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Difference between revisions of "2002 AMC 10A Problems"

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{{AMC10 Problems|year=2002|ab=A}}
 
==Problem 1==
 
==Problem 1==
  
−
Which of the following is closest to <math>\frac{{10}^{2000} + {10}^{2002}}{{10}^{2001} + {10}^{2001}}</math>?
+
The ratio <math>\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}</math> is closest to which of the following numbers?
  
 
<math>\text{(A)}\ 0.1 \qquad \text{(B)}\ 0.2 \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 10</math>
 
<math>\text{(A)}\ 0.1 \qquad \text{(B)}\ 0.2 \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 10</math>
Line 9: Line 10:
 
== Problem 2 ==
 
== Problem 2 ==
  
−
Given that a, b, and c are non-zero real numbers, define <math>(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}</math>, find <math>(2, 12, 9)</math>.
+
For the nonzero numbers <math>a</math>, <math>b</math>, <math>c</math>, define <math>(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}</math>. Find <math>(2, 12, 9)</math>.
  
 
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math>
 
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math>
Line 16: Line 17:
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
According to the standard convention for exponentiation,
 +
 +
<math>2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65,536</math>.
 +
 +
If the order in which the exponentiations are performed is changed, how many <u>other</u> values are possible?
 +
 +
 +
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4</math>
  
 
[[2002 AMC 10A Problems/Problem 3|Solution]]
 
[[2002 AMC 10A Problems/Problem 3|Solution]]
Line 21: Line 30:
 
== Problem 4 ==
 
== Problem 4 ==
  
−
For how many positive integers m is there at least 1 positive integer n such that <math>mn \le m + n</math>?
+
For how many positive integers <math>m</math> does there exist at least one positive integer <math>n</math> such that <math>mn \le m + n</math>?
  
−
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 12 \qquad \text{(E)}</math> Infinite
+
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 12 \qquad \text{(E)}</math> infinitely many
  
  
Line 29: Line 38:
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
 +
 +
<asy>
 +
unitsize(.3cm);
 +
path c=Circle((0,2),1);
 +
filldraw(Circle((0,0),3),grey,black);
 +
filldraw(Circle((0,0),1),white,black);
 +
filldraw(c,white,black);
 +
filldraw(rotate(60)*c,white,black);
 +
filldraw(rotate(120)*c,white,black);
 +
filldraw(rotate(180)*c,white,black);
 +
filldraw(rotate(240)*c,white,black);
 +
filldraw(rotate(300)*c,white,black);
 +
</asy>
 +
 +
<math>\text{(A)}\ \pi \qquad \text{(B)}\ 1.5\pi \qquad \text{(C)}\ 2\pi \qquad \text{(D)}\ 3\pi \qquad \text{(E)}\ 3.5\pi</math>
  
 
[[2002 AMC 10A Problems/Problem 5|Solution]]
 
[[2002 AMC 10A Problems/Problem 5|Solution]]
Line 34: Line 59:
 
== Problem 6 ==
 
== Problem 6 ==
  
−
From a starting number, Cindy was supposed to subtract 3, and then divide by 9, but instead, Cindy subtracted 9, then divided by 3, getting 43. If the correct instructions were followed, what would the result be?
+
Cindy was asked by her teacher to subtract <math>3</math> from a certain number and then divide the result by <math>9</math>. Instead, she subtracted <math>9</math> and then divided the result by <math>3</math>, giving an answer of <math>43</math>. What would her answer have been had she worked the problem correctly?
  
 
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 34 \qquad \text{(C)}\ 43 \qquad \text{(D)}\ 51 \qquad \text{(E)} 138</math>
 
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 34 \qquad \text{(C)}\ 43 \qquad \text{(D)}\ 51 \qquad \text{(E)} 138</math>
Line 42: Line 67:
 
== Problem 7 ==
 
== Problem 7 ==
  
−
A <math>45^\circ</math> arc of circle A is equal in length to a <math>30^\circ</math> arc of circle B. What is the ratio of circle A's area and circle B's area?
+
If an arc of <math>45^\circ</math> on circle <math>A</math> has the same length as an arc of <math>30^\circ</math> on circle <math>B</math>, then the ratio of the area of circle <math>A</math> to the area of circle <math>B</math> is
  
 
<math>\text{(A)}\ 4/9 \qquad \text{(B)}\ 2/3 \qquad \text{(C)}\ 5/6 \qquad \text{(D)}\ 3/2 \qquad \text{(E)}\ 9/4</math>
 
