Difference between revisions of "1983 AIME Problems"
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== Problem 4 == | == Problem 4 == | ||
A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is 50 cm, the length of <math>AB</math> is 6 cm, and that of <math>BC</math> is 2 cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B</math> to the center of the circle. | A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is 50 cm, the length of <math>AB</math> is 6 cm, and that of <math>BC</math> is 2 cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B</math> to the center of the circle. | ||
− | + | ||
+ | <asy> | ||
+ | size(150); defaultpen(linewidth(0.6)+fontsize(11)); | ||
+ | real r=10; | ||
+ | pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C; | ||
+ | path P=circle(O,r); | ||
+ | C=intersectionpoint(B--(B.x+r,B.y),P); | ||
+ | draw(P); | ||
+ | draw(C--B--O--A--B); | ||
+ | dot(O); dot(A); dot(B); dot(C); | ||
+ | label("$O$",O,SW); | ||
+ | label("$A$",A,NE); | ||
+ | label("$B$",B,S); | ||
+ | label("$C$",C,SE); | ||
+ | </asy> | ||
[[1983 AIME Problems/Problem 4|Solution]] | [[1983 AIME Problems/Problem 4|Solution]] | ||
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The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to the base and has length <math>2s</math>. All edges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid? | The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to the base and has length <math>2s</math>. All edges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid? | ||
− | + | <asy> | |
+ | size(170);import three; pathpen = black+linewidth(0.65); pointpen = black; | ||
+ | currentprojection = perspective(30,-20,10);real s = 6 * 2^.5; | ||
+ | triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6); | ||
+ | D(A--B--C--D--A--E--D); D(B--F--C); D(E--F); | ||
+ | MP("A",A);MP("B",B);MP("C",C);MP("D",D);MP("E",E,N);MP("F",F,N); | ||
+ | </asy> | ||
[[1983 AIME Problems/Problem 11|Solution]] | [[1983 AIME Problems/Problem 11|Solution]] | ||
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The length of diameter <math>AB</math> is a two digit integer. Reversing the digits gives the length of a perpendicular chord <math>CD</math>. The distance from their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</math>. | The length of diameter <math>AB</math> is a two digit integer. Reversing the digits gives the length of a perpendicular chord <math>CD</math>. The distance from their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</math>. | ||
− | + | <asy>pointpen=black; pathpen=black+linewidth(0.65); | |
+ | pair O=(0,0),A=(-65/2,0),B=(65/2,0); | ||
+ | pair H=(-((65/2)^2-28^2)^.5,0),C=(H.x,28),D=(H.x,-28); | ||
+ | D(CP(O,A));D(MP("A",A,W)--MP("B",B,E));D(MP("C",C,N)--MP("D",D)); | ||
+ | dot(MP("H",H,SE));dot(MP("O",O,SE));</asy> | ||
[[1983 AIME Problems/Problem 12|Solution]] | [[1983 AIME Problems/Problem 12|Solution]] | ||
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In the adjoining figure, two circles with radii <math>6</math> and <math>8</math> are drawn with their centers <math>12</math> units apart. At <math>P</math>, one of the points of intersection, a line is drawn in sich a way that the chords <math>QP</math> and <math>PR</math> have equal length. (<math>P</math> is the midpoint of <math>QR</math>) Find the square of the length of <math>QP</math>. | In the adjoining figure, two circles with radii <math>6</math> and <math>8</math> are drawn with their centers <math>12</math> units apart. At <math>P</math>, one of the points of intersection, a line is drawn in sich a way that the chords <math>QP</math> and <math>PR</math> have equal length. (<math>P</math> is the midpoint of <math>QR</math>) Find the square of the length of <math>QP</math>. | ||
− | [[Image: | + | [[Image:1983_AIME-14.png]] |
[[1983 AIME Problems/Problem 14|Solution]] | [[1983 AIME Problems/Problem 14|Solution]] | ||
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The adjoining figure shows two intersecting chords in a circle, with <math>B</math> on minor arc <math>AD</math>. Suppose that the radius of the circle is <math>5</math>, that <math>BC=6</math>, and that <math>AD</math> is bisected by <math>BC</math>. Suppose further that <math>AD</math> is the only chord starting at <math>A</math> which is bisected by <math>BC</math>. It follows that the sine of the minor arc <math>AB</math> is a rational number. If this fraction is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? | The adjoining figure shows two intersecting chords in a circle, with <math>B</math> on minor arc <math>AD</math>. Suppose that the radius of the circle is <math>5</math>, that <math>BC=6</math>, and that <math>AD</math> is bisected by <math>BC</math>. Suppose further that <math>AD</math> is the only chord starting at <math>A</math> which is bisected by <math>BC</math>. It follows that the sine of the minor arc <math>AB</math> is a rational number. If this fraction is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? | ||
− | [[Image: | + | [[Image:1983 AIME Problem 15.png]] |
[[1983 AIME Problems/Problem 15|Solution]] | [[1983 AIME Problems/Problem 15|Solution]] |
Revision as of 19:18, 25 April 2008
Contents
Problem 1
Let ,
, and
all exceed
, and let
be a positive number such that
,
, and
. Find
.
Problem 2
Let , where
. Determine the minimum value taken by
by
in the interval
.
Problem 3
What is the product of the real roots of the equation ?
Problem 4
A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is 50 cm, the length of is 6 cm, and that of
is 2 cm. The angle
is a right angle. Find the square of the distance (in centimeters) from
to the center of the circle.
Problem 5
Suppose that the sum of the squares of two complex numbers and
is
and the sum of the cubes is
. What is the largest real value of
can have?
Problem 6
Let equal
. Determine the remainder upon dividing
by
.
Problem 7
Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let be the probability that at least two of the three had been sitting next to each other. If
is written as a fraction in lowest terms, what is the sum of the numerator and the denominator?
Problem 8
What is the largest 2-digit prime factor of the integer ?
Problem 9
Find the minimum value of for
.
Problem 10
The numbers ,
, and
have something in common. Each is a four-digit number beginning with
that has exactly two identical digits. How many such numbers are there?
Problem 11
The solid shown has a square base of side length . The upper edge is parallel to the base and has length
. All edges have length
. Given that
, what is the volume of the solid?
size(170);import three; pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(30,-20,10);real s = 6 * 2^.5; triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6); D(A--B--C--D--A--E--D); D(B--F--C); D(E--F); MP("A",A);MP("B",B);MP("C",C);MP("D",D);MP("E",E,N);MP("F",F,N); (Error making remote request. Unknown error_msg)
Problem 12
The length of diameter is a two digit integer. Reversing the digits gives the length of a perpendicular chord
. The distance from their intersection point
to the center
is a positive rational number. Determine the length of
.
Problem 13
For and each of its non-empty subsets, an alternating sum is defined as follows. Arrange the number in the subset in decreasing order and then, beginning with the largest, alternately add and subtract succesive numbers. For example, the alternating sum for
is
and for
it is simply
. Find the sum of all such alternating sums for
.
Problem 14
In the adjoining figure, two circles with radii and
are drawn with their centers
units apart. At
, one of the points of intersection, a line is drawn in sich a way that the chords
and
have equal length. (
is the midpoint of
) Find the square of the length of
.
Problem 15
The adjoining figure shows two intersecting chords in a circle, with on minor arc
. Suppose that the radius of the circle is
, that
, and that
is bisected by
. Suppose further that
is the only chord starting at
which is bisected by
. It follows that the sine of the minor arc
is a rational number. If this fraction is expressed as a fraction
in lowest terms, what is the product
?