Difference between revisions of "1958 AHSME Problems"
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== Problem 1 == | == Problem 1 == | ||
The value of <math> [2 - 3(2 - 3)^{-1}]^{-1}</math> is: | The value of <math> [2 - 3(2 - 3)^{-1}]^{-1}</math> is: |
Revision as of 14:38, 19 February 2020
1958 AHSC (Answer Key) Printable version: | 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 Problem 41
- 42 Problem 42
- 43 Problem 43
- 44 Problem 44
- 45 Problem 45
- 46 Problem 46
- 47 Problem 47
- 48 Problem 48
- 49 Problem 49
- 50 Problem 50
- 51 See also
Problem 1
The value of is:
Problem 2
If , then equals:
Problem 3
Of the following expressions the one equal to is:
Problem 4
In the expression each is replaced by . The resulting expression, evaluated for , equals:
Problem 5
The expression equals:
Problem 6
The arithmetic mean between and , when , is:
Problem 7
A straight line joins the points and . Its -intercept is:
Problem 8
Which of these four numbers , is (are) rational:
Problem 9
A value of satisfying the equation is:
Problem 10
For what real values of , other than , does the equation have real roots?
Problem 11
The number of roots satisfying the equation is:
Problem 12
If then equals:
Problem 13
The sum of two numbers is ; their product is . The sum of their reciprocals is:
Problem 14
At a dance party a group of boys and girls exchange dances as follows: one boy dances with girls, a second boy dances with girls, and so on, the last boy dancing with all the girls. If represents the number of boys and the number of girls, then:
Problem 15
A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:
Problem 16
The area of a circle inscribed in a regular hexagon is . The area of hexagon is:
Problem 17
If is positive and , then:
Problem 18
The area of a circle is doubled when its radius is increased by . Then equals:
Problem 19
The sides of a right triangle are and and the hypotenuse is . A perpendicular from the vertex divides into segments and , adjacent respectively to and . If , then the ratio of to is:
Problem 20
If , then equals:
Problem 21
In the accompanying figure and are equal chords of a circle with center . Arc is a quarter-circle. Then the ratio of the area of triangle to the area of triangle is:
Problem 22
A particle is placed on the parabola at a point whose -coordinate is . It is allowed to roll along the parabola until it reaches the nearest point whose -coordinate is . The horizontal distance traveled by the particle (the numerical value of the difference in the -coordinates of and ) is:
Problem 23
If, in the expression , increases or decreases by a positive amount of , the expression changes by an amount:
Problem 24
A man travels feet due north at minutes per mile. He returns due south to his starting point at miles per minute. The average rate in miles per hour for the entire trip is:
Problem 25
If , then equals:
Problem 26
A set of numbers has the sum . Each number of the set is increased by , then multiplied by , and then decreased by . The sum of the numbers in the new set thus obtained is:
Problem 27
The points , , and are on the same straight line. The value(s) of is (are):
Problem 28
A -quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is:
Problem 29
In a general triangle (as shown) lines and are drawn. Which of the following angle relations is true?
Problem 30
If and , then equals:
Problem 31
The altitude drawn to the base of an isosceles triangle is , and the perimeter . The area of the triangle is:
Problem 32
With $ a rancher is to buy steers at $ each and cows at $ each. If the number of steers and the number of cows are both positive integers, then:
Problem 33
For one root of to be double the other, the coefficients must be related as follows:
Problem 34
The numerator of a fraction is , then denominator is , and can have any value between and , both included. The values of for which the numerator is greater than the denominator are:
Problem 35
A triangle is formed by joining three points whose coordinates are integers. If the -coordinate and the -coordinate each have a value of , then the area of the triangle, in square units:
Problem 36
The sides of a triangle are , , and units. If an altitude is dropped upon the side of length , the larger segment cut off on this side is:
Problem 37
The first term of an arithmetic series of consecutive integers is . The sum of terms of this series may be expressed as:
Problem 38
Let be the distance from the origin to a point with coordinates and . Designate the ratio by and the ratio by . Then the values of are limited to the numbers:
Problem 39
We may say concerning the solution of that:
Problem 40
Given , , and the general relation for . Then equals:
Problem 41
The roots of are and . For the roots of
to be and , must equal:
Problem 42
In a circle with center , chord equals chord . Chord cuts in . If and , then equals:
Problem 43
is the hypotenuse of a right triangle . Median has length and median has length . The length of is:
Problem 44
Given the true statements: (1) If is greater than , then is greater than (2) If is less than , then is greater than . A valid conclusion is:
Problem 45
A check is written for dollars and cents, and both two-digit numbers. In error it is cashed for dollars and cents, the incorrect amount exceeding the correct amount by . Then:
Problem 46
For values of less than but greater than , the expression has:
Problem 47
is a rectangle (see the accompanying diagram) with any point on . and . and . Then is equal to:
Problem 48
Diameter of a circle with center is units. is a point units from , and on . is a point units from , and on . is any point on the circle. Then the broken-line path from to to :
Problem 49
In the expansion of there are dissimilar terms. The number of dissimilar terms in the expansion of is:
Problem 50
In this diagram a scheme is indicated for associating all the points of segment with those of segment , and reciprocally. To described this association scheme analytically, let be the distance from a point on to and let be the distance from the associated point of to . Then for any pair of associated points, if equals:
See also
1958 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1957 AHSME |
Followed by 1959 AHSME | |
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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.