Difference between revisions of "1989 AIME Problems"

Latest revision as of 19:47, 14 December 2023

1989 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

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 $d^{2}_{}$ .

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 $\triangle ABC$ .

Solution

@@ Line 1: / Line 1: @@
+{{AIME Problems|year=1989}}
 == Problem 1 ==
+Compute <math>\sqrt{(31)(30)(29)(28)+1}</math>.
 [[1989 AIME Problems/Problem 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?
 [[1989 AIME Problems/Problem 2|Solution]]
 == Problem 3 ==
+Suppose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if
+<center><math>\frac{n}{810}=0.d25d25d25\ldots</math></center>
 [[1989 AIME Problems/Problem 3|Solution]]
 == 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>?
 [[1989 AIME Problems/Problem 4|Solution]]
 == 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>.
 [[1989 AIME Problems/Problem 5|Solution]]
 == 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?
+<center><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></center><!-- Minsoen's image: [[Image:AIME_1989_Problem_6.png]] -->
 [[1989 AIME Problems/Problem 6|Solution]]
 == Problem 7 ==
+If the integer <math>k^{}_{}</math> is added to each of the numbers <math>36^{}_{}</math>, <math>300^{}_{}</math>, and <math>596^{}_{}</math>, one obtains the squares of three consecutive terms of an arithmetic series. Find <math>k^{}_{}</math>.
 [[1989 AIME Problems/Problem 7|Solution]]
 == Problem 8 ==
+Assume that <math>x_1,x_2,\ldots,x_7</math> are real numbers such that
+<cmath>\begin{align*}
+x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\
+x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\
+x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123.
+\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]]
 == 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 <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]]
 == 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=1989c^2</math>, find
+<center><math>\frac{\cot \gamma}{\cot \alpha+\cot \beta}</math></center>
 [[1989 AIME Problems/Problem 10|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 <math>D^{}_{}</math> be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of <math>\lfloor D^{}_{}\rfloor</math>? (For real <math>x^{}_{}</math>, <math>\lfloor x^{}_{}\rfloor</math> is the greatest integer less than or equal to <math>x^{}_{}</math>.)
 [[1989 AIME Problems/Problem 11|Solution]]
 == Problem 12 ==
+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|center]]
 [[1989 AIME Problems/Problem 12|Solution]]
 == Problem 13 ==
+Let <math>S^{}_{}</math> be a subset of <math>\{1,2,3^{}_{},\ldots,1989\}</math> such that no two members of <math>S^{}_{}</math> differ by <math>4^{}_{}</math> or <math>7^{}_{}</math>. What is the largest number of elements <math>S^{}_{}</math> can have?
 [[1989 AIME Problems/Problem 13|Solution]]
 == 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>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>
+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
+<center><math>r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}</math></center>
+to denote the base <math>-n+i^{}_{}</math> expansion of <math>r+si^{}_{}</math>. There are only finitely many integers <math>k+0i^{}_{}</math> that have four-digit expansions
+<center><math>k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.</math></center>
+Find the sum of all such <math>k^{}_{}</math>.
 [[1989 AIME Problems/Problem 14|Solution]]
 == Problem 15 ==
+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]]
 [[1989 AIME Problems/Problem 15|Solution]]
 == See also ==
+{{AIME box|year=1989|before=[[1988 AIME Problems]]|after=[[1990 AIME Problems]]}}
 * [[American Invitational Mathematics Examination]]
 * [[AIME Problems and Solutions]]
 * [[Mathematics competition resources]]
+[[Category:AIME Problems|1989]]
+{{MAA Notice}}

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
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Difference between revisions of "1989 AIME Problems"

Latest revision as of 19:47, 14 December 2023

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