Difference between revisions of "1980 AHSME Problems"
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A positive number <math>x</math> satisfies the inequality <math>\sqrt{x} < 2x</math> if and only if | A positive number <math>x</math> satisfies the inequality <math>\sqrt{x} < 2x</math> if and only if | ||
− | <math>\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} \x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} \x < 4</math> | + | <math>\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} \ x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} \ x < 4</math> |
[[1980 AHSME Problems/Problem 6|Solution]] | [[1980 AHSME Problems/Problem 6|Solution]] |
Revision as of 22:28, 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
The largest whole number such that seven times the number is less than is
Problem 2
The degree of as a polynomial in is
Problem 3
If the ratio of to is , what is the ratio of to ?
Problem 4
In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of is
Problem 5
If and are perpendicular diameters of circle , in , and , then the length of divided by the length of is
Problem 6
A positive number satisfies the inequality if and only if
Problem 7
Sides and of convex polygon have lengths 3,4,12, and 13, respectively, and is a right angle. The area of the quadrilateral is
Problem 8
How many pairs of non-zero real numbers satisfy the equation
Problem 9
A man walks miles due west, turns to his left and walks 3 miles in the new direction. If he finishes a a point from his starting point, then is
Problem 10
The number of teeth in three meshed gears , , and are , , and , respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of , , and are in the proportion
Problem 11
If the sum of the first terms and the sum of the first terms of a given arithmetic progression are and , respectively, then the sum of first terms is:
Problem 12
The equations of and are and , respectively. Suppose makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does , and that has 4 times the slope of . If is not horizontal, then is
Problem 13
A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to . Then it makes a counterclockwise and travels a unit to . If it continues in this fashion, each time making a degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest?
Problem 14
If the function is defined by satisfies for all real numbers except , then is
Problem 15
A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly dollars where is a positive integer. The smallest value of is
Problem 16
Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron.
Problem 17
Given that , for how many integers is an integer?
Problem 18
If , , , and , then equals
Problem 19
Let and be three parallel chords of a circle on the same side of the center. The distance between and is the same as the distance between and . The lengths of the chords are , and . The radius of the circle is
Problem 20
A box contains pennies, nickels, and dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least cents?
Problem 21
In triangle , , is the midpoint of side , and is a point on side such that ; and intersect at . The ratio of the area of triangle to the area of quadrilateral is
Problem 22
For each real number , let be the minimum of the numbers , and . Then the maximum value of is
Problem 23
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths and , where is a real number such that . The length of the hypotenuse is
Problem 24
For some real number , the polynomial is divisible by . Which of the following numbers is closest to ?
Problem 25
In the non-decreasing sequence of odd integers each odd positive integer appears times. It is a fact that there are integers , and such that for all positive integers , , where denotes the largest integer not exceeding . The sum equals
Problem 26
Four balls of radius are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges have length , is circumscribed around the balls. Then equals
Problem 27
The sum equals
Problem 28
The polynomial is not divisible by if equals
Problem 29
How many ordered triples (x,y,z) of integers satisfy the system of equations below?
Problem 30
A six digit number (base 10) is squarish if it satisfies the following conditions:
(i) none of its digits are zero;
(ii) it is a perfect square; and
(iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers.
How many squarish numbers are there?
See also
1980 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1979 AHSME |
Followed by 1981 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.