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Difference between revisions of "1989 AIME Problems"

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== Problem 4 ==
 
== Problem 4 ==
−
If <math>a<b<c<d<e^{}_{}</math> are consecutive positive integers such that <math>b+c+d^{}_{}</math> is a perfect square and <math>a+b+c+d+e^{}_{}</math> is a perfect cube, what is the smallest possible value of <math>c^{}_{}</math>?
+
If <math>a<b<c<d<e^{}_{}</math> are consecutive positive integers such that <math>b+c+d</math> is a perfect square and <math>a+b+c+d+e^{}_{}</math> is a perfect cube, what is the smallest possible value of <math>c^{}_{}</math>?
  
 
[[1989 AIME Problems/Problem 4|Solution]]
 
[[1989 AIME Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
−
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to <math>0^{}_{}</math> and is the same as that of getting heads exactly twice. Let <math>\frac ij^{}_{}</math>, in lowest terms, be the probability that the coin comes up heads in exactly <math>3_{}^{}</math> out of <math>5^{}_{}</math> flips. Find <math>i+j^{}_{}</math>.
+
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to <math>0</math> and is the same as that of getting heads exactly twice. Let <math>\frac ij^{}_{}</math>, in lowest terms, be the probability that the coin comes up heads in exactly <math>3</math> out of <math>5</math> flips. Find <math>i+j^{}_{}</math>.
  
 
[[1989 AIME Problems/Problem 5|Solution]]
 
[[1989 AIME Problems/Problem 5|Solution]]
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== Problem 6 ==
 
== Problem 6 ==
 
Two skaters, Allie and Billie, are at points <math>A^{}_{}</math> and <math>B^{}_{}</math>, respectively, on a flat, frozen lake. The distance between <math>A^{}_{}</math> and <math>B^{}_{}</math> is <math>100^{}_{}</math> meters. Allie leaves <math>A^{}_{}</math> and skates at a speed of <math>8^{}_{}</math> meters per second on a straight line that makes a <math>60^\circ</math> angle with <math>AB^{}_{}</math>. At the same time Allie leaves <math>A^{}_{}</math>, Billie leaves <math>B^{}_{}</math> at a speed of <math>7^{}_{}</math> meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
 
Two skaters, Allie and Billie, are at points <math>A^{}_{}</math> and <math>B^{}_{}</math>, respectively, on a flat, frozen lake. The distance between <math>A^{}_{}</math> and <math>B^{}_{}</math> is <math>100^{}_{}</math> meters. Allie leaves <math>A^{}_{}</math> and skates at a speed of <math>8^{}_{}</math> meters per second on a straight line that makes a <math>60^\circ</math> angle with <math>AB^{}_{}</math>. At the same time Allie leaves <math>A^{}_{}</math>, Billie leaves <math>B^{}_{}</math> at a speed of <math>7^{}_{}</math> meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
−
 
+
<center><asy>
−
[[Image:AIME_1989_Problem_6.png]]
+
pointpen=black; pathpen=black+linewidth(0.7);
 +
pair A=(0,0),B=(10,0),C=6*expi(pi/3);
 +
D(B--A); D(A--C,EndArrow); MP("A",A,SW);MP("B",B,SE);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B)/2);
 +
</asy></center><!-- Minsoen's image: [[Image:AIME_1989_Problem_6.png]] -->
  
 
[[1989 AIME Problems/Problem 6|Solution]]
 
[[1989 AIME Problems/Problem 6|Solution]]
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== Problem 8 ==
 
== Problem 8 ==
 
Assume that <math>x_1,x_2,\ldots,x_7</math> are real numbers such that
 
Assume that <math>x_1,x_2,\ldots,x_7</math> are real numbers such that
−
<center><math>x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1^{}_{}</math></center>
+
<cmath>\begin{align*}
−
<center><math>4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12^{}_{}</math></center>
+
x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\
−
<center><math>9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123^{}_{}</math></center>
+
4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\
−
 
+
9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123.
−
Find the value of <math>16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7^{}</math>.
+
\end{align*}</cmath>
 +
Find the value of <math>16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7</math>.
  
 
[[1989 AIME Problems/Problem 8|Solution]]
 
[[1989 AIME Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
−
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer <math>n</math> such that <math>133^5+110^5+84^5+27^5=n^{5}_{}</math>. Find the value of <math>n^{}_{}</math>.
+
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that <cmath>133^5+110^5+84^5+27^5=n^{5}.</cmath> Find the value of <math>n</math>.
  
