Difference between revisions of "1974 AHSME Problems"
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[[1974 AHSME Problems/Problem 9|Solution]] | [[1974 AHSME Problems/Problem 9|Solution]] | ||
− | ==Problem | + | ==Problem== |
What is the smallest integral value of <math> k </math> such that | What is the smallest integral value of <math> k </math> such that | ||
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[[1974 AHSME Problems/Problem 10|Solution]] | [[1974 AHSME Problems/Problem 10|Solution]] | ||
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+ | ==Solution== | ||
+ | Expanding, we have <math> 2kx^2-8x-x^2+6=0 </math>, or <math> (2k-1)x^2-8x+6=0 </math>. For this quadratic not to have real roots, it must have a negative discriminant. Therefore, <math> (-8)^2-4(2k-1)(6)<0\implies 64-48k+24<0\implies k>\frac{11}{6} </math>. From here, we can easily see that the smallest integral value of <math> k </math> is <math> 2, \boxed{\text{B}} </math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{AHSME box|year=1974|num-b=9|num-a=11}} | ||
+ | |||
==Problem 11== | ==Problem 11== | ||
If <math> (a, b) </math> and <math> (c, d) </math> are two points on the line whose equation is <math> y=mx+k </math>, then the distance between <math> (a, b) </math> and <math> (c, d) </math>, in terms of <math> a, c, </math> and <math> m </math> is | If <math> (a, b) </math> and <math> (c, d) </math> are two points on the line whose equation is <math> y=mx+k </math>, then the distance between <math> (a, b) </math> and <math> (c, d) </math>, in terms of <math> a, c, </math> and <math> m </math> is |
Revision as of 17:05, 26 May 2012
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
- 11 Solution
- 12 See Also
- 13 Problem 11
- 14 Problem 12
- 15 Problem 13
- 16 Problem 14
- 17 Problem 15
- 18 Problem 16
- 19 Problem 17
- 20 Problem 18
- 21 Problem 19
- 22 Problem 20
- 23 Problem 21
- 24 Problem 22
- 25 Problem 23
- 26 Problem 24
- 27 Problem 25
- 28 Problem 26
- 29 Problem 27
- 30 Problem 28
- 31 Problem 29
- 32 Problem 30
- 33 See Also
Problem 1
If or and or , then is equivalent to
Problem 2
Let and be such that and , . Then equals
Problem 3
The coefficient of in the polynomial expansion of
is
Problem 4
What is the remainder when is divided by ?
Problem 5
Given a quadrilateral inscribed in a circle with side extended beyond to point , if and , find .
Problem 6
For positive real numbers and define ' then
Problem 7
A town's population increased by people, and then this new population decreased by . The town now had less people than it did before the increase. What is the original population?
Problem 8
What is the smallest prime number dividing the sum ?
Problem 9
The integers greater than one are arranged in five columns as follows:
(Four consecutive integers appear in each row; in the first, third and other odd numbered rows, the integers appear in the last four columns and increase from left to right; in the second, fourth and other even numbered rows, the integers appear in the first four columns and increase from right to left.)
In which column will the number fall?
Problem
What is the smallest integral value of such that
has no real roots?
Solution
Expanding, we have , or . For this quadratic not to have real roots, it must have a negative discriminant. Therefore, . From here, we can easily see that the smallest integral value of is .
See Also
1974 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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 |
Problem 11
If and are two points on the line whose equation is , then the distance between and , in terms of and is
Problem 12
If and when , then equals
Problem 13
Which of the following is equivalent to "If P is true, then Q is false."?
Problem 14
Which statement is correct?
Problem 15
If , then equals
Problem 16
A circle of radius is inscribed in a right isosceles triangle, and a circle of radius is circumscribed about the triangle. Then equals
Problem 17
If , then equals
Problem 18
If and , then, in terms of and , equals
Problem 19
In the adjoining figure is a square and is an equilateral triangle. If the area of is one square inch, then the area of in square inches is
Problem 20
Let
Then
Problem 21
In a geometric series of positive terms the difference between the fifth and fourth terms is , and the difference between the second and first terms is . What is the sum of the first five terms of this series?
Problem 22
The minimum of is attained when is
Problem 23
In the adjoining figure and are parallel tangents to a circle of radius , with and the points of tangency. is a third tangent with as a point of tangency. If and then is
Problem 24
A fair die is rolled six times. The probability of rolling at least a five at least five times is
Problem 25
In parallelogram of the accompanying diagram, line is drawn bisecting at and meeting (extended) at . From vertex , line is drawn bisecting side at and meeting (extended) at . Lines and meet at . If the area of parallelogram is , then the area of the triangle is equal to
Problem 26
The number of distinct positive integral divisors of excluding and is
Problem 27
If for all real , then the statement: " whenever and and " is true when
Problem 28
Which of the following is satisfied by all numbers of the form
where is or , is or ,..., is or ?
Problem 29
For let be the sum of the first terms of the arithmetic progression whose first term is and whose common difference is ; then is
Problem 30
A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If is the ratio of the lesser part to the greater part, then the value of
is