1970 AHSME Problems
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 fourth power of is:
Problem 2
A square and a circle have equal perimeters. The ratio of the area of the circle to the area of the square is:
Problem 3
If and , then in terms of is:
Problem 4
Let be the set of all numbers which are the sum of the squares of three consecutive integers. Then we can say that:
Problem 5
If , then , where , is equal to:
Problem 6
The smallest value of for real values of is:
Problem 7
Inside square with side , quarter-circle arcs with radii and centers at and are drawn. These arcs intersect at point inside the square. How far is from side ?
Problem 8
If and , then
Problem 9
Points and are on line segment , and both points are on the same side of the midpoint of . Point divides in the ratio and divides in the ratio . If , then the length of segment is
Problem 10
Let be an infinite repeating decimal with the digits and repeating. When is written as a fraction in lowest terms, the denominator exceeds the numerator by
Problem 11
If two factors of are and , the value of is
Problem 12
A circle with radius is tangent to sides , , and of rectangle and passes through the midpoint of diagonal . The area of the rectangle in terms of , is
Problem 13
Given the binary operation defined by for all positive numbers and . The for all positive , we have
Problem 14
Consider where and are positive numbers. If the roots of this equation differ by , then equals
Problem 15
Lines in the -plane are drawn through the point and the trisection points of the line segment joining the points and . One of these lines has the equation
Problem 16
If is a function such that , and such that for , then is equal to
Problem 17
If , then for all and such that and , we have
Problem 18
is equal to
Problem 19
The sum of an infinite geometric series with common ratio such that , is , and the sum of the squares of the terms of this series is . The first term of the series is
Problem 20
Lines and lie in a plane. is the midpoint of line segment , and and are perpendicular to . Then we
Problem 21
On an auto trip, the distance read from the instrument panel was miles. With snow tires on for the return trip over the same route, the reading was miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was inches.
Problem 22
If the sum of the first positive integers is more than the sum of the first positive integers, then the sum of the first positive integers is
Problem 23
The number ( is written in base ), when written in the base system, ends in exactly zeroes. The value of is
Problem 24
An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is , then the area of the hexagon is
Problem 25
For every real number , let be the greatest integer less than or equal to . If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing ounces is always
Problem 26
The number of distinct points in the -plane common to the graphs of and is
Problem 27
In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle?
Problem 28
In triangle , the median from vertex is perpendicular to the median from vertex . If the lengths of sides and are and respectively, then the length of side is
Problem 29
It is now between and o'clock, and six minutes form now, the minute hand of the watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now?
Problem 30
In the accompanying figure, segments and are parallel, the measure of is twice the measure of , and the measures of segments and are and respectively. Then the measure of is equal to
Problem 31
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to , what is the probability that this number is divisible by ?
Problem 32
and travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after has travelled yards, and meet a second time yards before completes one lap, then the circumference of the track in yards is
Problem 33
Find the sum of the digits of all numerals in the sequence .
Problem 34
The greatest integer that will divide , and and leave the same remainder is
Problem 35
A retiring employee receives and annual pension proportional to the square root of the number of years of his service. Had he served a years more, his pension would have been dollars greater, whereas, had he served years more , his pension would have been dollars greater than the original annual pension. Find his annual pension in terms of , and .
See also
1970 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1969 AHSME |
Followed by 1971 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.