Difference between revisions of "1960 AHSME Problems"
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+ | {{AHSC 40 Problems | ||
+ | |year = 1960 | ||
+ | }} | ||
== Problem 1== | == Problem 1== | ||
If <math>2</math> is a solution (root) of <math>x^3+hx+10=0</math>, then <math>h</math> equals: | If <math>2</math> is a solution (root) of <math>x^3+hx+10=0</math>, then <math>h</math> equals: | ||
<math>\textbf{(A)}10\qquad | <math>\textbf{(A)}10\qquad | ||
− | \textbf{(B )}9 \qquad | + | \textbf{(B)}9 \qquad |
− | \textbf{(C )}2\qquad | + | \textbf{(C)}2\qquad |
− | \textbf{(D )}-2\qquad | + | \textbf{(D)}-2\qquad |
− | \textbf{(E )}-9 </math> | + | \textbf{(E)}-9 </math> |
[[1960 AHSME Problems/Problem 1|Solution]] | [[1960 AHSME Problems/Problem 1|Solution]] | ||
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is increased by <math>x</math> inches as when <math>H</math> is increased by <math>x</math> inches. This condition is satisfied by: | is increased by <math>x</math> inches as when <math>H</math> is increased by <math>x</math> inches. This condition is satisfied by: | ||
− | <math>\textbf{(A)}\ \text{ | + | <math>\textbf{(A)}\ \text{no real value of x} \qquad \\ |
− | \textbf{(B)}\ \text{ | + | \textbf{(B)}\ \text{one integral value of x} \qquad \\ |
− | \textbf{(C)}\ \text{ | + | \textbf{(C)}\ \text{one rational, but not integral, value of x} \qquad \\ |
− | \textbf{(D)}\ \text{ | + | \textbf{(D)}\ \text{one irrational value of x}\qquad \\ |
− | \textbf{(E)}\ \text{ | + | \textbf{(E)}\ \text{two real values of x} </math> |
[[1960 AHSME Problems/Problem 23|Solution]] | [[1960 AHSME Problems/Problem 23|Solution]] | ||
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exterior angle at the same vertex. Then | exterior angle at the same vertex. Then | ||
− | <math>\textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad | + | <math>\textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad \\ |
− | \textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad | + | \textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad \\ |
− | \textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad | + | \textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad \\ |
− | \textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad | + | \textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad \\ |
\textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular} </math> | \textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular} </math> | ||
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label("$P$",P,SW);</asy> | label("$P$",P,SW);</asy> | ||
− | <math>\textbf{(A)} AP^2 = PB \times AB\qquad | + | <math>\textbf{(A)} AP^2 = PB \times AB\qquad \\ |
− | \textbf{(B)}\ AP \times DO = PB \times AD\qquad | + | \textbf{(B)}\ AP \times DO = PB \times AD\qquad \\ |
− | \textbf{(C)}\ AB^2 = AD \times DE\qquad | + | \textbf{(C)}\ AB^2 = AD \times DE\qquad \\ |
− | \textbf{(D)}\ AB \times AD = OB \times AO\qquad | + | \textbf{(D)}\ AB \times AD = OB \times AO\qquad \\ |
\textbf{(E)}\ \text{none of these} </math> | \textbf{(E)}\ \text{none of these} </math> | ||
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== Problem 37== | == Problem 37== | ||
− | The base of a triangle is of length <math>b</math>, and the | + | The base of a triangle is of length <math>b</math>, and the altitude is of length <math>h</math>. |
A rectangle of height <math>x</math> is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is: | A rectangle of height <math>x</math> is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is: | ||
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Given right <math>\triangle ABC</math> with legs <math>BC=3, AC=4</math>. Find the length of the shorter angle trisector from <math>C</math> to the hypotenuse: | Given right <math>\triangle ABC</math> with legs <math>BC=3, AC=4</math>. Find the length of the shorter angle trisector from <math>C</math> to the hypotenuse: | ||
+ | |||
<math> \textbf{(A)}\ \frac{32\sqrt{3}-24}{13}\qquad\textbf{(B)}\ \frac{12\sqrt{3}-9}{13}\qquad\textbf{(C)}\ 6\sqrt{3}-8\qquad\textbf{(D)}\ \frac{5\sqrt{10}}{6}\qquad\textbf{(E)}\ \frac{25}{12}\qquad</math> | <math> \textbf{(A)}\ \frac{32\sqrt{3}-24}{13}\qquad\textbf{(B)}\ \frac{12\sqrt{3}-9}{13}\qquad\textbf{(C)}\ 6\sqrt{3}-8\qquad\textbf{(D)}\ \frac{5\sqrt{10}}{6}\qquad\textbf{(E)}\ \frac{25}{12}\qquad</math> | ||
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* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
− | {{AHSME 40p box|year=1960|before=[[1959 AHSME]]|after=[[1961 AHSME]]}} | + | {{AHSME 40p box|year=1960|before=[[1959 AHSME|1959 AHSC]]|after=[[1961 AHSME|1961 AHSC]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 21:27, 30 December 2020
1960 AHSC (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
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 |
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
If is a solution (root) of , then equals:
Problem 2
It takes seconds for a clock to strike o'clock beginning at o'clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike o'clock?
