# Difference between revisions of "1996 AHSME Problems"

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==Problem 19== | ==Problem 19== | ||

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+ | The midpoints of the sides of a regular hexagon <math>ABCDEF</math> are joined to form a smaller hexagon. What fraction of the area of <math>ABCDEF</math> is enclosed by the smaller hexagon? | ||

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<asy> | <asy> | ||

size(120); | size(120); |

## Revision as of 12:59, 19 August 2011

## 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

The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?

$\begin{tabular}{r}&\ \texttt{6 4 1}\\ \texttt{8 5 2} &+\texttt{9 7 3}\\ \hline \texttt{2 4 5 6}\end{tabular}$ (Error compiling LaTeX. ! Extra alignment tab has been changed to \cr.)

## Problem 2

Each day Walter gets dollars for doing his chores or dollars for doing them exceptionally well. After days of doing his chores daily, Walter has received a total of dollars. On how many days did Walter do them exceptionally well?

## Problem 3

## Problem 4

Six numbers from a list of nine integers are and . The largest possible value of the median of all nine numbers in this list is

$\text{(A)}\ 5\qquad\text{(B)}\6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9$ (Error compiling LaTeX. ! Undefined control sequence.)

## Problem 5

Given that , which of the following is the largest?

## Problem 6

If , then

## Problem 7

A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges for the father and for each year of a child's age, where age is defined as the age at the most recent birthday. If the bill is , which of the following could be the age of the youngest child?

## Problem 8

If and , then

## Problem 9

Triangle and square are in perpendicular planes. Given that and , what is ?

## Problem 10

How many line segments have both their endpoints located at the vertices of a given cube?

## Problem 11

Given a circle of raidus , there are many line segments of length that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.

## Problem 12

A function from the integers to the integers is defined as follows:

Suppose is odd and . What is the sum of the digits of ?

## Problem 13

Sunny runs at a steady rate, and Moonbeam runs times as fast, where is a number greater than 1. If Moonbeam gives Sunny a head start of meters, how many meters must Moonbeam run to overtake Sunny?

## Problem 14

Let denote the sum of the even digits of . For example, . Find

## Problem 15

Two opposite sides of a rectangle are each divided into congruent segments, and the endpoints of one segment are joined to the center to form triangle . The other sides are each divided into congruent segments, and the endpoints of one of these segments are joined to the center to form triangle . [See figure for .] What is the ratio of the area of triangle to the area of triangle ?

## Problem 16

A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one "2" is tossed?

## Problem 17

In rectangle , angle is trisected by and , where is on , is on , and . Which of the following is closest to the area of the rectangle ?

## Problem 18

A circle of radius has center at . A circle of radius has center at . A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the -intercept of the line?

## Problem 19

The midpoints of the sides of a regular hexagon are joined to form a smaller hexagon. What fraction of the area of is enclosed by the smaller hexagon?