Difference between revisions of "1983 AIME Problems"

Latest revision as of 02:13, 27 May 2024

1983 AIME (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$ , inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Let $x$ , $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_xw=24$ , $\log_y w = 40$ and $\log_{xyz}w=12$ . Find $\log_zw$ .

Solution

Problem 2

Let $f(x)=|x-p|+|x-15|+|x-p-15|$ , where $0 < p < 15$ . Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \leq x\leq15$ .

Solution

Problem 3

What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$ ?

Solution

Problem 4

A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle.

$[asy] size(150); defaultpen(linewidth(0.6)+fontsize(11)); real r=10; pair O=(0,0), A=r*dir(45),B=(A.x,A.y-r); path P=circle(O,r); pair C=intersectionpoint(B--(B.x+r,B.y),P); // Drawing arc instead of full circle //draw(P); draw(arc(O, r, degrees(A), degrees(C))); draw(C--B--A--B); dot(A); dot(B); dot(C); label("$A$",A,NE); label("$B$",B,S); label("$C$",C,SE); [/asy]$

Solution

Problem 5

Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cubes is $10$ . What is the largest real value that $x + y$ can have?

Solution

Problem 6

Let $a_n=6^{n}+8^{n}$ . Determine the remainder on dividing $a_{83}$ by $49$ .

Solution

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 $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Solution

Problem 8

What is the largest $2$ -digit prime factor of the integer $n = {200\choose 100}$ ?

Solution

Problem 9

Find the minimum value of $\frac{9x^2\sin^2 x + 4}{x\sin x}$ for $0 < x < \pi$ .

Solution

Problem 10

The numbers $1447$ , $1005$ and $1231$ have something in common: each is a $4$ -digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there?

Solution

Problem 11

The solid shown has a square base of side length $s$ . The upper edge is parallel to the base and has length $2s$ . All other edges have length $s$ . Given that $s=6\sqrt{2}$ , what is the volume of the solid?

$[asy] import three; size(170); pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(30,-20,10); real s = 6 * 2^.5; triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6); draw(A--B--C--D--A--E--D); draw(B--F--C); draw(E--F); label("A",A, S); label("B",B, S); label("C",C, S); label("D",D, S); label("E",E,N); label("F",F,N); [/asy]$

Solution

Problem 12

Diameter $AB$ of a circle has length a $2$ -digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$ . The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$ .

Solution

Problem 13

For $\{1, 2, 3, \ldots, n\}$ 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 $\{1, 2, 3, 6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply $5$ . Find the sum of all such alternating sums for $n=7$ .

Solution

Problem 14

In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$ , one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$ .

$[asy]size(160); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); draw(O2--O1); dot(O1); dot(O2); draw(Q--R); label("$Q$",Q,NW); label("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);[/asy]$

Solution

Problem 15

The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$ . Suppose that the radius of the circle is $5$ , that $BC=6$ , and that $AD$ is bisected by $BC$ . Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$ . It follows that the sine of the central angle of minor arc $AB$ is a rational number. If this number is expressed as a fraction $\frac{m}{n}$ in lowest terms, what is the product $mn$ ? $[asy]size(140); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=1; pair O1=(0,0); pair A=(-0.91,-0.41); pair B=(-0.99,0.13); pair C=(0.688,0.728); pair D=(-0.25,0.97); path C1=Circle(O1,1); draw(C1); label("$A$",A,W); label("$B$",B,W); label("$C$",C,NE); label("$D$",D,N); draw(A--D); draw(B--C); pair F=intersectionpoint(A--D,B--C); add(pathticks(A--F,1,0.5,0,3.5)); add(pathticks(F--D,1,0.5,0,3.5)); [/asy]$

Solution

@@ Line 2: / Line 2: @@
 == 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>\log_y w = 40</math>, and <math>\log_{xyz}w=12</math>. Find <math>\log_zw</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]]
 == 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 < x \leq 15</math>.
+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]]
@@ Line 17: / Line 17: @@
 == Problem 4 ==
-A machine shop cutting tool is in 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 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 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.
 <asy>
-size(150); defaultpen(linewidth(0.65)+fontsize(11));
+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;
+pair O=(0,0), A=r*dir(45),B=(A.x,A.y-r);
 path P=circle(O,r);
-C=intersectionpoint(B--(B.x+r,B.y),P);
+pair C=intersectionpoint(B--(B.x+r,B.y),P);
-draw(P);
+// Drawing arc instead of full circle
-draw(C--B--O--A--B);
+//draw(P);
-dot(O); dot(A); dot(B); dot(C);
+draw(arc(O, r, degrees(A), degrees(C)));
-label("$O$",O,SW);
+draw(C--B--A--B);
+dot(A);
+dot(B);
+dot(C);
 label("$A$",A,NE);
 label("$B$",B,S);
@@ Line 42: / Line 46: @@
 == 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>.
+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]]
 == 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 the denominator?
+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]]
 == 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]]
@@ Line 62: / Line 66: @@
 == 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?
+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]]
 == 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>
-size(170);import three; pathpen = black+linewidth(0.65); pointpen = black;
+import three;
-currentprojection = perspective(30,-20,10);real s = 6 * 2^.5;
+size(170);
+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);
+draw(A--B--C--D--A--E--D);
-MP("A",A);MP("B",B);MP("C",C);MP("D",D);MP("E",E,N);MP("F",F,N);
+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>
@@ Line 80: / Line 95: @@
 == 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>.
+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>.
-<asy>pointpen=black; pathpen=black+linewidth(0.65);
+[[File:pdfresizer.com-pdf-convert-aimeq12.png]]
-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]]
 == 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>.
+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]]
 == 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 such 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>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]]
 == 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>?
+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]]
-[[Image:1983_AIME-15.png]]
+== See Also ==
-[[1983 AIME Problems/Problem 15|Solution]]
+{{AIME box|year=1983|before=First AIME|after=[[1984 AIME Problems]]}}
-== See also ==
-* [[1983 AIME]]
 * [[American Invitational Mathematics Examination]]
 * [[AIME Problems and Solutions]]
 * [[Mathematics competition resources]]
+{{MAA Notice}}
 [[Category:AIME Problems]]

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

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Difference between revisions of "1983 AIME Problems"

Latest revision as of 02:13, 27 May 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

See Also