# Difference between revisions of "1996 AHSME Problems"

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Triangle <math>PAB</math> and square <math>ABCD</math> are in perpendicular planes. Given that <math>PA = 3, PB = 4</math> and <math>AB = 5</math>, what is <math>PD</math>? | Triangle <math>PAB</math> and square <math>ABCD</math> are in perpendicular planes. Given that <math>PA = 3, PB = 4</math> and <math>AB = 5</math>, what is <math>PD</math>? | ||

+ | <asy> | ||

+ | real r=sqrt(2)/2; | ||

+ | draw(origin--(8,0)--(8,-1)--(0,-1)--cycle); | ||

+ | draw(origin--(8,0)--(8+r, r)--(r,r)--cycle); | ||

+ | filldraw(origin--(-6*r, -6*r)--(8-6*r, -6*r)--(8, 0)--cycle, white, black); | ||

+ | filldraw(origin--(8,0)--(8,6)--(0,6)--cycle, white, black); | ||

+ | pair A=(6,0), B=(2,0), C=(2,4), D=(6,4), P=B+1*dir(-65); | ||

+ | draw(A--P--B--C--D--cycle); | ||

+ | dot(A^^B^^C^^D^^P); | ||

+ | label("$A$", A, dir((4,2)--A)); | ||

+ | label("$B$", B, dir((4,2)--B)); | ||

+ | label("$C$", C, dir((4,2)--C)); | ||

+ | label("$D$", D, dir((4,2)--D)); | ||

+ | label("$P$", P, dir((4,2)--P));</asy> | ||

<math> \text{(A)}\ 5\qquad\text{(B)}\ \sqrt{34} \qquad\text{(C)}\ \sqrt{41}\qquad\text{(D)}\ 2\sqrt{13}\qquad\text{(E)}\ 8 </math> | <math> \text{(A)}\ 5\qquad\text{(B)}\ \sqrt{34} \qquad\text{(C)}\ \sqrt{41}\qquad\text{(D)}\ 2\sqrt{13}\qquad\text{(E)}\ 8 </math> | ||

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Two opposite sides of a rectangle are each divided into <math>n</math> congruent segments, and the endpoints of one segment are joined to the center to form triangle <math>A</math>. The other sides are each divided into <math>m</math> congruent segments, and the endpoints of one of these segments are joined to the center to form triangle <math>B</math>. [See figure for <math>n=5, m=7</math>.] What is the ratio of the area of triangle to the area of triangle ? | Two opposite sides of a rectangle are each divided into <math>n</math> congruent segments, and the endpoints of one segment are joined to the center to form triangle <math>A</math>. The other sides are each divided into <math>m</math> congruent segments, and the endpoints of one of these segments are joined to the center to form triangle <math>B</math>. [See figure for <math>n=5, m=7</math>.] What is the ratio of the area of triangle to the area of triangle ? | ||

+ | |||

+ | <asy> | ||

+ | int i; | ||

+ | for(i=0; i<8; i=i+1) { | ||

+ | dot((i,0)^^(i,5)); | ||

+ | } | ||

+ | for(i=1; i<5; i=i+1) { | ||

+ | dot((0,i)^^(7,i)); | ||

+ | } | ||

+ | draw(origin--(7,0)--(7,5)--(0,5)--cycle, linewidth(0.8)); | ||

+ | pair P=(3.5, 2.5); | ||

+ | draw((0,4)--P--(0,3)^^(2,0)--P--(3,0)); | ||

+ | label("$B$", (2.3,0), NE); | ||

+ | label("$A$", (0,3.7), SE); | ||

+ | </asy> | ||

<math> \text{(A)}\ 1\qquad\text{(B)}\ m/n\qquad\text{(C)}\ n/m\qquad\text{(D)}\ 2m/n\qquad\text{(E)}\ 2n/m </math> | <math> \text{(A)}\ 1\qquad\text{(B)}\ m/n\qquad\text{(C)}\ n/m\qquad\text{(D)}\ 2m/n\qquad\text{(E)}\ 2n/m </math> | ||

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In rectangle <math>ABCD</math>, angle <math>C</math> is trisected by <math>\overline{CF}</math> and <math>\overline{CE}</math>, where <math>E</math> is on <math>\overline{AB}</math>, <math>F</math> is on <math>\overline{AD}</math>, <math>BE=6</math> and <math>AF=2</math>. Which of the following is closest to the area of the rectangle <math>ABCD</math>? | In rectangle <math>ABCD</math>, angle <math>C</math> is trisected by <math>\overline{CF}</math> and <math>\overline{CE}</math>, where <math>E</math> is on <math>\overline{AB}</math>, <math>F</math> is on <math>\overline{AD}</math>, <math>BE=6</math> and <math>AF=2</math>. Which of the following is closest to the area of the rectangle <math>ABCD</math>? | ||

