Difference between revisions of "1972 AHSME Problems"
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− | A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle \theta is | + | A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle <math>\theta</math> is |
<math>\textbf{(A) }3\sec ^2\theta\csc\theta\qquad | <math>\textbf{(A) }3\sec ^2\theta\csc\theta\qquad | ||
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[[1972 AHSME Problems/Problem 30|Solution]] | [[1972 AHSME Problems/Problem 30|Solution]] | ||
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== Problem 31 == | == Problem 31 == | ||
Revision as of 21:35, 22 August 2017
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
The lengths in inches of the three sides of each of four triangles , and are as follows:
Of these four given triangles, the only right triangles are
Problem 2
If a dealer could get his goods for % less while keeping his selling price fixed, his profit, based on cost, would be increased to % from his present profit of %, which is
Problem 3
If where , then is equal to
Problem 4
The number of solutions to , where is a subset of is
Problem 5
From among those which have the greatest and the next to the greatest values, in that order, are
Problem 6
If , then the value of is
Problem 7
If , then is equal to
Problem 8
If where and are real, then
Problem 9
Ann and Sue bought identical boxes of stationery. Ann used hers to write -sheet letters and Sue used hers to write -sheet letters. Ann used all the envelopes and had sheets of paper left, while Sue used all of the sheets of paper and had envelopes left. The number of sheets of paper in each box was
Problem 10
For real, the inequality is equivalent to
Problem 11
The value(s) of for which the following pair of equations may have a real common solution, are
Problem 12
The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge expressed as a number of feet is
Problem 13
Inside square (See figure) with sides of length inches, segment is drawn where is the point on which is inches from . The perpendicular bisector of is drawn and intersects , and at points , and respectively. The ratio of segment to is
Problem 14
A triangle has angles of and . If the side opposite the angle has length , then the side opposite the angle has length
Problem 15
A contractor estimated that one of his two bricklayers would take hours to build a certain wall and the other hours. However, he knew from experience that when they worked together, their combined output fell by bricks per hour. Being in a hurry, he put both men on the job and found that it took exactly 5 hours to build the wall. The number of bricks in the wall was
Problem 16
There are two positive numbers that may be inserted between and such that the first three are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is
Problem 17
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is
Problem 18
Let be a trapezoid with the measure of base twice that of base , and let be the point of intersection of the diagonals. If the measure of diagonal is , then that of segment is equal to
Problem 19
The sum of the first terms of the sequence in terms of is
Problem 20
If where and , then is equal to
Problem 21
If the sum of the measures in degrees of angles and in the figure above is , then is equal to
Problem 22
If are imaginary roots of the equation where , and are real numbers, then in terms of and is
Problem 23
The radius of the smallest circle containing the symmetric figure composed of the 3 unit squares shown above is
Problem 24
A man walked a certain distance at a constant rate. If he had gone mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone mile per hour slower, he would have been hours longer on the road. The distance in miles he walked was
Problem 25
Inscribed in a circle is a quadrilateral having sides of lengths , and taken consecutively. The diameter of this circle has length
Problem 26
In the circle above, is the midpoint of arc and segment is perpendicular to chord at . If the measure of chord is and that of segment is , then segment has measure equal to
Problem 27
If the area of is square units and the geometric mean (mean proportional) between sides and is inches, then is equal to
Problem 28
A circular disc with diameter is placed on an checkerboard with width so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is
Problem 29
If for , then in terms of is
Problem 30
A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle is
Problem 31
When the number is divided by , the remainder in the division is
Problem 32
Chords and in the circle above intersect at E and are perpendicular to each other. If segments , and have measures , and respectively, then the length of the diameter of the circle is
Problem 33
The minimum value of the quotient of a (base ten) number of three different non-zero digits divided by the sum of its digits is
Problem 34
Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is
Problem 35
Equilateral triangle (see figure) with side of length inches is placed inside square with side of length inches so that is on side . The triangle is rotated clockwise about , then , and so on along the sides of the square until returns to its original position. The length of the path in inches traversed by vertex is equal to
See also
1972 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1971 AHSME |
Followed by 1973 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 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.