<math>\text{(A)}\ 4/9 \qquad \text{(B)}\ 2/3 \qquad \text{(C)}\ 5/6 \qquad \text{(D)}\ 3/2 \qquad \text{(E)}\ 9/4</math>
Line 49: Line 74:
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let <math>B</math> be the total area of the blue triangles, <math>W</math> the total area of the white squares, and <math>R</math> the area of the red square. Which of the following is correct?
 +
 +
<asy>
 +
unitsize(3mm);
 +
fill((-4,-4)--(-4,4)--(4,4)--(4,-4)--cycle,blue);
 +
fill((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,red);
 +
path onewhite=(-3,3)--(-2,4)--(-1,3)--(-2,2)--(-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle;
 +
path divider=(-2,2)--(-3,3)--cycle;
 +
fill(onewhite,white);
 +
fill(rotate(90)*onewhite,white);
 +
fill(rotate(180)*onewhite,white);
 +
fill(rotate(270)*onewhite,white);
 +
</asy>
 +
 +
<math>\text{(A)}\ B = W \qquad \text{(B)}\ W = R \qquad \text{(C)}\ B = R \qquad \text{(D)}\ 3B = 2R \qquad \text{(E)}\ 2R = W</math>
  
 
[[2002 AMC 10A Problems/Problem 8|Solution]]
 
[[2002 AMC 10A Problems/Problem 8|Solution]]
Line 56: Line 96:
 
There are 3 numbers A, B, and C, such that <math>1001C - 2002A = 4004</math>, and <math>1001B + 3003A = 5005</math>. What is the average of A, B, and C?
 
There are 3 numbers A, B, and C, such that <math>1001C - 2002A = 4004</math>, and <math>1001B + 3003A = 5005</math>. What is the average of A, B, and C?
  
−
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}</math> More than 1
+
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E) }\text{Not uniquely determined}</math>
 +
 
  
  
Line 63: Line 104:
 
== Problem 10 ==
 
== Problem 10 ==
  
−
What is the sum of all of the roots of <math>(2x + 3) (x - 4) + (2x + 3) (x - 6) = 0</math>?
+
Compute the sum of all the roots of <math>(2x + 3)(x - 4) + (2x + 3)(x - 6) = 0</math>.
  
 
<math>\text{(A)}\ 7/2 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 13</math>
 
<math>\text{(A)}\ 7/2 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 13</math>
Line 71: Line 112:
 
== Problem 11 ==
 
== Problem 11 ==
  
−
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
+
Jamal wants to store <math>30</math> computer files on floppy disks, each of which has a capacity of <math>1.44</math> megabytes (MB). Three of his files require <math>0.8</math> MB of memory each, <math>12</math> more require <math>0.7</math> MB each, and the remaining <math>15</math> require <math>0.4</math> MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?
  
 
<math>\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 15 \qquad \text{(E)} 16</math>
 
<math>\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 15 \qquad \text{(E)} 16</math>
Line 78: Line 119:
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
 +
Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages <math>40</math> miles per hour, he arrives at his workplace three minutes late. When he averages <math>60</math> miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?
 +
 +
<math>\text{(A)}\ 45 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 55 \qquad \text{(E)} 58</math>
  
 
[[2002 AMC 10A Problems/Problem 12|Solution]]
 
[[2002 AMC 10A Problems/Problem 12|Solution]]
Line 83: Line 128:
 
== Problem 13 ==
 
== Problem 13 ==
  
−
Give a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.
+
The sides of a triangle have lengths 15, 20, and 25. Find the length of the shortest altitude.
  
 
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 12.5 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15</math>
 
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 12.5 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15</math>
Line 91: Line 136:
 
== Problem 14 ==
 
== Problem 14 ==
  
−
The 2 roots of the quadratic <math>x^2 - 63x + k = 0</math> are both prime. How many values of k are there?
+
Both roots of the quadratic equation <math>x^2 - 63x + k = 0</math> are prime numbers. The number of possible values of <math>k</math> is
  
−
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 4 \qquad \text{(E)}&</math>More than 4
+
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 4 \qquad \text{(E) more than 4}</math>
  
 
[[2002 AMC 10A Problems/Problem 14|Solution]]
 
[[2002 AMC 10A Problems/Problem 14|Solution]]
Line 99: Line 144:
 
== Problem 15 ==
 
== Problem 15 ==
  
−
Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?
+
The digits <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, and <math>9</math> are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?
  
