1996 AHSME Problems: Difference between revisions

Revision as of 12:08, 19 August 2011

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. Unknown error_msg)

$\text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E)}\ 8$

Solution

Problem 2

Each day Walter gets $3$ dollars for doing his chores or $5$ dollars for doing them exceptionally well. After $10$ days of doing his chores daily, Walter has received a total of $36$ dollars. On how many days did Walter do them exceptionally well?

$\text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 7$

Solution

Problem 3

$\frac{(3!)!}{3!}=$

$\text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 6\qquad\text{(D)}\ 40\qquad\text{(E)}\ 120$

Solution

Problem 4

Six numbers from a list of nine integers are $7,8,3,5, 9$ and $5$ . 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)

Solution

Problem 5

Given that $0 < a < b < c < d$ , which of the following is the largest?

$\text{(A)}\ \frac{a+b}{c+d} \qquad\text{(B)}\ \frac{a+d}{b+c} \qquad\text{(C)}\ \frac{b+c}{a+d} \qquad\text{(D)}\ \frac{b+d}{a+c} \qquad\text{(E)}\ \frac{c+d}{a+b}$

Solution

Problem 6

If $f(x) = x^{(x+1)}(x+2)^{(x+3)}$ , then $f(0)+f(-1)+f(-2)+f(-3) =$

$\text{(A)}\ -\frac{8}{9}\qquad\text{(B)}\ 0\qquad\text{(C)}\ \frac{8}{9}\qquad\text{(D)}\ 1\qquad\text{(E)}\ \frac{10}{9}$

Solution

Problem 7

A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $4.95$ for the father and $0.45$ for each year of a child's age, where age is defined as the age at the most recent birthday. If the bill is $9.45$ , which of the following could be the age of the youngest child?

$\text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5$

Solution

Problem 8

If $3 = k\cdot 2^r$ and $15 = k\cdot 4^r$ , then $r =$

$\text{(A)}\ -\log_{2}5\qquad\text{(B)}\ \log_{5}2\qquad\text{(C)}\ \log_{10}5\qquad\text{(D)}\ \log_{2}5\qquad\text{(E)}\ \frac{5}{2}$

Solution

Problem 9

Triangle $PAB$ and square $ABCD$ are in perpendicular planes. Given that $PA = 3, PB = 4$ and $AB = 5$ , what is $PD$ ? $[asy] real r=sqrt(2)/2; draw(origin--(8,0)--(8,-1)--(0,-1)--cycle); draw(origin--(8,0)--(8+r, r)--(r,r)--cycle); filldraw(origin--(-6*r, -6*r)--(8-6*r, -6*r)--(8, 0)--cycle, white, black); filldraw(origin--(8,0)--(8,6)--(0,6)--cycle, white, black); pair A=(6,0), B=(2,0), C=(2,4), D=(6,4), P=B+1*dir(-65); draw(A--P--B--C--D--cycle); dot(A^^B^^C^^D^^P); label("$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$", P, dir((4,2)--P));[/asy]$

$\text{(A)}\ 5\qquad\text{(B)}\ \sqrt{34} \qquad\text{(C)}\ \sqrt{41}\qquad\text{(D)}\ 2\sqrt{13}\qquad\text{(E)}\ 8$

Solution

Problem 10

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

$\text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 24\qquad\text{(D)}\ 28\qquad\text{(E)}\ 56$

Solution

Problem 11

Given a circle of raidus $2$ , there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.

$\text{(A)}\ \frac{\pi} 4\qquad\text{(B)}\ 4-\pi\qquad\text{(C)}\ \frac{\pi} 2\qquad\text{(D)}\ \pi\qquad\text{(E)}\ 2\pi$

Solution

Problem 12

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

$f(x) =\begin{cases}n+3 &\text{if n is odd}\\ n/2 &\text{if n is even}\end{cases}$

Suppose $k$ is odd and $f(f(f(k))) = 27$ . What is the sum of the digits of $k$ ?

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15$

Solution

Problem 13

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

$\text{(A)}\ hm\qquad\text{(B)}\ \frac{h}{h+m}\qquad\text{(C)}\ \frac{h}{m-1}\qquad\text{(D)}\ \frac{hm}{m-1}\qquad\text{(E)}\ \frac{h+m}{m-1}$

Solution

Problem 14

Let $E(n)$ denote the sum of the even digits of $n$ . For example, $E(5681) = 6+8 = 14$ . Find $E(1)+E(2)+E(3)+\cdots+E(100)$

$\text{(A)}\ 200\qquad\text{(B)}\ 360\qquad\text{(C)}\ 400\qquad\text{(D)}\ 900\qquad\text{(E)}\ 2250$

Solution

Problem 15

Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$ . The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$ . [See figure for $n=5, m=7$ .] 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$", (0,3.7), SE); [/asy]$

$\text{(A)}\ 1\qquad\text{(B)}\ m/n\qquad\text{(C)}\ n/m\qquad\text{(D)}\ 2m/n\qquad\text{(E)}\ 2n/m$

Solution

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?

$\text{(A)}\ \frac{1}{6}\qquad\text{(B)}\ \frac{91}{216}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{8}{15}\qquad\text{(E)}\ \frac{7}{12}$

Solution

Problem 17

In rectangle $ABCD$ , angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$ , where $E$ is on $\overline{AB}$ , $F$ is on $\overline{AD}$ , $BE=6$ and $AF=2$ . Which of the following is closest to the area of the rectangle $ABCD$ ? $[asy] pair A=origin, B=(10,0), C=(10,7), D=(0,7), E=(5,0), F=(0,2); draw(A--B--C--D--cycle, linewidth(0.8)); draw(E--C--F); dot(A^^B^^C^^D^^E^^F); label("$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]$ $\text{(A)}\ 110\qquad\text{(B)}\ 120\qquad\text{(C)}\ 130\qquad\text{(D)}\ 140\qquad\text{(E)}\ 150$

Solution

Problem 18

Solution

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$", F, dir(origin--F)); [/asy]$ Solution

Problem 20

Solution

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$", E, dir(point--E)); [/asy]$ Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

Problem 26

Solution

Problem 27

Solution

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]$ Solution

Problem 29

Solution

Problem 30

Solution

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1996 AHSME Problems: Difference between revisions

Revision as of 12:08, 19 August 2011

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

@@ Line 70: / Line 70: @@
 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>
@@ Line 123: / Line 137: @@
 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>
@@ Line 139: / Line 168: @@
 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>
@@ Line 149: / Line 190: @@
 ==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]]
@@ Line 157: / Line 210: @@
 ==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]]
@@ Line 186: / Line 253: @@
 ==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]]