1968 AHSME Problems
1968 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
Let units be the increase in circumference of a circle resulting from an increase in units in the diameter. Then equals:
Problem 2
The real value of such that divided by equals is:
Problem 3
A straight line passing through the point is perpendicular to the line . Its equation is:
Problem 4
Define an operation for positive real numbers as . Then equals:
Problem 5
If , then equals:
Problem 6
Let side of convex quadrilateral be extended through , and let side be extended through , to meet in point Let be the degree-sum of angles and , and let represent the degree-sum of angles and If , then:
Problem 7
Let be the intersection point of medians and of triangle if is 3 inches, then , in inches, is:
Problem 8
A positive number is mistakenly divided by instead of being multiplied by Based on the correct answer, the error thus committed, to the nearest percent, is :
Problem 9
The sum of the real values of satisfying the equality is:
Problem 10
Assume that, for a certain school, it is true that
I: Some students are not honest. II: All fraternity members are honest.
A necessary conclusion is:
Problem 11
If an arc of on circle has the same length as an arc of on circle , the ratio of the area of circle to that of circle is:
Problem 12
A circle passes through the vertices of a triangle with side-lengths The radius of the circle is:
Problem 13
If and are the roots of , then the sum of the roots is:
Problem 14
If and are non-zero numbers such that and , then equals
Problem 15
Let be the product of any three consecutive positive odd integers. The largest integer dividing all such is:
Problem 16
If is such that and , then:
Problem 17
Let , where is a positive integer. If , the set of possible values of is:
Problem 18
Side of triangle has length 8 inches. Line is drawn parallel to so that is on segment , and is on segment . Line extended bisects angle . If has length inches, then the length of , in inches, is:
Problem 19
Let be the number of ways dollars can be changed into dimes and quarters, with at least one of each coin being used. Then equals:
Problem 20
The measures of the interior angles of a convex polygon of sides are in arithmetic progression. If the common difference is and the largest angle is , then equals:
Problem 21
If , then the units' digit in the value of S is:
Problem 22
A segment of length is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is:
Problem 23
If all the logarithms are real numbers, the equality is satisfied for:
Problem 24
A painting " X " is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is:
Problem 25
Ace runs with constant speed and Flash runs times as fast, . Flash gives Ace a head start of yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is:
Problem 26
Let , where is the smallest positive integer such that . Then the sum of the digits of is:
Problem 27
Let , where . Then equals:
Problem 28
If the arithmetic mean of and is double their geometric mean, with , then a possible value for the ratio , to the nearest integer, is:
Problem 29
Given the three numbers with . Arranged in order of increasing magnitude, they are:
Problem 30
Convex polygons and are drawn in the same plane with and sides, respectively, . If and do not have any line segment in common, then the maximum number of intersections of and is:
Problem 31
In this diagram, not drawn to scale, Figures and are equilateral triangular regions with respective areas of and square inches. Figure is a square region with area square inches. Let the length of segment be decreased by % of itself, while the lengths of and remain unchanged. The percent decrease in the area of the square is:
Problem 32
and move uniformly along two straight paths intersecting at right angles in point . When is at , is yards short of . In two minutes they are equidistant from , and in minutes more they are again equidistant from . Then the ratio of 's speed to 's speed is:
Problem 33
A number has three digits when expressed in base . When is expressed in base the digits are reversed. Then the middle digit is:
Problem 34
With members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in the passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the revote was of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time?
Problem 35
In this diagram the center of the circle is , the radius is inches, chord is parallel to chord . ,,, are collinear, and is the midpoint of . Let (sq. in.) represent the area of trapezoid and let (sq. in.) represent the area of rectangle Then, as and are translated upward so that increases toward the value , while always equals , the ratio becomes arbitrarily close to:
See also
1968 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1967 AHSC |
Followed by 1969 AHSC | |
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.