Difference between revisions of "1976 AHSME Problems"
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"If the pink elephant on planet alpha has purple eyes, then the wild pig on planet beta does not have a long nose"? | "If the pink elephant on planet alpha has purple eyes, then the wild pig on planet beta does not have a long nose"? | ||
− | < | + | <math>\begin{array}{l}\textbf{I. }\text{"If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha has purple eyes."}\\ |
− | \textbf{II. | + | \textbf{II. }\text{"If the pink elephant on planet alpha does not have purple eyes, then the wild pig on planet beta does not have a long nose."}\\ |
− | \textbf{III. | + | \textbf{III. }\text{"If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha does not have purple eyes."}\\ |
− | \textbf{IV. | + | \textbf{IV. }\text{"The pink elephant on planet alpha does not have purple eyes, or the wild pig on planet beta does not have a long nose."}\end{array}</math> |
<math> \textbf{(A) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad | <math> \textbf{(A) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad |
Revision as of 22:23, 13 March 2015
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 See also
Problem 1
If one minus the reciprocal of equals the reciprocal of , then equals
Problem 2
For how many real numbers is a real number?
Problem 3
The sum of the distances from one vertex of a square with sides of length to the midpoints of each of the sides of the square is
Problem 4
Let a geometric progression with n terms have first term one, common ratio and sum , where and are not zero. The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is
Problem 5
How many integers greater than and less than , written in base- notation, are increased by when their digits are reversed?
Problem 6
If is a real number and the negative of one of the solutions of is a solution of , then the solutions of are
Problem 7
If is a real number, then the quantity is positive if and only if
Problem 8
A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units?
Problem 9
In triangle , is the midpoint of ; is the midpoint of ; and is the midpoint of . If the area of is , then the area of is
Problem 10
If , and are real numbers and and , then the equation has a solution
Problem 11
Which of the following statements is (are) equivalent to the statement "If the pink elephant on planet alpha has purple eyes, then the wild pig on planet beta does not have a long nose"?
Problem 12
A supermarket has crates of apples. Each crate contains at least apples and at most apples. What is the largest integer such that there must be at least crates containing the same number of apples?
Problem 13
If cows give cans of milk in days, how many days will it take cows to give cans of milk?
Problem 14
The measures of the interior angles of a convex polygon are in arithmetic progression. If the smallest angle is , and the largest is , then the number of sides the polygon has is
Problem 15
If is the remainder when each of the numbers , and is divided by , where is an integer greater than , then equals
Problem 16
In triangles and , lengths , and are all equal. Length is twice the length of the altitude of from to . Which of the following statements is (are) true?
\[\textbf{I. }\angle ACB \text{ and }\angle DFE\text{ must be complementary.}\\ \textbf{II. }\angle ACB \text{ and }\angle DFE\text{ must be supplementary.}\\ \textbf{III. }\text{The area of }\triangle ABC\text{ must equal the area of }\triangle DEF.}\\ \textbf{IV. }\text{The area of }\triangle ABC\text{ must equal twice the area of }\triangle DEF.}\] (Error compiling LaTeX. ! Extra }, or forgotten $.)
Problem 17
If is an acute angle, and , then equals
Problem 18
In the adjoining figure, is tangent at to the circle with center ; point is interior to the circle; and intersects the circle at . If , , and , then the radius of the circle is
Problem 19
A polynomial has remainder three when divided by and remainder five when divided by . The remainder when is divided by is
Problem 20
Let , and be positive real numbers distinct from one. Then
Problem 21
What is the smallest positive odd integer such that the product is greater than ? (In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from to .)
Problem 22
Given an equilateral triangle with side of length , consider the locus of all points in the plane of the triangle such that the sum of the squares of the distances from to the vertices of the triangle is a fixed number . This locus
Problem 23
For integers and such that , let . Then is an integer
Problem 24
In the adjoining figure, circle has diameter ; circle is tangent to circle and to at the center of circle ; and circle tangent to circle , to circle and . The ratio of the area of circle to the area of circle is
Problem 25
For a sequence , define and, for all integer . If , then for all
Problem 26
In the adjoining figure, every point of circle is exterior to circle . Let and be the points of intersection of an internal common tangent with the two external common tangents. Then the length of is
$\textbf{(A) }\text{the average of the lengths of the internal and external common tangents}\qquad\\ \textbf{(B) }\text{equal to the length of an external common tangent if and only if circles }\mathit{O}\text{ and }\mathit{O'} have equal radii}\\ \textbf{(C) }\text{always equal to the length of an external common tangent}\qquad\\ \textbf{(D) }\text{greater than the length of an external common tangent}\qquad\\ \textbf{(E) }\text{the geometric mean of the lengths of the internal and external common tangents}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)
Problem 27
If , then equals
Problem 28
Lines are distinct. All lines a positive integer, are parallel to each other. All lines , a positive integer, pass through a given point . The maximum number of points of intersection of pairs of lines from the complete set is
Problem 29
Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is years, then Ann's age is
Problem 30
How many distinct ordered triples satisfy the equations
See also
1976 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1975 AHSME |
Followed by 1977 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.