Difference between revisions of "1983 AIME Problems"
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== Problem 1 == | == Problem 1 == | ||
+ | Let <math>x</math>,<math>y</math>, and <math>z</math> all exceed <math>1</math>, and let <math>w</math> be a positive number such that <math>\log_xw=24</math>, <math>\displaystyle \log_y w = 40</math>, and <math>\log_{xyz}w=12</math>. Find <math>\log_zw</math>. | ||
[[1983 AIME Problems/Problem 1|Solution]] | [[1983 AIME Problems/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
+ | Let <math>f(x)=|x-p|+|x-15|+|x-p-15|</math>, where <math>p \leq x \leq 15</math>. Determine the minimum value taken by <math>f(x)</math> by <math>x</math> in the interval <math>0 < p<15</math>. | ||
[[1983 AIME Problems/Problem 2|Solution]] | [[1983 AIME Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
+ | What is the product of the real roots of the equation <math>x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}</math>? | ||
[[1983 AIME Problems/Problem 3|Solution]] | [[1983 AIME Problems/Problem 3|Solution]] | ||
== 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. | ||
+ | [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=790&sid=cfd5dae222dd7b8944719b56de7b8bf7[/img] | ||
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[[1983 AIME Problems/Problem 4|Solution]] | [[1983 AIME Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
+ | Suppose that the sum of the squares of two complex numbers <math>x</math> and <math>y</math> is <math>7</math> and the sum of the cubes is <math>10</math>. What is the largest real value of <math>x + y</math> can have? | ||
[[1983 AIME Problems/Problem 5|Solution]] | [[1983 AIME Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
+ | Let <math>a_n</math> equal <math>6^{n}+8^{n}</math>. Determine the remainder upon dividing <math>a_ {83}</math> by <math>49</math>. | ||
[[1983 AIME Problems/Problem 6|Solution]] | [[1983 AIME Problems/Problem 6|Solution]] | ||
== Problem 7 == | == 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 of to slay a troublesome dragon. Let <math>P</math> be the brobability that at least two of the three had been sitting next to each other. If <math>P</math> is written as a fraction in lowest terms, what is the sum of the numerator and the denominator? | ||
[[1983 AIME Problems/Problem 7|Solution]] | [[1983 AIME Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
+ | What is the largest 2-digit prime factor of the integer <math>{200\choose 100}</math>? | ||
[[1983 AIME Problems/Problem 8|Solution]] | [[1983 AIME Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
+ | Find the minimum value of <math>\frac{9x^2\sin^2 x + 4}{x\sin x}</math> for <math>0 < x < \pi</math>. | ||
[[1983 AIME Problems/Problem 9|Solution]] | [[1983 AIME Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
+ | The numbers <math>1447</math>, <math>1005</math>, and <math>1231</math> have something in common. Each is a four-digit number beginning with <math>1</math> that has exactly two identical digits. How many such numbers are there? | ||
[[1983 AIME Problems/Problem 10|Solution]] | [[1983 AIME Problems/Problem 10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
+ | 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? | ||
+ | [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=791&sid=cfd5dae222dd7b8944719b56de7b8bf7[/img] | ||
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[[1983 AIME Problems/Problem 11|Solution]] | [[1983 AIME Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
+ | 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>. | ||
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+ | [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=792&sid=cfd5dae222dd7b8944719b56de7b8bf7[/img] | ||
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[[1983 AIME Problems/Problem 12|Solution]] | [[1983 AIME Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
+ | For <math>\{1, 2, 3, \ldots, n\}</math> 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 <math>\{1, 2, 3, 6,9\}</math> is <math>9-6+3-2+1=6</math> and for <math>\{5\}</math> it is simply <math>5</math>. Find the sum of all such alternating sums for <math>n=7</math>. | ||
[[1983 AIME Problems/Problem 13|Solution]] | [[1983 AIME Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | 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>. | ||
+ | [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=793[/img] | ||
+ | {{image}} | ||
[[1983 AIME Problems/Problem 14|Solution]] | [[1983 AIME Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
+ | 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>? | ||
+ | [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=794[/img] | ||
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+ | {{image}} | ||
[[1983 AIME Problems/Problem 15|Solution]] | [[1983 AIME Problems/Problem 15|Solution]] |
Revision as of 23:30, 23 July 2006
Contents
[hide]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. [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=790&sid=cfd5dae222dd7b8944719b56de7b8bf7[/img]
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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 of to slay a troublesome dragon. Let be the brobability 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? [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=791&sid=cfd5dae222dd7b8944719b56de7b8bf7[/img]
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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 .
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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 . [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=793[/img]
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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 ? [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=794[/img]
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