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Difference between revisions of "1986 AIME Problems"

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{{AIME Problems|year=1986}}
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== Problem 1 ==
 
== Problem 1 ==
−
What is the sum of the solutions to the equation <math>\displaystyle \sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}</math>?
+
What is the sum of the solutions to the equation <math>\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}</math>?
  
 
[[1986 AIME Problems/Problem 1|Solution]]
 
[[1986 AIME Problems/Problem 1|Solution]]
Line 10: Line 12:
  
 
== Problem 3 ==
 
== Problem 3 ==
−
If <math>\displaystyle \tan x+\tan y=25</math> and <math>\displaystyle \cot x + \cot y=30</math>, what is <math>\displaystyle \tan(x+y)</math>?
+
If <math>\tan x+\tan y=25</math> and <math>\cot x + \cot y=30</math>, what is <math>\tan(x+y)</math>?
  
 
[[1986 AIME Problems/Problem 3|Solution]]
 
[[1986 AIME Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
−
Determine <math>\displaystyle 3x_4+2x_5</math> if <math>\displaystyle x_1</math>, <math>\displaystyle x_2</math>, <math>\displaystyle x_3</math>, <math>\displaystyle x_4</math>, and <math>\displaystyle x_5</math> satisfy the system of equations below.
+
Determine <math>3x_4+2x_5</math> if <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, <math>x_4</math>, and <math>x_5</math> satisfy the system of equations below.
−
<center><math>\displaystyle 2x_1+x_2+x_3+x_4+x_5=6</math></center>  
+
<center><math>2x_1+x_2+x_3+x_4+x_5=6</math></center>  
−
<center><math>\displaystyle x_1+2x_2+x_3+x_4+x_5=12</math></center>
+
<center><math>x_1+2x_2+x_3+x_4+x_5=12</math></center>
−
<center><math>\displaystyle x_1+x_2+2x_3+x_4+x_5=24</math></center>
+
<center><math>x_1+x_2+2x_3+x_4+x_5=24</math></center>
−
<center><math>\displaystyle x_1+x_2+x_3+2x_4+x_5=48</math></center>
+
<center><math>x_1+x_2+x_3+2x_4+x_5=48</math></center>
−
<center><math>\displaystyle x_1+x_2+x_3+x_4+2x_5=96</math></center>
+
<center><math>x_1+x_2+x_3+x_4+2x_5=96</math></center>
  
 
[[1986 AIME Problems/Problem 4|Solution]]
 
[[1986 AIME Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
−
What is that largest [[positive integer]] <math>\displaystyle n</math> for which <math>\displaystyle n^3+100</math> is [[divisible]] by <math>\displaystyle n+10</math>?
+
What is that largest [[positive integer]] <math>n</math> for which <math>n^3+100</math> is [[divisible]] by <math>n+10</math>?
  
 
[[1986 AIME Problems/Problem 5|Solution]]
 
[[1986 AIME Problems/Problem 5|Solution]]
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== Problem 7 ==
 
== Problem 7 ==
−
The increasing sequence <math>1,3,4,9,10,12,13\cdots</math> consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the <math>\displaystyle 100^{\mbox{th}}</math> term of this sequence.
+
The increasing sequence <math>1,3,4,9,10,12,13\cdots</math> consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the <math>100^{\mbox{th}}</math> term of this sequence.
  
 
[[1986 AIME Problems/Problem 7|Solution]]
 
[[1986 AIME Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
−
Let <math>\displaystyle S</math> be the sum of the base <math>\displaystyle 10</math> logarithms of all the proper divisors of <math>\displaystyle 1000000</math>. What is the integer nearest to <math>\displaystyle S</math>?
+
Let <math>S</math> be the sum of the base <math>10</math> logarithms of all the proper divisors of <math>1000000</math>. What is the integer nearest to <math>S</math>?
  
 
[[1986 AIME Problems/Problem 8|Solution]]
 
[[1986 AIME Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
−
In <math>\displaystyle \triangle ABC</math>, <math>\displaystyle AB= 425</math>, <math>\displaystyle BC=450</math>, and <math>\displaystyle AC=510</math>. An interior point <math>\displaystyle P</math> is then drawn, and segments are drawn through <math>\displaystyle P</math> parallel to the sides of the triangle. If these three segments are of an equal length <math>\displaystyle d</math>, find <math>\displaystyle d</math>.
+
In <math>\triangle ABC</math>, <math>AB= 425</math>, <math>BC=450</math>, and <math>AC=510</math>. An interior point <math>P</math> is then drawn, and segments are drawn through <math>P</math> parallel to the sides of the triangle. If these three segments are of an equal length <math>d</math>, find <math>d</math>.
  
 
[[1986 AIME Problems/Problem 9|Solution]]
 
[[1986 AIME Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
−
In a parlor game, the magician asks one of the participants to think of a three digit number (abc) where a, b, and c represent digits in base 10 in the order indicated. The magician then asks this person to form the numbers (acb), (bca), (bac), (cab), and (cba), to add these five numbers, and to reveal their sum, <math>\displaystyle N</math>. If told the value of <math>\displaystyle N</math>, the magician can identify the original number, (abc). Play the role of the magician and determine the (abc) if <math>\displaystyle N= 3194</math>.
+
In a parlor game, the magician asks one of the participants to think of a three digit number (abc) where a, b, and c represent digits in base 10 in the order indicated. The magician then asks this person to form the numbers (acb), (bca), (bac), (cab), and (cba), to add these five numbers, and to reveal their sum, <math>N</math>. If told the value of <math>N</math>, the magician can identify the original number, (abc). Play the role of the magician and determine the (abc) if <math>N= 3194</math>.
  
