Difference between revisions of "1996 AHSME Problems"
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+ | {{AHSME Problems | ||
+ | |year = 1996 | ||
+ | }} | ||
==Problem 1== | ==Problem 1== | ||
The addition below is incorrect. What is the largest digit that can be changed to make the addition correct? | The addition below is incorrect. What is the largest digit that can be changed to make the addition correct? | ||
− | <math> \begin{tabular}{ | + | <math> \begin{tabular}{rr}&\ \texttt{6 4 1}\\ &\texttt{8 5 2}\\ +&\texttt{9 7 3}\\ \hline &\texttt{2 4 5 6}\end{tabular} </math> |
Line 31: | Line 34: | ||
Six numbers from a list of nine integers are <math>7,8,3,5, 9</math> and <math>5</math>. The largest possible value of the median of all nine numbers in this list is | Six numbers from a list of nine integers are <math>7,8,3,5, 9</math> and <math>5</math>. The largest possible value of the median of all nine numbers in this list is | ||
− | <math> \text{(A)}\ 5\qquad\text{(B)}\6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 </math> | + | <math> \text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 </math> |
[[1996 AHSME Problems/Problem 4|Solution]] | [[1996 AHSME Problems/Problem 4|Solution]] | ||
Line 70: | Line 73: | ||
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> | ||
Line 85: | Line 102: | ||
==Problem 11== | ==Problem 11== | ||
− | Given a circle of | + | Given a circle of radius <math>2</math>, there are many line segments of length <math>2</math> that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. |
<math> \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 </math> | <math> \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 </math> | ||
Line 92: | Line 109: | ||
==Problem 12== | ==Problem 12== | ||
+ | |||
+ | A function <math>f</math> from the integers to the integers is defined as follows: | ||
+ | |||
+ | <cmath> f(x) =\begin{cases}n+3 &\text{if n is odd}\\ n/2 &\text{if n is even}\end{cases} </cmath> | ||
+ | |||
+ | Suppose <math>k</math> is odd and <math>f(f(f(k))) = 27</math>. What is the sum of the digits of <math>k</math>? | ||
+ | |||
+ | <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15 </math> | ||
+ | |||
[[1996 AHSME Problems/Problem 12|Solution]] | [[1996 AHSME Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
+ | |||
+ | Sunny runs at a steady rate, and Moonbeam runs <math>m</math> times as fast, where <math>m</math> is a number greater than 1. If Moonbeam gives Sunny a head start of <math>h</math> meters, how many meters must Moonbeam run to overtake Sunny? | ||
+ | |||
+ | <math> \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} </math> | ||
[[1996 AHSME Problems/Problem 13|Solution]] | [[1996 AHSME Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
+ | |||
+ | Let <math>E(n)</math> denote the sum of the even digits of <math>n</math>. For example, <math> E(5681) = 6+8 = 14 </math>. Find <math> E(1)+E(2)+E(3)+\cdots+E(100) </math> | ||
+ | |||
+ | <math> \text{(A)}\ 200\qquad\text{(B)}\ 360\qquad\text{(C)}\ 400\qquad\text{(D)}\ 900\qquad\text{(E)}\ 2250 </math> | ||
[[1996 AHSME Problems/Problem 14|Solution]] | [[1996 AHSME Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
+ | |||
+ | 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 <math>A</math> to the area of triangle <math>B</math>? | ||
+ | |||
+ | <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> | ||
[[1996 AHSME Problems/Problem 15|Solution]] | [[1996 AHSME Problems/Problem 15|Solution]] | ||
==Problem 16== | ==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? | ||
+ | |||
+ | <math> \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} </math> | ||
[[1996 AHSME Problems/Problem 16|Solution]] | [[1996 AHSME Problems/Problem 16|Solution]] | ||
==Problem 17== | ==Problem 17== | ||
+ | |||
+ | 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> | ||
[[1996 AHSME Problems/Problem 17|Solution]] | [[1996 AHSME Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
+ | |||
+ | A circle of radius <math>2</math> has center at <math>(2,0)</math>. A circle of radius <math>1</math> has center at <math>(5,0)</math>. A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the <math>y</math>-intercept of the line? | ||
+ | |||
+ | <math> \text{(A)}\ \sqrt{2}/4\qquad\text{(B)}\ 8/3\qquad\text{(C)}\ 1+\sqrt 3\qquad\text{(D)}\ 2\sqrt 2\qquad\text{(E)}\ 3 </math> | ||
[[1996 AHSME Problems/Problem 18|Solution]] | [[1996 AHSME Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
+ | |||
+ | 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? | ||
+ | |||
+ | <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> | ||
