# Difference between revisions of "1987 AHSME Problems"

m |
Mathgeek2006 (talk | contribs) m (→Problem 11) |
||

Line 166: | Line 166: | ||

Let <math>c</math> be a constant. The simultaneous equations | Let <math>c</math> be a constant. The simultaneous equations | ||

− | <cmath>\begin{align*}x-y = &\ 2 \\cx+y = &\ 3 \\\end{align}</cmath> | + | <cmath>\begin{align*}x-y = &\ 2 \\cx+y = &\ 3 \\\end{align*}</cmath> |

have a solution <math>(x, y)</math> inside Quadrant I if and only if | have a solution <math>(x, y)</math> inside Quadrant I if and only if | ||

## Revision as of 20:12, 10 March 2015

## Contents

- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30

## Problem 1

equals

## Problem 2

A triangular corner with side lengths is cut from equilateral triangle ABC of side length . The perimeter of the remaining quadrilateral is

## Problem 3

How many primes less than have as the ones digit? (Assume the usual base ten representation)

## Problem 4

equals

## Problem 5

A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. However, the student neglected to indicate , the total number of measurements. What is the smallest possible value of ?

## Problem 6

In the shown, is some interior point, and are the measures of angles in degrees. Solve for in terms of and .

## Problem 7

If , which of the four quantities is the largest?

## Problem 8

In the figure the sum of the distances and is

## Problem 9

The first four terms of an arithmetic sequence are . The ratio of to is

## Problem 10

How many ordered triples of non-zero real numbers have the property that each number is the product of the other two?

## Problem 11

Let be a constant. The simultaneous equations have a solution inside Quadrant I if and only if

## Problem 12

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. If there are five letters in all, and the boss delivers them in the order , which of the following could not be the order in which the secretary types them?

## Problem 13

A long piece of paper cm wide is made into a roll for cash registers by wrapping it times around a cardboard tube of diameter cm, forming a roll cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms concentric circles with diameters evenly spaced from cm to cm.)

## Problem 14

is a square and and are the midpoints of and respectively. Then

## Problem 15

If is a solution to the system , find .

## Problem 16

A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base . Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base and the elements of the set . Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as , respectively. What is the base- expression for the integer coded as ?

## Problem 17

In a mathematics competition, the sum of the scores of Bill and Dick equalled the sum of the scores of Ann and Carol. If the scores of Bill and Carol had been interchanged, then the sum of the scores of Ann and Carol would have exceeded the sum of the scores of the other two. Also, Dick's score exceeded the sum of the scores of Bill and Carol. Determine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative.

## Problem 18

It takes algebra books (all the same thickness) and geometry books (all the same thickness, which is greater than that of an algebra book) to completely fill a certain shelf. Also, of the algebra books and of the geometry books would fill the same shelf. Finally, of the algebra books alone would fill this shelf. Given that are distinct positive integers, it follows that is

## Problem 19

Which of the following is closest to ?

## Problem 20

Evaluate

## Problem 21

There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is . What is the area (in ) of the square inscribed in the same as shown in Figure 2 below?

## Problem 22

A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole cm across as the top and cm deep. What was the radius of the ball (in centimeters)?

## Problem 23

If is a prime and both roots of are integers, then

## Problem 24

How many polynomial functions of degree satisfy ?

## Problem 25

is a triangle: and both the coordinates of are integers. What is the minimum area can have?

## Problem 26

The amount is split into two nonnegative real numbers uniformly at random, for instance, into and , or into and . Then each number is rounded to its nearest integer, for instance, and in the first case above, and in the second. What is the probability that the two integers sum to ?

## Problem 27

A cube of cheese is cut along the planes and . How many pieces are there? (No cheese is moved until all three cuts are made.)

## Problem 28

Let be real numbers. Suppose that all the roots of are complex numbers lying on a circle in the complex plane centered at and having radius . The sum of the reciprocals of the roots is necessarily

## Problem 29

Consider the sequence of numbers defined recursively by and for by when is even and by when is odd. Given that , the sum of the digits of is

## Problem 30

In the figure, has and . A line , with on and , divides into two pieces of equal area. (Note: the figure may not be accurate; perhaps is on instead of .) The ratio is