Difference between revisions of "1964 AHSME Problems"
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Given the distinct points <math>P(x_1, y_1), Q(x_2, y_2)</math> and <math>R(x_1+x_2, y_1+y_2)</math>. | Given the distinct points <math>P(x_1, y_1), Q(x_2, y_2)</math> and <math>R(x_1+x_2, y_1+y_2)</math>. | ||
− | Line segments are drawn connecting these points to each other and to the origin <math> | + | Line segments are drawn connecting these points to each other and to the origin <math>O</math>. |
Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure <math>OPRQ</math>, | Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure <math>OPRQ</math>, | ||
depending upon the location of the points <math>P, Q</math>, and <math>R</math>, can be: | depending upon the location of the points <math>P, Q</math>, and <math>R</math>, can be: |
Revision as of 20:34, 23 July 2019
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
What is the value of ?
Problem 2
The graph of is:
Problem 3
When a positive integer is divided by a positive integer , the quotient is and the remainder is , where and are integers. What is the remainder when is divided by ?
Problem 4
The expression
where and , is equivalent to:
Problem 5
If varies directly as , and if when , the value of when is:
Problem 6
If are in geometric progression, the fourth term is:
Problem 7
Let n be the number of real values of for which the roots of are equal. Then n equals:
Problem 8
The smaller root of the equation is:
Problem 9
A jobber buys an article at less . He then wishes to sell the article at a gain of of his cost after allowing a discount on his marked price. At what price, in dollars, should the article be marked?
Problem 10
Given a square side of length . On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is:
Problem 11
Given and , find the value of
Problem 12
Which of the following is the negation of the statement: For all of a certain set, ?
Problem 13
A circle is inscribed in a triangle with side lengths , and . Let the segments of the side of length , made by a point of tangency, be and , with . What is the ratio ?
Problem 14
A farmer bought sheep. He sold of them for the price paid for the sheep. The remaining sheep were sold at the same price per head as the other . Based on the cost, the percent gain on the entire transaction is:
Problem 15
A line through the point cuts from the second quadrant a triangular region with area . The equation of the line is:
Problem 16
Let and let be the set of integers . The number of members of such that has remainder zero when divided by is:
Problem 17
Given the distinct points and . Line segments are drawn connecting these points to each other and to the origin . Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure , depending upon the location of the points , and , can be:
Problem 18
Let be the number of pairs of values of and such that and have the same graph. Then is:
Problem 19
If and , the numerical value of is:
Problem 20
The sum of the numerical coefficients of all the terms in the expansion of is:
Problem 21
If , then equals:
Problem 22
Given parallelogram with the midpoint of diagonal . Point is connected to a point in so that . What is the ratio of the area of to the area of quadrilateral ?
Problem 23
Two numbers are such that their difference, their sum, and their product are to one another as . The product of the two numbers is:
Problem 24
Let constants. For what value of is a minimum?
Problem 25
The set of values of for which has two factors, with integer coefficients, which are linear in and , is precisely:
Problem 26
In a ten-mile race beats by miles and beats by miles. If the runners maintain constant speeds throughout the race, by how many miles does beat ?
Problem 27
If is a real number and where , then:
Problem 28
The sum of terms of an arithmetic progression is , and the common difference is . If the first term is an integer, and , then the number of possible values for is:
Problem 29
In this figure inches, inches, inches, inches. The length of , in inches, is:
Problem 30
If , the larger root minus the smaller root is:
Problem 31
Let . Then , expressed in terms of , equals:
Problem 32
If , then:
Problem 33
is a point interior to rectangle and such that inches, inches, and inches. Then , in inches, equals:
Problem 34
If is a multiple of , the sum , where , equals:
Problem 35
The sides of a triangle are of lengths and . The altitudes of the triangle meet at point . If is the altitude to the side length , what is the ratio ?
Problem 36
In this figure the radius of the circle is equal to the altitude of the equilateral triangle . The circle is made to roll along the side , remaining tangent to it at a variable point and intersecting lines and in variable points and , respectively. Let be the number of degrees in arc . Then , for all permissible positions of the circle:
Problem 37
Given two positive number such that , let be their arithmetic mean and let be their positive geometric mean. Then minus is always less than:
Problem 38
The sides and of are respectively of lengths inches, and inches. The median is inches. Then , in inches, is:
Problem 39
The magnitudes of the sides of are , and , as shown, with . Through interior point and the vertices , lines are drawn meeting the opposite sides in , respectively. Let . Then, for all positions of point , is less than:
Problem 40
A watch loses minutes per day. It is set right at P.M. on March . Let be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows A.M. on March , equals:
See also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1963 AHSME |
Followed by 1965 AHSME | |
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.