 
<math>\text{(A)}\ 150 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 170 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 190</math>
 
<math>\text{(A)}\ 150 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 170 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 190</math>
Line 107: Line 152:
 
== Problem 16 ==
 
== Problem 16 ==
  
−
Let <math>\text{a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5}</math>. What is <math>\text{a + b + c + d}</math>?
+
If <math>a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5</math>, then <math>a + b + c + d</math> is
  
−
<math>\text{(A)}\ -5 \qquad \text{(B)}\ -7/3 \qquad \text{(C)}\ -7/3 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 5</math>
+
<math>\text{(A)}\ -5 \qquad \text{(B)}\ -10/3 \qquad \text{(C)}\ -7/3 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 5</math>
  
 
[[2002 AMC 10A Problems/Problem 16|Solution]]
 
[[2002 AMC 10A Problems/Problem 16|Solution]]
  
 
== Problem 17 ==
 
== Problem 17 ==
 +
 +
Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
 +
 +
<math>\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 3/8 \qquad \text{(D)}\ 2/5 \qquad \text{(E)} 1/2 </math>
  
 
[[2002 AMC 10A Problems/Problem 17|Solution]]
 
[[2002 AMC 10A Problems/Problem 17|Solution]]
  
 
== Problem 18 ==
 
== Problem 18 ==
 +
 +
A <math>3 \times 3 \times 3</math> cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the <math>3 \times 3 \times 3</math> cube is
 +
 +
<math>\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)} 96</math>
  
 
[[2002 AMC 10A Problems/Problem 18|Solution]]
 
[[2002 AMC 10A Problems/Problem 18|Solution]]
  
 
== Problem 19 ==
 
== Problem 19 ==
 +
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?
 +
 +
<math>\text{(A)}\ 2\pi/3 \qquad \text{(B)}\ 2\pi \qquad \text{(C)}\ 5\pi/2 \qquad \text{(D)}\ 8\pi/3 \qquad \text{(E)}\ 3\pi</math>
  
 
[[2002 AMC 10A Problems/Problem 19|Solution]]
 
[[2002 AMC 10A Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== Problem 20 ==
 +
Points <math>A,B,C,D,E</math> and <math>F</math> lie, in that order, on <math>\overline{AF}</math>, dividing it into five segments, each of length 1. Point <math>G</math> is not on line <math>AF</math>. Point <math>H</math> lies on <math>\overline{GD}</math>, and point <math>J</math> lies on <math>\overline{GF}</math>. The line segments <math>\overline{HC}, \overline{JE},</math> and <math>\overline{AG}</math> are parallel. Find <math>HC/JE</math>.
 +
 +
<math>\text{(A)}\ 5/4 \qquad \text{(B)}\ 4/3 \qquad \text{(C)}\ 3/2 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 2</math>
  
 
[[2002 AMC 10A Problems/Problem 20|Solution]]
 
[[2002 AMC 10A Problems/Problem 20|Solution]]
Line 131: Line 190:
 
== Problem 21 ==
 
== Problem 21 ==
  
−
There are 8 integers, whose average, median, unique mode, and range are all 8. Which of the following cannot be the largest of the 8 numbers?
+
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is
  
 
<math>\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15</math>
 
<math>\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15</math>
Line 145: Line 204:
  
 
== Problem 23 ==
 
== Problem 23 ==
 +
Points <math>A,B,C</math> and <math>D</math> lie on a line, in that order, with <math>AB = CD</math> and <math>BC = 12</math>. Point <math>E</math> is not on the line, and <math>BE = CE = 10</math>. The perimeter of <math>\triangle AED</math> is twice the perimeter of <math>\triangle BEC</math>. Find <math>AB</math>.
 +
 +
<math>\text{(A)}\ 15/2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 17/2 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 19/2</math>
  
 
[[2002 AMC 10A Problems/Problem 23|Solution]]
 
[[2002 AMC 10A Problems/Problem 23|Solution]]
Line 150: Line 212:
 
== Problem 24 ==
 
== Problem 24 ==
  
−
Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, ..., 10}. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
+
Tina randomly selects two distinct numbers from the set <math>\{1, 2, 3, 4, 5\}</math>, and Sergio randomly selects a number from the set <math>\{1, 2, ..., 10\}</math>. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is
  
 
<math>\text{(A)}\ 2/5 \qquad \text{(B)}\ 9/20 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 11/20 \qquad \text{(E)}\ 24/25</math>
 