 
[[1986 AIME Problems/Problem 10|Solution]]
 
[[1986 AIME Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
−
The polynomial <math>1-x+x^2-x^3+\cdots+x^{16}-x^{17}</math> may be written in the form <math>a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}</math>, where <math>\displaystyle y=x+1</math> and thet <math>\displaystyle a_i</math>'s are constants. Find the value of <math>\displaystyle a_2</math>.
+
The polynomial <math>1-x+x^2-x^3+\cdots+x^{16}-x^{17}</math> may be written in the form <math>a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}</math>, where <math>y=x+1</math> and thet <math>a_i</math>'s are constants. Find the value of <math>a_2</math>.
  
 
[[1986 AIME Problems/Problem 11|Solution]]
 
[[1986 AIME Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
−
Let the sum of a set of numbers be the sum of its elements. Let <math>\displaystyle S</math> be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of <math>\displaystyle S</math> have the same sum. What is the largest sum a set <math>\displaystyle S</math> with these properties can have?
+
Let the sum of a set of numbers be the sum of its elements. Let <math>S</math> be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of <math>S</math> have the same sum. What is the largest sum a set <math>S</math> with these properties can have?
  
 
[[1986 AIME Problems/Problem 12|Solution]]
 
[[1986 AIME Problems/Problem 12|Solution]]
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== Problem 14 ==
 
== Problem 14 ==
−
The shortest distances between an interior diagonal of a rectangular parallelepiped, <math>\displaystyle P</math>, and the edges it does not meet are <math>\displaystyle 2\sqrt{5}</math>, <math>\displaystyle \frac{30}{\sqrt{13}}</math>, and <math>\displaystyle \frac{15}{\sqrt{10}}</math>. Determine the volume of <math>\displaystyle P</math>.
+
The shortest distances between an interior diagonal of a rectangular parallelepiped, <math>P</math>, and the edges it does not meet are <math>2\sqrt{5}</math>, <math>\frac{30}{\sqrt{13}}</math>, and <math>\frac{15}{\sqrt{10}}</math>. Determine the volume of <math>P</math>.
  
 
[[1986 AIME Problems/Problem 14|Solution]]
 
[[1986 AIME Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
−
Let triangle <math>\displaystyle ABC</math> be a right triangle in the xy-plane with a right angle at <math>\displaystyle C_{}</math>. Given that the length of the hypotenuse <math>\displaystyle AB</math> is <math>\displaystyle 60</math>, and that the medians through <math>\displaystyle A</math> and <math>\displaystyle B</math> lie along the lines <math>\displaystyle y=x+3</math> and <math>\displaystyle y=2x+4</math> respectively, find the area of triangle <math>\displaystyle ABC</math>.
+
Let triangle <math>ABC</math> be a right triangle in the xy-plane with a right angle at <math>C_{}</math>. Given that the length of the hypotenuse <math>AB</math> is <math>60</math>, and that the medians through <math>A</math> and <math>B</math> lie along the lines <math>y=x+3</math> and <math>y=2x+4</math> respectively, find the area of triangle <math>ABC</math>.
  
 
[[1986 AIME Problems/Problem 15|Solution]]
 
[[1986 AIME Problems/Problem 15|Solution]]

Revision as of 17:18, 2 January 2009

1986 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

What is the sum of the solutions to the equation $\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}$?

Solution

Problem 2

Evaluate the product $(\sqrt 5+\sqrt6+\sqrt7)(-\sqrt 5+\sqrt6+\sqrt7)(\sqrt 5-\sqrt6+\sqrt7)(\sqrt 5+\sqrt6-\sqrt7)$.

Solution

Problem 3

If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$?

Solution

Problem 4

Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.

$2x_1+x_2+x_3+x_4+x_5=6$
$x_1+2x_2+x_3+x_4+x_5=12$
$x_1+x_2+2x_3+x_4+x_5=24$
$x_1+x_2+x_3+2x_4+x_5=48$
$x_1+x_2+x_3+x_4+2x_5=96$

Solution

Problem 5

What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?

Solution

Problem 6

The pages of a book are numbered $1_{}^{}$ through $n_{}^{}$. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of $1986_{}^{}$. What was the number of the page that was added twice?

Solution

Problem 7

The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\mbox{th}}$ term of this sequence.

Solution

Problem 8

Let $S$ be the sum of the base $10$ logarithms of all the proper divisors of $1000000$. What is the integer nearest to $S$?

Solution

Problem 9

In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.

Solution

Problem 10

In a parlor game, the magician asks one of the participants to think of a three digit number (abc) where a, b, and c represent digits in base 10 in the order indicated. The magician then asks this person to form the numbers (acb), (bca), (bac), (cab), and (cba), to add these five numbers, and to reveal their sum, $N$. If told the value of $N$, the magician can identify the original number, (abc). Play the role of the magician and determine the (abc) if $N= 3194$.

Solution

Problem 11

The polynomial $1-x+x^2-x^3+\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and thet $a_i$'s are constants. Find the value of $a_2$.

Solution

Problem 12

Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have?

Solution

Problem 13

In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTHHTHTTTHHTTH of 15 coin tosses we observe that there are two HH, three HT, four TH, and five TT subsequences. How many different sequences of 15 coin tosses will contain exactly two HH, three HT, four TH, and five TT subsequences?

Solution

Problem 14

The shortest distances between an interior diagonal of a rectangular parallelepiped, $P$, and the edges it does not meet are $2\sqrt{5}$, $\frac{30}{\sqrt{13}}$, and $\frac{15}{\sqrt{10}}$. Determine the volume of $P$.

Solution

Problem 15

Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C_{}$. Given that the length of the hypotenuse $AB$ is $60$, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.

Solution

See also

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