+ | |||
+ | <math> \text{(A)}\ \frac{1}{2}\qquad\text{(B)}\ \frac{\sqrt 3}{3}\qquad\text{(C)}\ \frac{2}{3}\qquad\text{(D)}\ \frac{3}{4}\qquad\text{(E)}\ \frac{\sqrt 3}{2} </math> | ||
[[1996 AHSME Problems/Problem 19|Solution]] | [[1996 AHSME Problems/Problem 19|Solution]] | ||
==Problem 20== | ==Problem 20== | ||
+ | |||
+ | In the xy-plane, what is the length of the shortest path from <math>(0,0)</math> to <math>(12,16)</math> that does not go inside the circle <math> (x-6)^{2}+(y-8)^{2}= 25 </math>? | ||
+ | |||
+ | <math> \text{(A)}\ 10\sqrt 3\qquad\text{(B)}\ 10\sqrt 5\qquad\text{(C)}\ 10\sqrt 3+\frac{ 5\pi}{3}\qquad\text{(D)}\ 40\frac{\sqrt{3}}{3}\qquad\text{(E)}\ 10+5\pi </math> | ||
[[1996 AHSME Problems/Problem 20|Solution]] | [[1996 AHSME Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
+ | |||
+ | Triangles <math>ABC</math> and <math>ABD</math> are isosceles with <math>AB=AC=BD</math>, and <math>BD</math> intersects <math>AC</math> at <math>E</math>. If <math>BD</math> is perpendicular to <math>AC</math>, then <math> \angle C+\angle D </math> is | ||
+ | |||
+ | <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> | ||
+ | |||
+ | <math> \text{(A)}\ 115^\circ\qquad\text{(B)}\ 120^\circ\qquad\text{(C)}\ 130^\circ\qquad\text{(D)}\ 135^\circ\qquad\text{(E)}\ \text{not uniquely determined} </math> | ||
[[1996 AHSME Problems/Problem 21|Solution]] | [[1996 AHSME Problems/Problem 21|Solution]] | ||
==Problem 22== | ==Problem 22== | ||
+ | |||
+ | Four distinct points, <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>, are to be selected from <math>1996</math> points | ||
+ | evenly spaced around a circle. All quadruples are equally likely to be chosen. | ||
+ | What is the probability that the chord <math>AB</math> intersects the chord <math>CD</math>? | ||
+ | |||
+ | <math> \text{(A)}\ \frac{1}{4}\qquad\text{(B)}\ \frac{1}{3}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{2}{3}\qquad\text{(E)}\ \frac{3}{4} </math> | ||
[[1996 AHSME Problems/Problem 22|Solution]] | [[1996 AHSME Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
+ | |||
+ | The sum of the lengths of the twelve edges of a rectangular box is <math>140</math>, and | ||
+ | the distance from one corner of the box to the farthest corner is <math>21</math>. The total | ||
+ | surface area of the box is | ||
+ | |||
+ | <math> \text{(A)}\ 776\qquad\text{(B)}\ 784\qquad\text{(C)}\ 798\qquad\text{(D)}\ 800\qquad\text{(E)}\ 812 </math> | ||
[[1996 AHSME Problems/Problem 23|Solution]] | [[1996 AHSME Problems/Problem 23|Solution]] | ||
==Problem 24== | ==Problem 24== | ||
+ | |||
+ | The sequence <math> 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\ldots </math> consists of <math>1</math>’s separated by blocks of <math>2</math>’s with <math>n</math> <math>2</math>’s in the <math>n^{th}</math> block. The sum of the first <math>1234</math> terms of this sequence is | ||
+ | |||
+ | <math> \text{(A)}\ 1996\qquad\text{(B)}\ 2419\qquad\text{(C)}\ 2429\qquad\text{(D)}\ 2439\qquad\text{(E)}\ 2449 </math> | ||
[[1996 AHSME Problems/Problem 24|Solution]] | [[1996 AHSME Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
+ | |||
+ | Given that <math>x^2 + y^2 = 14x + 6y + 6</math>, what is the largest possible value that <math>3x + 4y</math> can have? | ||
+ | |||
+ | <math> \text{(A)}\ 72\qquad\text{(B)}\ 73\qquad\text{(C)}\ 74\qquad\text{(D)}\ 75\qquad\text{(E)}\ 76 </math> | ||
[[1996 AHSME Problems/Problem 25|Solution]] | [[1996 AHSME Problems/Problem 25|Solution]] | ||
==Problem 26== | ==Problem 26== | ||
+ | |||
+ | An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely: | ||
+ | |||
+ | (a) the selection of four red marbles; | ||
+ | |||
+ | (b) the selection of one white and three red marbles; | ||
+ | |||
+ | (c) the selection of one white, one blue, and two red marbles; and | ||
+ | |||
+ | (d) the selection of one marble of each color. | ||
+ | |||
+ | What is the smallest number of marbles satisfying the given condition? | ||
+ | |||
+ | <math> \text{(A)}\ 19\qquad\text{(B)}\ 21\qquad\text{(C)}\ 46\qquad\text{(D)}\ 69\qquad\text{(E)}\ \text{more than 69} </math> | ||
[[1996 AHSME Problems/Problem 26|Solution]] | [[1996 AHSME Problems/Problem 26|Solution]] | ||
==Problem 27== | ==Problem 27== | ||
+ | |||
+ | Consider two solid spherical balls, one centered at <math> (0, 0,\frac{21}{2}) </math> with radius <math>6</math>, and the other centered at <math> (0, 0, 1) </math> with radius <math>\frac{9}{2}</math>. How many points with only integer coordinates (lattice points) are there in the intersection of the balls? | ||
+ | |||
