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Difference between revisions of "1984 AIME Problems"
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{{AIME Problems|year=1984}}
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== Problem 1 ==
 
== Problem 1 ==
−
Find the value of <math>\displaystyle a_2+a_4+a_6+a_8+\ldots+a_{98}</math> if <math>\displaystyle a_1</math>, <math>\displaystyle a_2</math>, <math>\displaystyle a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>\displaystyle a_1+a_2+a_3+\ldots+a_{98}=137</math>.
+
Find the value of <math>a_2+a_4+a_6+a_8+\ldots+a_{98}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>a_1+a_2+a_3+\ldots+a_{98}=137</math>.
  
 
[[1984 AIME Problems/Problem 1|Solution]]
 
[[1984 AIME Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
−
The integer <math>\displaystyle n</math> is the smallest positive multiple of <math>\displaystyle 15</math> such that every digit of <math>\displaystyle n</math> is either <math>\displaystyle 8</math> or <math>\displaystyle 0</math>. Compute <math>\frac{n}{15}</math>.
+
The integer <math>n</math> is the smallest positive multiple of <math>15</math> such that every digit of <math>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>.
  
 
[[1984 AIME Problems/Problem 2|Solution]]
 
[[1984 AIME Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
−
A point <math>\displaystyle P</math> is chosen in the interior of <math>\displaystyle \triangle ABC</math> such that when lines are drawn through <math>\displaystyle P</math> parallel to the sides of <math>\displaystyle \triangle ABC</math>, the resulting smaller triangles <math>\displaystyle t_{1}</math>, <math>\displaystyle t_{2}</math>, and <math>\displaystyle t_{3}</math> in the figure, have areas <math>\displaystyle 4</math>, <math>\displaystyle 9</math>, and <math>\displaystyle 49</math>, respectively. Find the area of <math>\displaystyle \triangle ABC</math>.
+
A point <math>P</math> is chosen in the interior of <math>\triangle ABC</math> such that when lines are drawn through <math>P</math> parallel to the sides of <math>\triangle ABC</math>, the resulting smaller triangles <math>t_{1}</math>, <math>t_{2}</math>, and <math>t_{3}</math> in the figure, have areas <math>4</math>, <math>9</math>, and <math>49</math>, respectively. Find the area of <math>\triangle ABC</math>.
 +
 
 +
<asy>
 +
size(200);
 +
pathpen=black+linewidth(0.65);pointpen=black;
 +
pair A=(0,0),B=(12,0),C=(4,5);
 +
D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12);
 +
MP("A",C,N);MP("B",A,SW);MP("C",B,SE); /* sorry mixed up points according to resources diagram. */
 +
MP("t_3",(A+B+(B-A)*3/4+(A-B)*5/6)/2+(-1,0.8),N);
 +
MP("t_2",(B+C+(B-C)*5/12+(C-B)*5/6)/2+(-0.3,0.1),WSW);
 +
MP("t_1",(A+C+(C-A)*3/4+(A-C)*5/12)/2+(0,0.15),ESE);
 +
</asy>
  
 
[[1984 AIME Problems/Problem 3|Solution]]
 
[[1984 AIME Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
−
Let <math>\displaystyle S</math> be a list of positive integers - not necessarily distinct - in which the number <math>\displaystyle 68</math> appears. The arithmetic mean of the numbers in <math>\displaystyle S</math> is <math>\displaystyle 56</math>. However, if <math>\displaystyle 68</math> is removed, the arithmetic mean of the numbers is <math>\displaystyle 55</math>. What's the largest number that can appear in <math>\displaystyle S</math>?
+
Let <math>S</math> be a list of positive integers--not necessarily distinct--in which the number <math>68</math> appears. The average (arithmetic mean) of the numbers in <math>S</math> is <math>56</math>. However, if <math>68</math> is removed, the average of the remaining numbers drops to <math>55</math>. What is the largest number that can appear in <math>S</math>?
  