<math>\text{(A)}\ 2/5 \qquad \text{(B)}\ 9/20 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 11/20 \qquad \text{(E)}\ 24/25</math>
Line 157: Line 219:
  
 
== Problem 25 ==
 
== Problem 25 ==
−
In [[trapezoid]] <math>ABCD</math> with bases <math>AB</math> and <math>CD</math>, we have <math>AB = 52</math>, <math>BC = 12</math>, <math>CD = 39</math>, and <math>DA = 5</math>. The area of <math>ABCD</math> is
+
<asy>
 +
pair A,B,C,D;
 +
A=(0,0);
 +
B=(52,0);
 +
C=(38,20);
 +
D=(5,20);
 +
dot(A);
 +
dot(B);
 +
dot(C);
 +
dot(D);
 +
draw(A--B--C--D--cycle);
 +
label("$A$",A,S);
 +
label("$B$",B,S);
 +
label("$C$",C,N);
 +
label("$D$",D,N);
 +
label("52",(A+B)/2,S);
 +
label("39",(C+D)/2,N);
 +
label("12",(B+C)/2,E);
 +
label("5",(D+A)/2,W);
 +
</asy>
 +
In trapezoid <math>ABCD</math> with bases <math>AB</math> and <math>CD</math>, we have <math>AB = 52</math>, <math>BC = 12</math>, <math>CD = 39</math>, and <math>DA = 5</math>. The area of <math>ABCD</math> is
  
 
<math>\text{(A)}\ 182 \qquad \text{(B)}\ 195 \qquad \text{(C)}\ 210 \qquad \text{(D)}\ 234 \qquad \text{(E)}\ 260</math>
 
<math>\text{(A)}\ 182 \qquad \text{(B)}\ 195 \qquad \text{(C)}\ 210 \qquad \text{(D)}\ 234 \qquad \text{(E)}\ 260</math>
Line 164: Line 246:
  
 
== See also ==
 
== See also ==
−
* [[AMC Problems and Solutions]]
+
{{AMC10 box|year=2002|ab=A|before=[[2001 AMC 10 Problems]]|after=[[2002 AMC 10B Problems]]}}
 +
* [[AMC 10]]
 +
* [[AMC 10 Problems and Solutions]]
 +
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Latest revision as of 20:39, 19 July 2024

2002 AMC 10A (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
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

Problem 1

The ratio $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which of the following numbers?

$\text{(A)}\ 0.1 \qquad \text{(B)}\ 0.2 \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 10$

Solution

Problem 2

For the nonzero numbers $a$, $b$, $c$, define $(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}$. Find $(2, 12, 9)$.

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

Solution

Problem 3

According to the standard convention for exponentiation,

$2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65,536$.

If the order in which the exponentiations are performed is changed, how many other values are possible?


$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$

Solution

Problem 4

For how many positive integers $m$ does there exist at least one positive integer $n$ such that $mn \le m + n$?

$\text{(A)}\ 4 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 12 \qquad \text{(E)}$ infinitely many


Solution

Problem 5

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

[asy] unitsize(.3cm); path c=Circle((0,2),1); filldraw(Circle((0,0),3),grey,black); filldraw(Circle((0,0),1),white,black); filldraw(c,white,black); filldraw(rotate(60)*c,white,black); filldraw(rotate(120)*c,white,black); filldraw(rotate(180)*c,white,black); filldraw(rotate(240)*c,white,black); filldraw(rotate(300)*c,white,black); [/asy]

$\text{(A)}\ \pi \qquad \text{(B)}\ 1.5\pi \qquad \text{(C)}\ 2\pi \qquad \text{(D)}\ 3\pi \qquad \text{(E)}\ 3.5\pi$

Solution

Problem 6

Cindy was asked by her teacher to subtract $3$ from a certain number and then divide the result by $9$. Instead, she subtracted $9$ and then divided the result by $3$, giving an answer of $43$. What would her answer have been had she worked the problem correctly?

$\text{(A)}\ 15 \qquad \text{(B)}\ 34 \qquad \text{(C)}\ 43 \qquad \text{(D)}\ 51 \qquad \text{(E)} 138$

Solution

Problem 7

If an arc of $45^\circ$ on circle $A$ has the same length as an arc of $30^\circ$ on circle $B$, then the ratio of the area of circle $A$ to the area of circle $B$ is

$\text{(A)}\ 4/9 \qquad \text{(B)}\ 2/3 \qquad \text{(C)}\ 5/6 \qquad \text{(D)}\ 3/2 \qquad \text{(E)}\ 9/4$

Solution

Problem 8

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?

