Difference between revisions of "1990 AIME Problems"

(7 intermediate revisions by 5 users not shown)
Line 1: Line 1:
 +
{{AIME Problems|year=1990}}
 +
 
== Problem 1 ==
 
== Problem 1 ==
 
The [[increasing sequence]] <math>2,3,5,6,7,10,11,\ldots</math> consists of all [[positive integer]]s that are neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence.
 
The [[increasing sequence]] <math>2,3,5,6,7,10,11,\ldots</math> consists of all [[positive integer]]s that are neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence.
Line 10: Line 12:
  
 
== Problem 3 ==
 
== Problem 3 ==
Let <math>P_1^{}</math> be a regular <math>n~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that each interior angle of <math>P_1^{}</math> is <math>\frac{59}{58}</math> as large as each interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>?
+
Let <math>P_1^{}</math> be a regular <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that each interior angle of <math>P_1^{}</math> is <math>\frac{59}{58}</math> as large as each interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>?
  
 
[[1990 AIME Problems/Problem 3|Solution]]
 
[[1990 AIME Problems/Problem 3|Solution]]
Line 21: Line 23:
  
 
== Problem 5 ==
 
== Problem 5 ==
Let <math>n^{}_{}</math> be the smallest positive integer that is a multiple of <math>75_{}^{}</math> and has exactly <math>75_{}^{}</math> positive integral divisors, including <math>1_{}^{}</math> and itself. Find <math>n/75^{}_{}</math>.
+
Let <math>n^{}_{}</math> be the smallest positive integer that is a multiple of <math>75_{}^{}</math> and has exactly <math>75_{}^{}</math> positive integral divisors, including <math>1_{}^{}</math> and itself. Find <math>\frac{n}{75}</math>.
  
 
[[1990 AIME Problems/Problem 5|Solution]]
 
[[1990 AIME Problems/Problem 5|Solution]]
Line 52: Line 54:
  
 
== Problem 10 ==
 
== Problem 10 ==
The sets <math>\displaystyle A = \{z : z^{18} = 1\}</math> and <math>\displaystyle B = \{w : w^{48} = 1\}</math> are both sets of complex roots of unity.  The set <math>C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}</math> is also a set of complex roots of unity.  How many distinct elements are in <math>C^{}_{}</math>?
+
The sets <math>A = \{z : z^{18} = 1\}</math> and <math>B = \{w : w^{48} = 1\}</math> are both sets of complex roots of unity.  The set <math>C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}</math> is also a set of complex roots of unity.  How many distinct elements are in <math>C^{}_{}</math>?
  
 
[[1990 AIME Problems/Problem 10|Solution]]
 
[[1990 AIME Problems/Problem 10|Solution]]
Line 82: Line 84:
 
== Problem 15 ==
 
== Problem 15 ==
 
Find <math>a_{}^{}x^5 + b_{}y^5</math> if the real numbers <math>a_{}^{}</math>, <math>b_{}^{}</math>, <math>x_{}^{}</math>, and <math>y_{}^{}</math> satisfy the equations
 
Find <math>a_{}^{}x^5 + b_{}y^5</math> if the real numbers <math>a_{}^{}</math>, <math>b_{}^{}</math>, <math>x_{}^{}</math>, and <math>y_{}^{}</math> satisfy the equations
<center><math>ax + by = 3^{}_{},</math></center>
+
<cmath>ax + by = 3^{}_{},</cmath>
<center><math>ax^2 + by^2 = 7^{}_{},</math></center>
+
<cmath>ax^2 + by^2 = 7^{}_{},</cmath>
<center><math>ax^3 + by^3 = 16^{}_{},</math></center>
+
<cmath>ax^3 + by^3 = 16^{}_{},</cmath>
<center><math>ax^4 + by^4 = 42^{}_{}.</math></center>
+
<cmath>ax^4 + by^4 = 42^{}_{}.</cmath>
  
  
Line 91: Line 93:
  
 
== See also ==
 
== See also ==
 +
 +
{{AIME box|year=1990|before=[[1989 AIME Problems]]|after=[[1991 AIME Problems]]}}
 +
 
* [[American Invitational Mathematics Examination]]
 
* [[American Invitational Mathematics Examination]]
 
* [[AIME Problems and Solutions]]
 
* [[AIME Problems and Solutions]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
 +
[[Category:AIME Problems]]
 +
{{MAA Notice}}

Revision as of 07:36, 7 September 2018

1990 AIME (Answer Key)
Printable version | AoPS Contest CollectionsPDF

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

The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence.

Solution

Problem 2

Find the value of $(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}$.

Solution

Problem 3

Let $P_1^{}$ be a regular $r~\mbox{gon}$ and $P_2^{}$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each interior angle of $P_1^{}$ is $\frac{59}{58}$ as large as each interior angle of $P_2^{}$. What's the largest possible value of $s_{}^{}$?

Solution

Problem 4

Find the positive solution to

$\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0$

Solution

Problem 5

Let $n^{}_{}$ be the smallest positive integer that is a multiple of $75_{}^{}$ and has exactly $75_{}^{}$ positive integral divisors, including $1_{}^{}$ and itself. Find $\frac{n}{75}$.

Solution

Problem 6

A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these fish are no longer in the lake on September 1 (because of death and emigrations), that 40% of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?

Solution

Problem 7

A triangle has vertices $P_{}^{}=(-8,5)$, $Q_{}^{}=(-15,-19)$, and $R_{}^{}=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0_{}^{}$. Find $a+c_{}^{}$.

Solution

Problem 8

In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:

1) The marksman first chooses a column from which a target is to be broken.

2) The marksman must then break the lowest remaining target in the chosen column.

If the rules are followed, in how many different orders can the eight targets be broken?

Solution

Problem 9

A fair coin is to be tossed $10_{}^{}$ times. Let $i/j^{}_{}$, in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j_{}^{}$.

Solution

Problem 10

The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C^{}_{}$?

Solution

Problem 11

Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n^{}_{}$ for which $n^{}_{}!$ can be expressed as the product of $n - 3_{}^{}$ consecutive positive integers.

Solution

Problem 12

A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form

$a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},$

where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers. Find $a + b + c + d^{}_{}$.

Solution

Problem 13

Let $T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}$. Given that $9^{4000}_{}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T_{}^{}$ have 9 as their leftmost digit?

Solution

Problem 14

The rectangle $ABCD^{}_{}$ below has dimensions $AB^{}_{} = 12 \sqrt{3}$ and $BC^{}_{} = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P^{}_{}$. If triangle $ABP^{}_{}$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.

AIME 1990 Problem 14.png

Solution

Problem 15

Find $a_{}^{}x^5 + b_{}y^5$ if the real numbers $a_{}^{}$, $b_{}^{}$, $x_{}^{}$, and $y_{}^{}$ satisfy the equations \[ax + by = 3^{}_{},\] \[ax^2 + by^2 = 7^{}_{},\] \[ax^3 + by^3 = 16^{}_{},\] \[ax^4 + by^4 = 42^{}_{}.\]


Solution

See also

1990 AIME (ProblemsAnswer KeyResources)
Preceded by
1989 AIME Problems
Followed by
1991 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