− | + | <asy> | |

+ | pair A=origin, B=(10,0), C=(10,7), D=(0,7), E=(5,0), F=(0,2); | ||

+ | draw(A--B--C--D--cycle, linewidth(0.8)); | ||

+ | draw(E--C--F); | ||

+ | dot(A^^B^^C^^D^^E^^F); | ||

+ | label("$A$", A, dir((5, 3.5)--A)); | ||

+ | label("$B$", B, dir((5, 3.5)--B)); | ||

+ | label("$C$", C, dir((5, 3.5)--C)); | ||

+ | label("$D$", D, dir((5, 3.5)--D)); | ||

+ | label("$E$", E, dir((5, 3.5)--E)); | ||

+ | label("$F$", F, dir((5, 3.5)--F)); | ||

+ | label("$2$", (0,1), dir(0)); | ||

+ | label("$6$", (7.5,0), N);</asy> | ||

<math> \text{(A)}\ 110\qquad\text{(B)}\ 120\qquad\text{(C)}\ 130\qquad\text{(D)}\ 140\qquad\text{(E)}\ 150 </math> | <math> \text{(A)}\ 110\qquad\text{(B)}\ 120\qquad\text{(C)}\ 130\qquad\text{(D)}\ 140\qquad\text{(E)}\ 150 </math> | ||

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

− | + | <asy> | |

+ | size(120); | ||

+ | draw(rotate(30)*polygon(6)); | ||

+ | draw(scale(2/sqrt(3))*polygon(6)); | ||

+ | pair A=2/sqrt(3)*dir(120), B=2/sqrt(3)*dir(180), C=2/sqrt(3)*dir(240), D=2/sqrt(3)*dir(300), E=2/sqrt(3)*dir(0), F=2/sqrt(3)*dir(60); | ||

+ | dot(A^^B^^C^^D^^E^^F); | ||

+ | label("$A$", A, dir(origin--A)); | ||

+ | label("$B$", B, dir(origin--B)); | ||

+ | label("$C$", C, dir(origin--C)); | ||

+ | label("$D$", D, dir(origin--D)); | ||

+ | label("$E$", E, dir(origin--E)); | ||

+ | label("$F$", F, dir(origin--F)); | ||

+ | </asy> | ||

[[1996 AHSME Problems/Problem 19|Solution]] | [[1996 AHSME Problems/Problem 19|Solution]] | ||

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==Problem 21== | ==Problem 21== | ||

− | + | <asy> | |

+ | size(120); | ||

+ | pair B=origin, A=1*dir(70), M=foot(A, B, (3,0)), C=reflect(A, M)*B, E=foot(B, A, C), D=1*dir(20); | ||

+ | dot(A^^B^^C^^D^^E); | ||

+ | draw(A--D--B--A--C--B); | ||

+ | markscalefactor=0.005; | ||

+ | draw(rightanglemark(A, E, B)); | ||

+ | dot(A^^B^^C^^D^^E); | ||

+ | pair point=midpoint(A--M); | ||

+ | label("$A$", A, dir(point--A)); | ||

+ | label("$B$", B, dir(point--B)); | ||

+ | label("$C$", C, dir(point--C)); | ||

+ | label("$D$", D, dir(point--D)); | ||

+ | label("$E$", E, dir(point--E)); | ||

+ | </asy> | ||

[[1996 AHSME Problems/Problem 21|Solution]] | [[1996 AHSME Problems/Problem 21|Solution]] | ||

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==Problem 28== | ==Problem 28== | ||

+ | <asy> | ||

+ | size(120); | ||

+ | import three; | ||

+ | currentprojection=orthographic(1, 4/5, 1/3); | ||

+ | draw(box(O, (4,4,3))); | ||

+ | triple A=(0,4,3), B=(0,0,0) , C=(4,4,0), D=(0,4,0); | ||

+ | draw(A--B--C--cycle, linewidth(0.9)); | ||

+ | label("$A$", A, NE); | ||

+ | label("$B$", B, NW); | ||

+ | label("$C$", C, S); | ||

+ | label("$D$", D, E); | ||

+ | label("$4$", (4,2,0), SW); | ||

+ | label("$4$", (2,4,0), SE); | ||

+ | label("$3$", (0, 4, 1.5), E); | ||

+ | </asy> | ||

[[1996 AHSME Problems/Problem 28|Solution]] | [[1996 AHSME Problems/Problem 28|Solution]] | ||

## Revision as of 12:08, 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 ?