Difference between revisions of "1983 AIME Problems"
m (→Problem 11) |
Sevenoptimus (talk | contribs) m (Fixed problem statement of Problem 6) |
||
(37 intermediate revisions by 17 users not shown) | |||
Line 2: | Line 2: | ||
== Problem 1 == | == Problem 1 == | ||
− | Let <math>x</math>,<math>y</math> | + | 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>\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 | + | Let <math>f(x)=|x-p|+|x-15|+|x-p-15|</math>, where <math>0 < p < 15</math>. Determine the [[minimum]] value taken by <math>f(x)</math> for <math>x</math> in the [[interval]] <math>p \leq x\leq15</math>. |
[[1983 AIME Problems/Problem 2|Solution]] | [[1983 AIME Problems/Problem 2|Solution]] | ||
Line 17: | Line 17: | ||
== Problem 4 == | == Problem 4 == | ||
− | A machine shop cutting tool | + | A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is <math>\sqrt{50}</math> cm, the length of <math>AB</math> is <math>6</math> cm and that of <math>BC</math> is <math>2</math> 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. |
− | + | [[File:pdfresizer.com-pdf-convert-aimeq4.png]] | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
[[1983 AIME Problems/Problem 4|Solution]] | [[1983 AIME Problems/Problem 4|Solution]] | ||
Line 42: | Line 29: | ||
== Problem 6 == | == Problem 6 == | ||
− | Let <math>a_n | + | Let <math>a_n=6^{n}+8^{n}</math>. Determine the remainder on 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 off to slay a troublesome dragon. Let <math>P</math> be the probability 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 | + | 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 <math>P</math> be the probability 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 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>? | + | What is the largest <math>2</math>-digit prime factor of the integer <math>n = {200\choose 100}</math>? |
[[1983 AIME Problems/Problem 8|Solution]] | [[1983 AIME Problems/Problem 8|Solution]] | ||
Line 62: | Line 49: | ||
== Problem 10 == | == Problem 10 == | ||
− | The numbers <math>1447</math>, <math>1005</math> | + | The numbers <math>1447</math>, <math>1005</math> and <math>1231</math> have something in common: each is a <math>4</math>-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? | + | 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 other edges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid? |
<asy> | <asy> | ||
+ | import three; | ||
size(170); | size(170); | ||
− | |||
pathpen = black+linewidth(0.65); | pathpen = black+linewidth(0.65); | ||
pointpen = black; | pointpen = black; | ||
Line 77: | Line 64: | ||
real s = 6 * 2^.5; | 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); | 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); | ||
− | + | draw(A--B--C--D--A--E--D); | |
− | + | draw(B--F--C); | |
− | + | draw(E--F); | |
− | + | label("A",A, S); | |
− | + | label("B",B, S); | |
− | + | label("C",C, S); | |
− | + | label("D",D, S); | |
− | + | label("E",E,N); | |
+ | label("F",F,N); | ||
</asy> | </asy> | ||
Line 90: | Line 78: | ||
== Problem 12 == | == Problem 12 == | ||
− | + | Diameter <math>AB</math> of a circle has length a <math>2</math>-digit integer (base ten). Reversing the digits gives the length of the 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>. | |
− | + | [[File:pdfresizer.com-pdf-convert-aimeq12.png]] | |
− | |||
− | |||
− | |||
− | |||
[[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 | + | For <math>\{1, 2, 3, \ldots, n\}</math> and each of its nonempty subsets a unique '''alternating sum''' is defined as follows. Arrange the numbers 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=5</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> | + | In the adjoining figure, two circles with radii <math>8</math> and <math>6</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 such a way that the chords <math>QP</math> and <math>PR</math> have equal length. Find the square of the length of <math>QP</math>. |
− | [[Image:1983_AIME-14.png]] | + | <!-- [[Image:1983_AIME-14.png]] --> |
+ | <asy>size(160); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(11pt)); | ||
+ | dotfactor=3; | ||
+ | pair O1=(0,0), O2=(12,0); | ||
+ | path C1=Circle(O1,8), C2=Circle(O2,6); | ||
+ | pair P=intersectionpoints(C1,C2)[0]; | ||
+ | path C3=Circle(P,sqrt(130)); | ||
+ | pair Q=intersectionpoints(C3,C1)[0]; | ||
+ | pair R=intersectionpoints(C3,C2)[1]; | ||
+ | draw(C1); | ||
+ | draw(C2); | ||
+ | draw(O2--O1); | ||
+ | dot(O1); | ||
+ | dot(O2); | ||
+ | draw(Q--R); | ||
+ | label("$Q$",Q,NW); | ||
+ | label("$P$",P,1.5*dir(80)); | ||
+ | label("$R$",R,NE); | ||
+ | label("12",waypoint(O1--O2,0.4),S);</asy> | ||
[[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 | + | 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 central angle of minor arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? |
+ | <asy>size(140); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(11pt)); | ||
+ | dotfactor=1; | ||
+ | pair O1=(0,0); | ||
+ | pair A=(-0.91,-0.41); | ||
+ | pair B=(-0.99,0.13); | ||
+ | pair C=(0.688,0.728); | ||
+ | pair D=(-0.25,0.97); | ||
+ | path C1=Circle(O1,1); | ||
+ | draw(C1); | ||
+ | label("$A$",A,W); | ||
+ | label("$B$",B,W); | ||
+ | label("$C$",C,NE); | ||
+ | label("$D$",D,N); | ||
+ | draw(A--D); | ||
+ | draw(B--C); | ||
+ | pair F=intersectionpoint(A--D,B--C); | ||
+ | add(pathticks(A--F,1,0.5,0,3.5)); | ||
+ | add(pathticks(F--D,1,0.5,0,3.5)); | ||
+ | </asy> | ||
+ | <!-- [[Image:1983_AIME-15.png]] --> | ||
+ | |||
+ | [[1983 AIME Problems/Problem 15|Solution]] | ||
− | + | == See Also == | |
− | [[ | + | {{AIME box|year=1983|before=First AIME|after=[[1984 AIME Problems]]}} |
− | |||
− | |||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} | ||
[[Category:AIME Problems]] | [[Category:AIME Problems]] |
Latest revision as of 19:23, 15 February 2019
1983 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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 for in the interval .
Problem 3
What is the product of the real roots of the equation ?
Problem 4
A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is cm, the length of is cm and that of is 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 that can have?
Problem 6
Let . Determine the remainder on 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 denominator?
Problem 8
What is the largest -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 -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 other edges have length . Given that , what is the volume of the solid?
Problem 12
Diameter of a circle has length a -digit integer (base ten). Reversing the digits gives the length of the 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 nonempty subsets a unique alternating sum is defined as follows. Arrange the numbers 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 such a way that the chords and have equal length. 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 central angle of minor arc is a rational number. If this number is expressed as a fraction in lowest terms, what is the product ?
See Also
1983 AIME (Problems • Answer Key • Resources) | ||
Preceded by First AIME |
Followed by 1984 AIME Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.