Difference between revisions of "1966 AHSME Problems"
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(Problems 11 and 28.) |
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Circle I is circumscribed about a given square and circle II is inscribed in the given square. If <math>r</math> is the ratio of the area of circle <math>I</math> to that of circle <math>II</math>, then <math>r</math> equals: | Circle I is circumscribed about a given square and circle II is inscribed in the given square. If <math>r</math> is the ratio of the area of circle <math>I</math> to that of circle <math>II</math>, then <math>r</math> equals: | ||
− | <math>\text{(A)} | + | <math>\text{(A)} \sqrt 2 \qquad \text{(B)} 2 \qquad \text{(C)} \sqrt 3 \qquad \text{(D)} 2\sqrt 2 \qquad \text{(E)} 2\sqrt 3</math> |
[[1966 AHSME Problems/Problem 4|Solution]] | [[1966 AHSME Problems/Problem 4|Solution]] | ||
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== Problem 11 == | == Problem 11 == | ||
+ | The sides of triangle <math>BAC</math> are in the ratio <math>2: 3: 4</math>. <math>BD</math> is the angle-bisector drawn to the shortest side <math>AC</math>, dividing it into segments <math>AD</math> and <math>CD</math>. If the length of <math>AC</math> is <math>10</math>, then the length of the longer segment of <math>AC</math> is: | ||
+ | <math>\text{(A)} \ 3\frac12 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 5\frac57 \qquad \text{(D)} \ 6 \qquad \text{(E)} \ 7\frac12</math> | ||
[[1966 AHSME Problems/Problem 11|Solution]] | [[1966 AHSME Problems/Problem 11|Solution]] | ||
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== Problem 28 == | == Problem 28 == | ||
+ | Five points <math>O,A,B,C,D</math> are taken in order on a straight line with distances <math>OA = a</math>, <math>OB = b</math>, <math>OC = c</math>, and <math>OD = d</math>. <math>P</math> is a point on the line between <math>B</math> and <math>C</math> and such that <math>AP: PD = BP: PC</math>. Then <math>OP</math> equals: | ||
+ | <math>\textbf{(A)} \frac {b^2 - bc}{a - b + c - d} \qquad \textbf{(B)} \frac {ac - bd}{a - b + c - d} \\ | ||
+ | \textbf{(C)} - \frac {bd + ac}{a - b + c - d} \qquad \textbf{(D)} \frac {bc + ad}{a + b + c + d} \qquad \textbf{(E)} \frac {ac - bd}{a + b + c + d}</math> | ||
[[1966 AHSME Problems/Problem 28|Solution]] | [[1966 AHSME Problems/Problem 28|Solution]] |
Revision as of 15:29, 7 February 2009
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
Given that the ratio of to
is constant, and
when
, then, when
,
equals:
Problem 2
Problem 3
If the arithmetic mean of two numbers is and their geometric mean is
, then an equation with the given two numbers as roots is:
Problem 4
Circle I is circumscribed about a given square and circle II is inscribed in the given square. If is the ratio of the area of circle
to that of circle
, then
equals:
Problem 5
The number of values of satisfying the equation
\[
\frac {2x^2 - 10x}{x^2 - 5x} = x - 3
\]
is:
Problem 6
is the diameter of a circle centered at
.
is a point on the circle such that angle
is
. If the diameter of the circle is
inches, the length of chord
, expressed in inches, is:
Problem 7
Let be an identity in
. The numerical value of
is:
Problem 8
The length of the common chord of two intersecting circles is feet. If the radii are
feet and
feet, a possible value for the distance between the centers of teh circles, expressed in feet, is:
Problem 9
If , then
equals:
Problem 10
If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is:
Problem 11
The sides of triangle are in the ratio
.
is the angle-bisector drawn to the shortest side
, dividing it into segments
and
. If the length of
is
, then the length of the longer segment of
is:
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Five points are taken in order on a straight line with distances
,
,
, and
.
is a point on the line between
and
and such that
. Then
equals: