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Difference between revisions of "1988 AIME Problems"
m (→‎Problem 8)
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== Problem 8 ==
 
== Problem 8 ==
 
The function <math>f</math>, defined on the set of ordered pairs of positive integers, satisfies the following properties:
 
The function <math>f</math>, defined on the set of ordered pairs of positive integers, satisfies the following properties:
−
<center><math>\begin{eqnarray*} f(x,x) &=& x, \\ f(x,y) &=& f(y,x), \quad \text{and} \\ (x + y) f(x,y) &=& yf(x,x + y). \end{eqnarray*}</math></center>
+
<cmath>
 +
\begin{align*}
 +
f(x,x) &=x, \\
 +
f(x,y) &=f(y,x), \quad \text{and} \\
 +
(x + y) f(x,y) &= yf(x,x + y).  
 +
\end{align*}
 +
</cmath>
 
Calculate <math>f(14,52)</math>.
 
Calculate <math>f(14,52)</math>.
  

Revision as of 17:26, 10 March 2015

1988 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

One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has $\{1, 2, 3, 6, 9\}$ as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?

1988-1.png

Solution

Problem 2

For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$.

Solution

Problem 3

Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.

Solution

Problem 4

Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$. Suppose further that

$|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|$.

What is the smallest possible value of $n$?

Solution

Problem 5

Let $m/n$, in lowest terms, be the probability that a randomly chosen positive divisor of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$.

Solution

Problem 6

It is possible to place positive integers into the vacant twenty-one squares of the 5 times 5 square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*).

AIME 1988 Problem 06.png

Solution

Problem 7

In triangle $ABC$, $\tan \angle CAB = 22/7$, and the altitude from $A$ divides $BC$ into segments of length $3$ and $17$. What is the area of triangle $ABC$?

Solution

Problem 8

The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \begin{align*} f(x,x) &=x, \\ f(x,y) &=f(y,x), \quad \text{and} \\ (x + y) f(x,y) &= yf(x,x + y). \end{align*} Calculate $f(14,52)$.

Solution

Problem 9

Find the smallest positive integer whose cube ends in $888$.

Solution

Problem 10

A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?

Solution

Problem 11

Let $w_1, w_2, \dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that

$\sum_{k = 1}^n (z_k - w_k) = 0.$

For the numbers $w_1 = 32 + 170i$, $w_2 = -7 + 64i$, $w_3 = -9 +200i$, $w_4 = 1 + 27i$, and $w_5 = -14 + 43i$, there is a unique mean line with y-intercept $3$. Find the slope of this mean line.

Solution

Problem 12

Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.

AIME 1988 Problem 12.png

Solution

Problem 13

Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$.

Solution

Problem 14

Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form \[12x^2 + bxy + cy^2 + d = 0.\] Find the product $bc$.

Solution

Problem 15

In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss delivers them in the order 1, 2, 3, 4, 5, 6, 7, 8, 9.

While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing else about the morning's typing. The colleague wonders which of the nine letters remain to be typed after lunch and in what order they will be typed. Based upon the above information, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.)

Solution

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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