1984 AHSME Problems
1984 AHSME (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 See also
Problem 1
equals
Problem 2
If , and are not , then
equals
Problem 3
Let be the smallest nonprime integer greater than with no prime factor less than . Then
Problem 4
A rectangle intersects a circle as shown: , and . Then equals
Problem 5
The largest integer for which is
Problem 6
In a certain school, there are three times as many boys as girls and nine times as many girls as teachers. Using the letters to represent the number of boys, girls and teachers, respectively, then the total number of boys, girls and teachers can be represented by the expression
Problem 7
When Dave walks to school, he averages steps per minute, each of his steps cm long. It takes him minutes to get to school. His brother, Jack, going to the same school by the same route, averages steps per minute, but his steps are only cm long. How long does it take Jack to get to school?
Problem 8
Figure is a trapezoid with , , , and . The length of is
Problem 9
The number of digits in (when written in the usual base form) is
Problem 10
Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are and . The fourth number is
Problem 11
A calculator has a key that replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let be the final result when one starts with an entry and alternately squares and reciprocates times each. Assuming the calculator is completely accurate (e.g. no roundoff or overflow), then equals
Problem 12
If the sequence is defined by
then equals
Problem 13
equals
Problem 14
The product of all real roots of the equation is
Problem 15
If , then one value for is
Problem 16
The function satisfies for all real numbers . If the equation has exactly four distinct real roots, then the sum of these roots is
Problem 17
A right triangle with hypotenuse has side . Altitude divides into segments and , with . The area of is
Problem 18
A point is to be chosen in the coordinate plane so that it is equally distant from the -axis, the -axis, and the line . Then is
Problem 19
A box contains balls, numbered . If balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd?
Problem 20
The number of the distinct solutions of the equation is
Problem 21
The number of triples of positive integers which satisfy the simultaneous equations
is
Problem 22
Let and be fixed positive numbers. For each real number let be the vertex of the parabola . If the set of the vertices for all real values of is graphed on the plane, the graph is
Problem 23
equals
Problem 24
If and are positive real numbers and each of the equations and has real roots, then the smallest possible value of is
Problem 25
The total area of all the faces of a rectangular solid is , and the total length of all its edges is . Then the length in cm of any one of its interior diagonals is
Problem 26
In the obtuse triangle , , , . If the area of is , then the area of is
Problem 27
In , is on and is on . Also, , , and . Find .
Problem 28
The number of distinct pairs of integers such that and is
Problem 29
Find the largest value for for pairs of real numbers which satisfy .
Problem 30
For any complex number , is defined to be the real number . If , then equals
See also
1984 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1983 AHSME |
Followed by 1985 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.