1966 AHSME Problems
1966 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 Problem 36
- 37 Problem 37
- 38 Problem 38
- 39 Problem 39
- 40 Problem 40
- 41 See also
Problem 1
Given that the ratio of to is constant, and when , then, when , equals:
Problem 2
When the base of a triangle is increased 10% and the altitude to this base is decreased 10%, the change in area is
Problem 3
If the arithmetic mean of two numbers is and their geometric mean is , then an equation with the given two numbers as roots is:
Problem 4
Circle I is circumscribed about a given square and circle II is inscribed in the given square. If is the ratio of the area of circle to that of circle , then equals:
Problem 5
The number of values of satisfying the equation
is:
Problem 6
is the diameter of a circle centered at . is a point on the circle such that angle is . If the diameter of the circle is inches, the length of chord , expressed in inches, is:
Problem 7
Let be an identity in . The numerical value of is:
Problem 8
The length of the common chord of two intersecting circles is feet. If the radii are feet and feet, a possible value for the distance between the centers of the circles, expressed in feet, is:
Problem 9
If , then equals:
Problem 10
If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is:
Problem 11
The sides of triangle are in the ratio . is the angle-bisector drawn to the shortest side , dividing it into segments and . If the length of is , then the length of the longer segment of is:
Problem 12
The number of real values of that satisfy the equation is:
Problem 13
The number of points with positive rational coordinates selected from the set of points in the -plane such that , is:
Problem 14
The length of rectangle is 5 inches and its width is 3 inches. Diagonal is divided into three equal segments by points and . The area of triangle , expressed in square inches, is:
Problem 15
If and , then
Problem 16
If and , and real numbers, then equals:
Problem 17
The number of distinct points common to the curves and is:
Problem 18
In a given arithmetic sequence the first term is , the last term is , and the sum of all the terms is . The common difference is:
Problem 19
Let be the sum of the first terms of the arithmetic sequence and let be the sum of the first terms of the arithmetic sequence . Assume . Then for:
Problem 20
The negation of the proposition "For all pairs of real numbers , if , then " is: There are real numbers such that
Problem 21
An "-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively ; for all values of , sides and are non-parallel, sides and being respectively identical with sides and ; prolong the pairs of sides numbered and until they meet. (A figure is shown for the case ).
Let be the degree-sum of the interior angles at the points of the star; then equals:
Problem 22
Consider the statements: (I), (II) , (III) , (IV) , where we allow and to be real or complex numbers. Those statements for which there exist solutions other than and , are:
Problem 23
If is real and , then the complete set of values of for which is real, is:
Problem 24
If , then equals:
Problem 25
If for and , then equals:
Problem 26
Let be a positive integer and let the lines and intersect in a point whose coordinates are integers. Then m can be:
Problem 27
At his usual rate a man rows 15 miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream's current is:
Problem 28
Five points are taken in order on a straight line with distances , , , and . is a point on the line between and and such that . Then equals:
Problem 29
The number of positive integers less than divisible by neither nor is:
Problem 30
If three of the roots of are , , and then the value of is:
Problem 31
Triangle is inscribed in a circle with center . A circle with center is inscribed in triangle . is drawn, and extended to intersect the larger circle in . Then we must have:
Problem 32
Let be the midpoint of side of triangle . Let be a point on between and , and let be drawn parallel to and intersecting at . If the ratio of the area of triangle to that of triangle is denoted by , then
Problem 33
If and , the number of distinct values of satisfying the equation
is:
Problem 34
Let be the speed in miles per hour at which a wheel, feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by of a second, the speed is increased by miles per hour. Then is:
Problem 35
Let be an interior point of triangle , and let . If , then
Problem 36
Let be an identity in . If we let , then equals:
Problem 37
Three men, Alpha, Beta, and Gamma, working together, do a job in 6 hours less time than Alpha alone, in 1 hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let be the number of hours needed by Alpha and Beta, working together, to do the job. Then equals:
Problem 38
In triangle the medians and to sides and , respectively, intersect in point . is the midpoint of side , and intersects in . If the area of triangle is , then the area of triangle is:
Problem 39
In base the expanded fraction becomes , and the expanded fraction becomes . In base fraction , when expanded, becomes , while the fraction becomes . The sum of and , each written in the base ten, is:
Problem 40
In this figure is a diameter of a circle, centered at , with radius . A chord is drawn and extended to meet the tangent to the circle at in point . Point is taken on so the . Denoting the distances of from the tangent through and from the diameter by and , respectively, we can deduce the relation:
See also
1966 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1965 AHSC |
Followed by 1967 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.