[asy] unitsize(3mm); fill((-4,-4)--(-4,4)--(4,4)--(4,-4)--cycle,blue); fill((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,red); path onewhite=(-3,3)--(-2,4)--(-1,3)--(-2,2)--(-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle; path divider=(-2,2)--(-3,3)--cycle; fill(onewhite,white); fill(rotate(90)*onewhite,white); fill(rotate(180)*onewhite,white); fill(rotate(270)*onewhite,white); [/asy]

$\text{(A)}\ B = W \qquad \text{(B)}\ W = R \qquad \text{(C)}\ B = R \qquad \text{(D)}\ 3B = 2R \qquad \text{(E)}\ 2R = W$

Solution

Problem 9

There are 3 numbers A, B, and C, such that $1001C - 2002A = 4004$, and $1001B + 3003A = 5005$. What is the average of A, B, and C?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E) }\text{Not uniquely determined}$


Solution

Problem 10

Compute the sum of all the roots of $(2x + 3)(x - 4) + (2x + 3)(x - 6) = 0$.

$\text{(A)}\ 7/2 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 13$

Solution

Problem 11

Jamal wants to store $30$ computer files on floppy disks, each of which has a capacity of $1.44$ megabytes (MB). Three of his files require $0.8$ MB of memory each, $12$ more require $0.7$ MB each, and the remaining $15$ require $0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?

$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 15 \qquad \text{(E)} 16$

Solution

Problem 12

Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages $40$ miles per hour, he arrives at his workplace three minutes late. When he averages $60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?

$\text{(A)}\ 45 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 55 \qquad \text{(E)} 58$

Solution

Problem 13

The sides of a triangle have lengths 15, 20, and 25. Find the length of the shortest altitude.

$\text{(A)}\ 6 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 12.5 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$

Solution

Problem 14

Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. The number of possible values of $k$ is

$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 4 \qquad \text{(E) more than 4}$

Solution

Problem 15

The digits $1$, $2$, $3$, $4$, $5$, $6$, $7$, and $9$ are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?

$\text{(A)}\ 150 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 170 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 190$

Solution

Problem 16

If $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$, then $a + b + c + d$ is

$\text{(A)}\ -5 \qquad \text{(B)}\ -10/3 \qquad \text{(C)}\ -7/3 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 5$

Solution

Problem 17

Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?

$\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 3/8 \qquad \text{(D)}\ 2/5 \qquad \text{(E)} 1/2$

Solution

Problem 18

A $3 \times 3 \times 3$ cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the $3 \times 3 \times 3$ cube is

$\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)} 96$

Solution

Problem 19

Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?

$\text{(A)}\ 2\pi/3 \qquad \text{(B)}\ 2\pi \qquad \text{(C)}\ 5\pi/2 \qquad \text{(D)}\ 8\pi/3 \qquad \text{(E)}\ 3\pi$

Solution

Problem 20

Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$.

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

Solution

Problem 21

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is

$\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$

Solution

Problem 22

A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?

$\text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20$

Solution

Problem 23

Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\triangle AED$ is twice the perimeter of $\triangle BEC$. Find $AB$.

$\text{(A)}\ 15/2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 17/2 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 19/2$

Solution

Problem 24

Tina randomly selects two distinct numbers from the set $\{1, 2, 3, 4, 5\}$, and Sergio randomly selects a number from the set $\{1, 2, ..., 10\}$. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is

$\text{(A)}\ 2/5 \qquad \text{(B)}\ 9/20 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 11/20 \qquad \text{(E)}\ 24/25$

Solution

Problem 25

[asy] pair A,B,C,D; A=(0,0); B=(52,0); C=(38,20); D=(5,20); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--D--cycle); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",D,N); label("52",(A+B)/2,S); label("39",(C+D)/2,N); label("12",(B+C)/2,E); label("5",(D+A)/2,W); [/asy] In trapezoid $ABCD$ with bases $AB$ and $CD$, we have $AB = 52$, $BC = 12$, $CD = 39$, and $DA = 5$. The area of $ABCD$ is

$\text{(A)}\ 182 \qquad \text{(B)}\ 195 \qquad \text{(C)}\ 210 \qquad \text{(D)}\ 234 \qquad \text{(E)}\ 260$

Solution

See also

2002 AMC 10A (Problems • Answer Key • Resources)
Preceded by
2001 AMC 10 Problems 		Followed by
2002 AMC 10B Problems
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All AMC 10 Problems and Solutions

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