 
[[1989 AIME Problems/Problem 9|Solution]]
 
[[1989 AIME Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
−
Let <math>a_{}^{}</math>, <math>b_{}^{}</math>, <math>c_{}^{}</math> be the three sides of a triangle, and let <math>\alpha_{}^{}</math>, <math>\beta_{}^{}</math>, <math>\gamma_{}^{}</math>, be the angles opposite them. If <math>a^2+b^2=1989^{}_{}c^2</math>, find
+
Let <math>a</math>, <math>b</math>, <math>c</math> be the three sides of a triangle, and let <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math>, be the angles opposite them. If <math>a^2+b^2=1989c^2</math>, find
 
<center><math>\frac{\cot \gamma}{\cot \alpha+\cot \beta}</math></center>
 
<center><math>\frac{\cot \gamma}{\cot \alpha+\cot \beta}</math></center>
  
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Let <math>ABCD^{}_{}</math> be a tetrahedron with <math>AB=41^{}_{}</math>, <math>AC=7^{}_{}</math>, <math>AD=18^{}_{}</math>, <math>BC=36^{}_{}</math>, <math>BD=27^{}_{}</math>, and <math>CD=13^{}_{}</math>, as shown in the figure. Let <math>d^{}_{}</math> be the distance between the midpoints of edges <math>AB^{}_{}</math> and <math>CD^{}_{}</math>. Find <math>d^{2}_{}</math>.
 
Let <math>ABCD^{}_{}</math> be a tetrahedron with <math>AB=41^{}_{}</math>, <math>AC=7^{}_{}</math>, <math>AD=18^{}_{}</math>, <math>BC=36^{}_{}</math>, <math>BD=27^{}_{}</math>, and <math>CD=13^{}_{}</math>, as shown in the figure. Let <math>d^{}_{}</math> be the distance between the midpoints of edges <math>AB^{}_{}</math> and <math>CD^{}_{}</math>. Find <math>d^{2}_{}</math>.
  
−
[[Image:AIME_1989_Problem_12.png]]
+
[[Image:AIME_1989_Problem_12.png|center]]
  
 
[[1989 AIME Problems/Problem 12|Solution]]
 
[[1989 AIME Problems/Problem 12|Solution]]
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== Problem 14 ==
 
== Problem 14 ==
−
Given a positive integer <math>n^{}_{}</math>, it can be shown that every complex number of the form <math>r+si^{}_{}</math>, where <math>r^{}_{}</math> and <math>s^{}_{}</math> are integers, can be uniquely expressed in the base <math>-n+i^{}_{}</math> using the integers <math>1,2^{}_{},\ldots,n^2</math> as digits. That is, the equation
+
Given a positive integer <math>n^{}_{}</math>, it can be shown that every complex number of the form <math>r+si^{}_{}</math>, where <math>r^{}_{}</math> and <math>s^{}_{}</math> are integers, can be uniquely expressed in the base <math>-n+i^{}_{}</math> using the integers <math>0,1,2^{}_{},\ldots,n^2</math> as digits. That is, the equation
 
<center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center>
 
<center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center>
 
is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m</math> chosen from the set <math>\{0^{}_{},1,2,\ldots,n^2\}</math>, with <math>a_m\ne 0^{}){}</math>. We write
 
is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m</math> chosen from the set <math>\{0^{}_{},1,2,\ldots,n^2\}</math>, with <math>a_m\ne 0^{}){}</math>. We write
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== Problem 15 ==
 
== Problem 15 ==
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Point <math>P^{}_{}</math> is inside <math>\triangle ABC^{}_{}</math>. Line segments <math>APD^{}_{}</math>, <math>BPE^{}_{}</math>, and <math>CPF^{}_{}</math> are drawn with <math>D^{}_{}</math> on <math>BC^{}_{}</math>, <math>E^{}_{}</math> on <math>AC^{}_{}</math>, and <math>F{}{}^{}_{}</math> on <math>AB^{}_{}</math> (see the figure at right). Given that <math>AP=6^{}_{}</math>, <math>BP=9^{}_{}</math>, <math>PD=6^{}_{}</math>, <math>PE=3^{}_{}</math>, and <math>CF=20^{}_{}</math>, find the area of <math>\triangle ABC^{}_{}</math>.
+
Point <math>P</math> is inside <math>\triangle ABC</math>. Line segments <math>APD</math>, <math>BPE</math>, and <math>CPF</math> are drawn with <math>D</math> on <math>BC</math>, <math>E</math> on <math>AC</math>, and <math>F</math> on <math>AB</math> (see the figure below). Given that <math>AP=6</math>, <math>BP=9</math>, <math>PD=6</math>, <math>PE=3</math>, and <math>CF=20</math>, find the area of <math>\triangle ABC</math>.
−
 
+
[[Image:AIME_1989_Problem_15.png|center]]
−
[[Image:AIME_1989_Problem_15.png]]
 
  
 
[[1989 AIME Problems/Problem 15|Solution]]
 
[[1989 AIME Problems/Problem 15|Solution]]
  
 
== See also ==
 
== See also ==
 +
 +
{{AIME box|year=1989|before=[[1988 AIME Problems]]|after=[[1990 AIME Problems]]}}
 +
 
* [[American Invitational Mathematics Examination]]
 
* [[American Invitational Mathematics Examination]]
 
* [[AIME Problems and Solutions]]
 
* [[AIME Problems and Solutions]]

Latest revision as of 19:47, 14 December 2023

1989 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

Compute $\sqrt{(31)(30)(29)(28)+1}$.