+ | <math> \text{(A)}\ 7\qquad\text{(B)}\ 9\qquad\text{(C)}\ 11\qquad\text{(D)}\ 13\qquad\text{(E)}\ 15 </math> | ||
[[1996 AHSME Problems/Problem 27|Solution]] | [[1996 AHSME Problems/Problem 27|Solution]] | ||
==Problem 28== | ==Problem 28== | ||
+ | |||
+ | On a <math> 4\times 4\times 3 </math> rectangular parallelepiped, vertices <math>A</math>, <math>B</math>, and <math>C</math> are adjacent to vertex <math>D</math>. The perpendicular distance from <math>D</math> to the plane containing | ||
+ | <math>A</math>, <math>B</math>, and <math>C</math> is closest to | ||
+ | |||
+ | <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> | ||
+ | |||
+ | <math> \text{(A)}\ 1.6\qquad\text{(B)}\ 1.9\qquad\text{(C)}\ 2.1\qquad\text{(D)}\ 2.7\qquad\text{(E)}\ 2.9 </math> | ||
[[1996 AHSME Problems/Problem 28|Solution]] | [[1996 AHSME Problems/Problem 28|Solution]] | ||
==Problem 29== | ==Problem 29== | ||
+ | |||
+ | If <math>n</math> is a positive integer such that <math>2n</math> has <math>28</math> positive divisors and <math>3n</math> has <math>30</math> positive divisors, then how many positive divisors does <math>6n</math> have? | ||
+ | |||
+ | <math> \text{(A)}\ 32\qquad\text{(B)}\ 34\qquad\text{(C)}\ 35\qquad\text{(D)}\ 36\qquad\text{(E)}\ 38 </math> | ||
[[1996 AHSME Problems/Problem 29|Solution]] | [[1996 AHSME Problems/Problem 29|Solution]] | ||
==Problem 30== | ==Problem 30== | ||
+ | |||
+ | A hexagon inscribed in a circle has three consecutive sides each of length <math>3</math> and three consecutive sides each of length <math>5</math>. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length <math>3</math> and the other with three sides each of length <math>5</math>, has length equal to <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
+ | |||
+ | <math> \text{(A)}\ 309\qquad\text{(B)}\ 349\qquad\text{(C)}\ 369\qquad\text{(D)}\ 389\qquad\text{(E)}\ 409 </math> | ||
[[1996 AHSME Problems/Problem 30|Solution]] | [[1996 AHSME Problems/Problem 30|Solution]] | ||
+ | |||
+ | == See also == | ||
+ | |||
+ | * [[AMC 12 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{AHSME box|year=1996|before=[[1995 AHSME]]|after=[[1997 AHSME]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 12:39, 19 February 2020
1996 AHSME (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 |
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
- 31 See also
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
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 radius , 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?
Problem 20
In the xy-plane, what is the length of the shortest path from to that does not go inside the circle ?
Problem 21
Triangles and are isosceles with , and intersects at . If is perpendicular to , then is
Problem 22
Four distinct points, , , , and , are to be selected from points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord intersects the chord ?
Problem 23
The sum of the lengths of the twelve edges of a rectangular box is , and the distance from one corner of the box to the farthest corner is . The total surface area of the box is
Problem 24
The sequence consists of ’s separated by blocks of ’s with ’s in the block. The sum of the first terms of this sequence is
Problem 25
Given that , what is the largest possible value that can have?
Problem 26
An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:
(a) the selection of four red marbles;
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; and
(d) the selection of one marble of each color.
What is the smallest number of marbles satisfying the given condition?
Problem 27
Consider two solid spherical balls, one centered at with radius , and the other centered at with radius . How many points with only integer coordinates (lattice points) are there in the intersection of the balls?
Problem 28
On a rectangular parallelepiped, vertices , , and are adjacent to vertex . The perpendicular distance from to the plane containing , , and is closest to
Problem 29
If is a positive integer such that has positive divisors and has positive divisors, then how many positive divisors does have?
Problem 30
A hexagon inscribed in a circle has three consecutive sides each of length and three consecutive sides each of length . The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length and the other with three sides each of length , has length equal to , where and are relatively prime positive integers. Find .
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1995 AHSME |
Followed by 1997 AHSME | |
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All AHSME Problems and Solutions |
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