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> | + | 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> | + | 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> | + | 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> | + | 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]] | ||
Line 25: | Line 38: | ||
== Problem 6 == | == Problem 6 == | ||
− | Three circles, each of radius <math> | + | 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]] | ||
Line 39: | Line 52: | ||
</math> | </math> | ||
− | Find <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> | + | 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> | + | 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> | + | 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> | + | 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> | + | 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> | + | 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]] | ||
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== 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 01:15, 19 June 2022
1984 AIME (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Find the value of if
,
,
is an arithmetic progression with common difference 1, and
.
Problem 2
The integer is the smallest positive multiple of
such that every digit of
is either
or
. Compute
.
Problem 3
A point is chosen in the interior of
such that when lines are drawn through
parallel to the sides of
, the resulting smaller triangles
,
, and
in the figure, have areas
,
, and
, respectively. Find the area of
.
Problem 4
Let be a list of positive integers--not necessarily distinct--in which the number
appears. The average (arithmetic mean) of the numbers in
is
. However, if
is removed, the average of the remaining numbers drops to
. What is the largest number that can appear in
?
Problem 5
Determine the value of if
and
.
Problem 6
Three circles, each of radius , are drawn with centers at
,
, and
. A line passing through
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?
Problem 7
The function f is defined on the set of integers and satisfies
Find .
Problem 8
The equation has complex roots with argument
between
and
in the complex plane. Determine the degree measure of
.
Problem 9
In tetrahedron , edge
has length 3 cm. The area of face
is
and the area of face
is
. These two faces meet each other at a
angle. Find the volume of the tetrahedron in
.
Problem 10
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over . 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
, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of
multiple choice problems and that one's score,
, is computed by the formula
, where
is the number of correct answers and
is the number of wrong answers. Students are not penalized for problems left unanswered.)
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 in lowest terms be the probability that no two birch trees are next to one another. Find
.
Problem 12
A function is defined for all real numbers and satisfies
and
for all
. If
is a root for
, what is the least number of roots
must have in the interval
?
Problem 13
Find the value of
Problem 14
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
Problem 15
Determine if
![$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$](http://latex.artofproblemsolving.com/3/f/a/3faf3bd2051d28e84510522f1060d255643f46fe.png)
![$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$](http://latex.artofproblemsolving.com/9/d/2/9d236639c64e0cd9eeb0bc55dcadef28a7fbb48a.png)
![$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$](http://latex.artofproblemsolving.com/d/9/0/d907da2778c251c183225c514cdab8b8f6a62e97.png)
![$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$](http://latex.artofproblemsolving.com/c/3/f/c3fed85bdaf16d0b43eb458a87be68b5d851b31b.png)
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 |
- 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.