correct to decimal places, then the digit in the fifth decimal place is
Problem 14
The number is exactly divisible by two numbers between and . These numbers are
Problem 15
An aquarium on a level table has rectangular faces and is inches wide and inches high. When it was tilted, the water in it covered an end but only of the rectangular bottom. The depth of the water when the bottom was again made level, was
Problem 16
After finding the average of scores, a student carelessly included the average with the scores and found the average of these numbers. The ratio of the second average to the true average was
Problem 17
A circular disk is divided by equally spaced radii() and one secant line. The maximum number of non-overlapping areas into which the disk can be divided is
Problem 18
The current in a river is flowing steadily at miles per hour. A motor boat which travels at a constant rate in still water goes downstream miles and then returns to its starting point. The trip takes one hour, excluding the time spent in turning the boat around. The ratio of the downstream to the upstream rate is
Problem 19
If the line intersects the ellipse exactly once, then the value of is
Problem 20
The sum of the squares of the roots of the equation is . The absolute value of is equal to
Problem 21
If , then the sum is equal to
Problem 22
If is one of the imaginary roots of the equation , then the product is equal to
Problem 23
Teams and are playing a series of games. If the odds for either to win any game are even and Team must win two or Team three games to win the series, then the odds favoring Team to win the series are
Problem 24
Pascal's triangle is an array of positive integers(See figure), in which the first row is , the second row is two 's, each row begins and ends with , and the number in any row when it is not , is the sum of the and numbers in the immediately preceding row. The quotient of the number of numbers in the first rows which are not 's and the number of 's is
Problem 25
A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get . The sum of their ages was
Problem 26
In , point divides side in the ratio . Let be the point of intersection of side and where is the midpoints of . The point divides side in the ratio
Problem 27
A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number which are white or blue is at least . The minimum number of red chips is
Problem 28
Nine lines parallel to the base of a triangle divide the other sides each into equal segments and the area into distinct parts. If the area of the largest of these parts is , then the area of the original triangle is
Problem 29
Given the progression . The least positive integer such that the product of the first terms of the progression exceeds is
Problem 30
Given the linear fractional transformation of into . Define for . Assuming that , it follows that is equal to
Problem 31
Quadrilateral is inscribed in a circle with side , a diameter of length . If sides and each have length , then side has length
Problem 32
If , then is equal to
Problem 33
If is the product of quantities in Geometric Progression, their sum, and the sum of their reciprocals, then in terms of , and is
Problem 34
An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position( o'clock, etc.) but only every minutes. At time and one-half for overtime, the extra pay to which a per hour worker should be entitled after working a normal hour day by that slow running clock, is
Problem 35
Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. The ratio of the area of the first circle to the sum of areas of all other circles in the sequence, is
See also
1971 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1970 AHSC |
Followed by 1972 AHSC | |
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All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.