 
[[1984 AIME Problems/Problem 4|Solution]]
 
[[1984 AIME Problems/Problem 4|Solution]]
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== Problem 6 ==
 
== Problem 6 ==
−
Three circles, each of radius <math>\displaystyle 3</math>, are drawn with centers at <math>\displaystyle (14, 92)</math>, <math>\displaystyle (17, 76)</math>, and <math>\displaystyle (19, 84)</math>. A line passing through <math>\displaystyle (17,76)</math> is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
+
Three circles, each of radius <math>3</math>, are drawn with centers at <math>(14, 92)</math>, <math>(17, 76)</math>, and <math>(19, 84)</math>. A line passing through <math>(17,76)</math> is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
  
 
[[1984 AIME Problems/Problem 6|Solution]]
 
[[1984 AIME Problems/Problem 6|Solution]]
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</math>
 
</math>
  
−
Find <math>\displaystyle f(84)</math>.
+
Find <math>f(84)</math>.
  
 
[[1984 AIME Problems/Problem 7|Solution]]
 
[[1984 AIME Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
−
The equation <math>\displaystyle z^6+z^3+1</math> has complex roots with argument <math>\theta</math> between <math>90^\circ</math> and <math>180^\circ</math> in thet complex plane. Determine the degree measure of <math>\theta</math>.
+
The equation <math>z^6+z^3+1=0</math> has complex roots with argument <math>\theta</math> between <math>90^\circ</math> and <math>180^\circ</math> in the complex plane. Determine the degree measure of <math>\theta</math>.
  
 
[[1984 AIME Problems/Problem 8|Solution]]
 
[[1984 AIME Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
−
In tetrahedron <math>\displaystyle ABCD</math>, edge <math>\displaystyle ABC</math> has length 3 cm. The area of face <math>\displaystyle AMC</math> is <math>\displaystyle 15\mbox{cm}^2</math> and the area of face <math>\displaystyle ABD</math> is <math>\displaystyle 12 \mbox { cm}^2</math>. These two faces meet each other at a <math>30^\circ</math> angle. Find the volume of the tetrahedron in <math>\displaystyle \mbox{cm}^3</math>.
+
In tetrahedron <math>ABCD</math>, edge <math>AB</math> has length 3 cm. The area of face <math>ABC</math> is <math>15\mbox{cm}^2</math> and the area of face <math>ABD</math> is <math>12 \mbox { cm}^2</math>. These two faces meet each other at a <math>30^\circ</math> angle. Find the volume of the tetrahedron in <math>\mbox{cm}^3</math>.
  
 
[[1984 AIME Problems/Problem 9|Solution]]
 
[[1984 AIME Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
−
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over <math>\displaystyle 80</math>. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over <math>\displaystyle 80</math>, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of <math>\displaystyle 30</math> multiple choice problems and that one's score, <math>\displaystyle s</math>, is computed by the formula <math>\displaystyle s=30+4c-w</math>, where <math>\displaystyle c</math> is the number of correct answers and <math>\displaystyle w</math> is the number of wrong answers. Students are not penalized for problems left unanswered.)
+
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over <math>80</math>. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over <math>80</math>, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of <math>30</math> multiple choice problems and that one's score, <math>s</math>, is computed by the formula <math>s=30+4c-w</math>, where <math>c</math> is the number of correct answers and <math>w</math> is the number of wrong answers. Students are not penalized for problems left unanswered.)
  
 
[[1984 AIME Problems/Problem 10|Solution]]
 
[[1984 AIME Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
−
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let <math>\frac m n</math> in lowest terms be the probability that no two birch trees are next to one another. Find <math>\displaystyle m+n</math>.
+
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let <math>\frac m n</math> in lowest terms be the probability that no two birch trees are next to one another. Find <math>m+n</math>.
  
 
[[1984 AIME Problems/Problem 11|Solution]]
 
[[1984 AIME Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
−
A function <math>\displaystyle f</math> is defined for all real numbers and satisfies <math>\displaystyle f(2+x)=f(2-x)</math> and <math>\displaystyle f(7+x)=f(7-x)</math> for all <math>\displaystyle x</math>. If <math>\displaystyle x=0</math> is a root for <math>\displaystyle f(x)=0</math>, what is the least number of roots <math>\displaystyle f(x)=0</math> must have in the interval <math>\displaystyle -1000\leq x \leq 1000</math>?
+
A function <math>f</math> is defined for all real numbers and satisfies <math>f(2+x)=f(2-x)</math> and <math>f(7+x)=f(7-x)</math> for all <math>x</math>. If <math>x=0</math> is a root for <math>f(x)=0</math>, what is the least number of roots <math>f(x)=0</math> must have in the interval <math>-1000\leq x \leq 1000</math>?
  
