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
[hide]- 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?
$
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. Unknown error_msg)
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 ?