Problem 3
Applied to a bill for the difference between a discount of % and two successive discounts of % and %, expressed in dollars, is:
Problem 4
Each of two angles of a triangle is and the included side is inches. The area of the triangle, in square inches, is:
Problem 5
The number of distinct points common to the graphs of and is:
Problem 6
The circumference of a circle is inches. The side of a square inscribed in this circle, expressed in inches, is:
Problem 7
Circle passes through the center of, and is tangent to, circle . The area of circle is square inches. Then the area of circle , in square inches, is:
Problem 8
The number can be written as a fraction. When reduced to lowest terms the sum of the numerator and denominator of this fraction is:
Problem 9
The fraction is (with suitable restrictions of the values of a, b, and c):
Problem 10
Given the following six statements:
The statement that negates statement is:
Problem 11
For a given value of the product of the roots of is . The roots may be characterized as:
Problem 12
The locus of the centers of all circles of given radius , in the same plane, passing through a fixed point, is:
Problem 13
The polygon(s) formed by , and , is (are):
Problem 14
If and are real numbers, the equation has a unique solution [The symbol means that is different from zero]:
Problem 15
Triangle is equilateral with side , perimeter , area , and circumradius (radius of the circumscribed circle). Triangle is equilateral with side , perimeter , area , and circumradius . If is different from , then:
Problem 16
In the numeration system with base , counting is as follows: . The number whose description in the decimal system is , when described in the base system, is a number with:
Problem 17
The formula gives, for a certain group, the number of individuals whose income exceeds dollars. The lowest income, in dollars, of the wealthiest individuals is at least:
Problem 18
The pair of equations and has:
Problem 19
Consider equation where , and are positive integers, and equation , where , and are positive integers. Then
Problem 20
The coefficient of in the expansion of is:
Problem 21
The diagonal of square is . The perimeter of square with twice the area of is:
Problem 22
The equality , where and are unequal non-zero constants, is satisfied by , where:
Problem 23
The radius of a cylindrical box is inches, the height is inches. The volume is to be increased by the same fixed positive amount when is increased by inches as when is increased by inches. This condition is satisfied by:
Problem 24
If , where is real, then is:
Problem 25
Let and be any two odd numbers, with less than . The largest integer which divides all possible numbers of the form is:
Problem 26
Find the set of -values satisfying the inequality . [The symbol means if is positive, if is negative, if is zero. The notation means that a can have any value between and , excluding and . ]
Problem 27
Let be the sum of the interior angles of a polygon for which each interior angle is times the exterior angle at the same vertex. Then
Problem 28
The equation has:
Problem 29
Five times 's money added to 's money is more than . Three times 's money minus 's money is . If represents 's money in dollars and represents 's money in dollars, then:
Problem 30
Given the line and a point on this line equidistant from the coordinate axes. Such a point exists in:
Problem 31
For to be a factor of , the values of and must be, respectively:
Problem 32
In this figure the center of the circle is . , is a straight line, , and has a length twice the radius. Then:
Problem 33
You are given a sequence of terms; each term has the form where stands for the product of all prime numbers less than or equal to , and takes, successively, the values . Let be the number of primes appearing in this sequence. Then is:
Problem 34
Two swimmers, at opposite ends of a -foot pool, start to swim the length of the pool, one at the rate of feet per second, the other at feet per second. They swim back and forth for minutes. Allowing no loss of times at the turns, find the number of times they pass each other.
Problem 35
From point outside a circle, with a circumference of units, a tangent is drawn. Also from a secant is drawn dividing the circle into unequal arcs with lengths and . It is found that , the length of the tangent, is the mean proportional between and . If and are integers, then may have the following number of values:
Problem 36
Let be the respective sums of terms of the same arithmetic progression with as the first term and as the common difference. Let . Then is dependent on:
Problem 37
The base of a triangle is of length , and the altitude is of length . A rectangle of height is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:
Problem 38
In this diagram and are the equal sides of an isosceles , in which is inscribed equilateral . Designate by , by , and by . Then:
Problem 39
To satisfy the equation , and must be:
Problem 40
Given right with legs . Find the length of the shorter angle trisector from to the hypotenuse:
See also
1960 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1959 AHSC |
Followed by 1961 AHSC | |
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.