1979 AHSME Problems
1979 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
If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is
Problem 2
For all non-zero real numbers and such that equals
Problem 3
In the adjoining figure, is a square, is an equilateral triangle and point is outside square . What is the measure of in degrees?
Problem 4
For all real numbers
Problem 5
Find the sum of the digits of the largest even three digit number (in base ten representation) which is not changed when its units and hundreds digits are interchanged.
Problem 6
Problem 7
The square of an integer is called a perfect square. If is a perfect square, the next larger perfect square is
Problem 8
Find the area of the smallest region bounded by the graphs of and .
Problem 9
The product of and equals
Problem 10
If is a regular hexagon whose apothem (distance from the center to midpoint of a side) is , and is the midpoint of side for , then the area of quadrilateral is
Problem 11
Find a positive integral solution to the equation
Problem 12
In the adjoining figure, is the diameter of a semi-circle with center . Point lies on the extension of past ; point lies on the semi-circle, and is the point of intersection (distinct from ) of line segment with the semi-circle. If length equals length , and the measure of is , then the measure of is
Problem 13
The inequality is satisfied if and only if
Problem 14
In a certain sequence of numbers, the first number is , and, for all , the product of the first numbers in the sequence is . The sum of the third and the fifth numbers in the sequence is
Problem 15
Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being in one jar and in the other jar. If the entire contents of the two jars are mixed together, the ratio of the volume of alcohol to the volume of water in the mixture is
Problem 16
A circle with area is contained in the interior of a larger circle with area . If the radius of the larger circle is , and if is an arithmetic progression, then the radius of the smaller circle is
Problem 17
Points , and are distinct and lie, in the given order, on a straight line. Line segments , and have lengths , and , respectively. If line segments and may be rotated about points and , respectively, so that points and coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied?
Problem 18
To the nearest thousandth, is and is . Which of the following is the best approximation of ?
Problem 19
Find the sum of the squares of all real numbers satisfying the equation .
Problem 20
If and then the radian measure of equals
Problem 21
The length of the hypotenuse of a right triangle is , and the radius of the inscribed circle is . The ratio of the area of the circle to the area of the triangle is
Problem 22
Find the number of pairs of integers which satisfy the equation .
Problem 23
The edges of a regular tetrahedron with vertices , and each have length one. Find the least possible distance between a pair of points and , where is on edge and is on edge .
Problem 24
Sides , and of (simple*) quadrilateral have lengths , and , respectively. If vertex angles and are obtuse and , then side has length
- A polygon is called “simple” if it is not self intersecting.
Problem 25
If and are the quotient and remainder, respectively, when the polynomial is divided by , and if and are the quotient and remainder, respectively, when is divided by , then equals
Problem 26
The function satisfies the functional equation for every pair of real numbers. If , then the number of integers for which is
Problem 27
An ordered pair of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation will not have distinct positive real roots?
Problem 28
Circles with centers , and each have radius , where . The distance between each pair of centers is . If is the point of intersection of circle and circle which is outside circle , and if is the point of intersection of circle and circle which is outside circle , then length equals
Problem 29
For each positive number , let . The minimum value of is
Problem 30
In , is the midpoint of side and is on side . If the length of is and and , then the area of plus twice the area of equals
See also
1979 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1978 AHSME |
Followed by 1980 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.