1961 AHSME Problems
1961 AHSC (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 |
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 Problem 36
- 37 Problem 37
- 38 Problem 38
- 39 Problem 39
- 40 Problem 40
- 41 See also
Problem 1
When simplified, becomes:
Problem 2
An automobile travels feet in seconds. If this rate is maintained for minutes, how many yards does it travel in minutes?
Problem 3
If the graphs of and are to meet at right angles, the value of is:
Problem 4
Let the set consisting of the squares of the positive integers be called ; thus is the set . If a certain operation on one or more members of the set always yields a member of the set, we say that the set is closed under that operation. Then is closed under:
Problem 5
Let . Then equals:
Problem 6
When simplified, becomes:
Problem 7
When simplified, the third term in the expansion of is:
Problem 8
Let the two base angles of a triangle be and , with larger than . The altitude to the base divides the vertex angle into two parts, and , with adjacent to side . Then:
Problem 9
Let be the result of doubling both the base and exponent of , and does not equal to . If equals the product of by , then equals:
Problem 10
Each side of is units. is the foot of the perpendicular dropped from on , and is the midpoint of . The length of , in the same unit, is:
Problem 11
Two tangents are drawn to a circle from an exterior point ; they touch the circle at points and respectively. A third tangent intersects segment in and in , and touches the circle at . If , then the perimeter of is
Problem 12
The first three terms of a geometric progression are . Find the fourth term.
Problem 13
The symbol means is a positive number or zero, and if is a negative number. For all real values of the expression is equal to?
Problem 14
A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus is terms of , where is the area of the rhombus in square inches.
Problem 15
If men working hours a day for days produce articles, then the number of articles (not necessarily an integer) produced by men working hours a day for days is:
Problem 16
An altitude of a triangle is increased by a length . How much must be taken from the corresponding base so that the area of the new triangle is one-half that of the original triangle?
Problem 17
In the base ten number system the number means . In the Land of Mathesis, however, numbers are written in the base . Jones purchases an automobile there for monetary units (abbreviated m.u). He gives the salesman a m.u bill, and receives, in change, m.u. The base is:
Problem 18
The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is:
Problem 19
Consider the graphs of and . We may say that:
Problem 20
The set of points satisfying the pair of inequalities and is contained entirely in quadrants:
Problem 21
Medians and of intersect in . The midpoint of is . Let the area of be times the area of . Then equals:
Problem 22
If is divisible by , then it is also divisible by:
Problem 23
Points and are both in the line segment and on the same side of its midpoint. divides in the ratio , and divides in the ratio . If , then the length of is:
Problem 24
Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by from that of each adjacent book. For the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then:
Problem 25
is isosceles with base . Points and are respectively in and and such that . The number of degrees in is:
Problem 26
For a given arithmetic series the sum of the first terms is , and the sum of the next terms is . The first term in the series is:
Problem 27
Given two equiangular polygons and with different numbers of sides; each angle of is degrees and each angle of is degrees, where is an integer greater than . The number of possibilities for the pair is:
Problem 28
If is multiplied out, the units' digit in the final product in the final product is:
Problem 29
Let the roots of be and . The equation with roots and is:
Problem 30
If and , then ?
Problem 31
In the ratio is . The bisector of the exterior angle at intersects extended at ( is between and ). The ratio is:
Problem 32
A regular polygon of sides is inscribed in a circle of radius . The area of the polygon is . Then equals:
Problem 33
The number of solutions of , in which and are integers, is:
Problem 34
Let S be the set of values assumed by the fraction . When is any member of the interval . If there exists a number such that no number of the set is greater than , then is an upper bound of . If there exists a number such that such that no number of the set is less than , then is a lower bound of . We may then say:
Problem 35
The number is to be written with a factorial base of numeration, that is, where are integers such that and means . Find
Problem 36
In the median from is given perpendicular to the median from . If and , find the length of .
Problem 37
In racing over a distance at uniform speed, can beat by yards, can beat by yards, and can beat by yards. Then , in yards, equals:
Problem 38
is inscribed in a semicircle of radius so that its base coincides with diameter . Point does not coincide with either or . Let . Then, for all permissible positions of :
Problem 39
Any five points are taken inside or on a square with side length . Let a be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than . Then is:
Problem 40
Find the minimum value of if .
See also
1961 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1960 AHSC |
Followed by 1962 AHSC | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.