# 1989 AIME Problems

 1989 AIME (Answer Key) Printable version | AoPS Contest Collections 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 $\sqrt{(31)(30)(29)(28)+1}$.

## 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?

## Problem 3

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

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

## Problem 4

If $a 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 $c^{}_{}$?

## 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 $i+j^{}_{}$.

## 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?

## 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 $k^{}_{}$.

## Problem 8

Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that

$x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1^{}_{}$
$4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12^{}_{}$
$9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123^{}_{}$

Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7^{}$.

## Problem 9

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

## Problem 10

Let $a_{}^{}$, $b_{}^{}$, $c_{}^{}$ be the three sides of a triangle, and let $\alpha_{}^{}$, $\beta_{}^{}$, $\gamma_{}^{}$, be the angles opposite them. If $a^2+b^2=1989^{}_{}c^2$, find

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

## 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 $x^{}_{}$.)

## 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 $d^{2}_{}$.

## 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?

## 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 $1,2^{}_{},\ldots,n^2$ 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 $k^{}_{}$.

## 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 at right). Given that $AP=6^{}_{}$, $BP=9^{}_{}$, $PD=6^{}_{}$, $PE=3^{}_{}$, and $CF=20^{}_{}$, find the area of $\triangle ABC^{}_{}$.