 
[[1984 AIME Problems/Problem 12|Solution]]
 
[[1984 AIME Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
−
Find the value of <math>\displaystyle 10\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).</math>
+
Find the value of <math>10\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).</math>
  
 
[[1984 AIME Problems/Problem 13|Solution]]
 
[[1984 AIME Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
  
 
[[1984 AIME Problems/Problem 14|Solution]]
 
[[1984 AIME Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
Determine <math>w^2+x^2+y^2+z^2</math> if
 +
 +
<center><math> \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 </math></center>
 +
<center><math> \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 </math></center>
 +
<center><math> \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 </math></center>
 +
<center><math> \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 </math></center>
  
 
[[1984 AIME Problems/Problem 15|Solution]]
 
[[1984 AIME Problems/Problem 15|Solution]]
  
 
== See also ==
 
== See also ==
 +
 +
{{AIME box|year=1984|before=[[1983 AIME Problems]]|after=[[1985 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]]

Latest revision as of 00:15, 19 June 2022

1984 AIME (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. 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.
  2. 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

Find the value of $a_2+a_4+a_6+a_8+\ldots+a_{98}$ if $a_1$, $a_2$, $a_3\ldots$ is an arithmetic progression with common difference 1, and $a_1+a_2+a_3+\ldots+a_{98}=137$.

Solution

Problem 2

The integer $n$ is the smallest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.

Solution

Problem 3

A point $P$ is chosen in the interior of $\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles $t_{1}$, $t_{2}$, and $t_{3}$ in the figure, have areas $4$, $9$, and $49$, respectively. Find the area of $\triangle ABC$.

[asy] size(200); pathpen=black+linewidth(0.65);pointpen=black; pair A=(0,0),B=(12,0),C=(4,5); D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12); MP("A",C,N);MP("B",A,SW);MP("C",B,SE); /* sorry mixed up points according to resources diagram. */ MP("t_3",(A+B+(B-A)*3/4+(A-B)*5/6)/2+(-1,0.8),N); MP("t_2",(B+C+(B-C)*5/12+(C-B)*5/6)/2+(-0.3,0.1),WSW); MP("t_1",(A+C+(C-A)*3/4+(A-C)*5/12)/2+(0,0.15),ESE); [/asy]

Solution

Problem 4

Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$?

Solution

Problem 5

Determine the value of $ab$ if $\log_8a+\log_4b^2=5$ and $\log_8b+\log_4a^2=7$.

Solution

Problem 6

Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?

Solution

Problem 7

The function f is defined on the set of integers and satisfies $f(n)= \begin{cases} n-3 & \mbox{if }n\ge 1000 \\ f(f(n+5)) & \mbox{if }n<1000 \end{cases}$

Find $f(84)$.

Solution

Problem 8

The equation $z^6+z^3+1=0$ has complex roots with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane. Determine the degree measure of $\theta$.

Solution

Problem 9

In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is $15\mbox{cm}^2$ and the area of face $ABD$ is $12 \mbox { cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\mbox{cm}^3$.

Solution

Problem 10

Mary told John her score on the American High School Mathematics Examination (AHSME), which was over $80$. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over $80$, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of $30$ multiple choice problems and that one's score, $s$, is computed by the formula $s=30+4c-w$, where $c$ is the number of correct answers and $w$ is the number of wrong answers. Students are not penalized for problems left unanswered.)

Solution

Problem 11

A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.

Solution

Problem 12

A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$. If $x=0$ is a root for $f(x)=0$, what is the least number of roots $f(x)=0$ must have in the interval $-1000\leq x \leq 1000$?

Solution

Problem 13

Find the value of $10\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).$

Solution

Problem 14

What is the largest even integer that cannot be written as the sum of two odd composite numbers?

Solution

Problem 15

Determine $w^2+x^2+y^2+z^2$ if

$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$
$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$
$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$
$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$

Solution

See also

1984 AIME (Problems • Answer Key • Resources)
Preceded by
1983 AIME Problems 		Followed by
1985 AIME Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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