Solution

Problem 2

Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?

Solution

Problem 3

Suppose $n_{}^{}$ is a positive integer and $d_{}^{}$ is a single digit in base 10. Find $n_{}^{}$ if

$\frac{n}{810}=0.d25d25d25\ldots$

Solution

Problem 4

If $a<b<c<d<e^{}_{}$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e^{}_{}$ is a perfect cube, what is the smallest possible value of $c^{}_{}$?

Solution

Problem 5

When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij^{}_{}$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j^{}_{}$.

Solution

Problem 6

Two skaters, Allie and Billie, are at points $A^{}_{}$ and $B^{}_{}$, respectively, on a flat, frozen lake. The distance between $A^{}_{}$ and $B^{}_{}$ is $100^{}_{}$ meters. Allie leaves $A^{}_{}$ and skates at a speed of $8^{}_{}$ meters per second on a straight line that makes a $60^\circ$ angle with $AB^{}_{}$. At the same time Allie leaves $A^{}_{}$, Billie leaves $B^{}_{}$ at a speed of $7^{}_{}$ meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?

[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=6*expi(pi/3); D(B--A); D(A--C,EndArrow); MP("A",A,SW);MP("B",B,SE);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B)/2); [/asy]

Solution

Problem 7

If the integer $k^{}_{}$ is added to each of the numbers $36^{}_{}$, $300^{}_{}$, and $596^{}_{}$, one obtains the squares of three consecutive terms of an arithmetic series. Find $k^{}_{}$.

Solution

Problem 8

Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that \begin{align*} x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\ 4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\ 9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123. \end{align*} Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$.

Solution

Problem 9

One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that \[133^5+110^5+84^5+27^5=n^{5}.\] Find the value of $n$.

Solution

Problem 10

Let $a$, $b$, $c$ be the three sides of a triangle, and let $\alpha$, $\beta$, $\gamma$, be the angles opposite them. If $a^2+b^2=1989c^2$, find

$\frac{\cot \gamma}{\cot \alpha+\cot \beta}$

Solution

Problem 11

A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D^{}_{}$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D^{}_{}\rfloor$? (For real $x^{}_{}$, $\lfloor x^{}_{}\rfloor$ is the greatest integer less than or equal to $x^{}_{}$.)

Solution

Problem 12

Let $ABCD^{}_{}$ be a tetrahedron with $AB=41^{}_{}$, $AC=7^{}_{}$, $AD=18^{}_{}$, $BC=36^{}_{}$, $BD=27^{}_{}$, and $CD=13^{}_{}$, as shown in the figure. Let $d^{}_{}$ be the distance between the midpoints of edges $AB^{}_{}$ and $CD^{}_{}$. Find $d^{2}_{}$.

AIME 1989 Problem 12.png

Solution

Problem 13

Let $S^{}_{}$ be a subset of $\{1,2,3^{}_{},\ldots,1989\}$ such that no two members of $S^{}_{}$ differ by $4^{}_{}$ or $7^{}_{}$. What is the largest number of elements $S^{}_{}$ can have?

Solution

Problem 14

Given a positive integer $n^{}_{}$, it can be shown that every complex number of the form $r+si^{}_{}$, where $r^{}_{}$ and $s^{}_{}$ are integers, can be uniquely expressed in the base $-n+i^{}_{}$ using the integers $0,1,2^{}_{},\ldots,n^2$ as digits. That is, the equation

$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$

is true for a unique choice of non-negative integer $m^{}_{}$ and digits $a_0,a_1^{},\ldots,a_m$ chosen from the set $\{0^{}_{},1,2,\ldots,n^2\}$, with $a_m\ne 0^{}){}$. We write

$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$

to denote the base $-n+i^{}_{}$ expansion of $r+si^{}_{}$. There are only finitely many integers $k+0i^{}_{}$ that have four-digit expansions

$k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.$

Find the sum of all such $k^{}_{}$.

Solution

Problem 15

Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$.

AIME 1989 Problem 15.png

Solution

See also

1989 AIME (Problems • Answer Key • Resources)
Preceded by
1988 AIME Problems 		Followed by
1990 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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