Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
GET READY FOR THE AMC 10 WITH AoPS
Learn with outstanding instructors and top-scoring students from around the world in our AMC 10 Problem Series online course.
CHECK SCHEDULE

Difference between revisions of "2010 AMC 10B Problems"
Line 1: Line 1:
−
18.Positive integers a, b, and c are randomly and independently chosen with replacement from the set {1, 2, 3, ..., 2010}. What is the probability that abc+ab+a is divisible by 3?
 
−
(A)1/3 (B)6 (C)4sqrt(3) (D)12 (E)18
 
−
 
−
19. A circle with center O has area 156pi. Triangle ABC is equilateral, BC is a chord on the circle, OA=4sqrt(3), and point O is outside triangle ABC. What is the side length of triangle ABC?
 
−
(A)2sqrt(3) (B)6 (C)4sqrt(3) (D)12 (E)18
 
−
 
−
20. 2 circles lie outside of regular hexagon ABCDEF. The first is tangent to Ab, and the second is tangent to DE. Both are tangent to lines BC and FA. What is the ratio of the area of the second circle to the area of the first circle?
 
−
(A)18 (B)27 (C)36 (D)81 (E)108
 
−
 
−
21.A palindrom between 1000 and 10000 is chosen at random. WHat is the probability that it is divisible by 7?
 
−
(A)1/10 (B)1/9 (C)1/7 (D)1/6 (E)1/5
 
−
 
−
22.Seven distinct pieces of candy are to be stored among 3 bags. The red bad and the blue bag must recieve at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
 
−
(A)1930 (B)1931 (C)1932 (D)1933 (E)1934
 
−
 
−
23.The entries in a 3x3 array include all digits from 1 to 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
 
−
(A)18 (B)24 (C)36 (D)42 (E)60
 
−
 
 
24.A high school b-ball game between the R's and the W's was tied at the end of the first quarter. The number of point the R's scored in each of the four quarters formed an increasing geometric sequence, and the number of points the W's scored in each of the 4 quarters formed an increasing arithmetic sequence. At the end of the 4th quarter, the the R's had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?
 
24.A high school b-ball game between the R's and the W's was tied at the end of the first quarter. The number of point the R's scored in each of the four quarters formed an increasing geometric sequence, and the number of points the W's scored in each of the 4 quarters formed an increasing arithmetic sequence. At the end of the 4th quarter, the the R's had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?
 
(A)30 (B)31 (C)32 (D)33 (E)34
 
(A)30 (B)31 (C)32 (D)33 (E)34
Line 254: Line 236:
 
== Problem 18 ==
 
== Problem 18 ==
  
 +
Positive integers <math>a</math>, <math>b</math>, and <math>c</math> are randomly and independently selected with replacement from the set <math>\{1, 2, 3,\dots, 2010\}</math>. What is the probability that <math>abc + ab + a</math> is divisible by <math>3</math>?
  
−
 
+
<math>\textbf{(A)}\ \dfrac{1}{3} \qquad \textbf{(B)}\ \dfrac{29}{81} \qquad \textbf{(C)}\ \dfrac{31}{81} \qquad \textbf{(D)}\ \dfrac{11}{27} \qquad \textbf{(E)}\ \dfrac{13}{27}</math>
−
<math>
 
−
\mathrm{(A)}\  
 
−
\qquad
 
−
\mathrm{(B)}\  
 
−
\qquad
 
−
\mathrm{(C)}\  
 
−
\qquad
 
−
\mathrm{(D)}\  
 
−
\qquad
 
−
\mathrm{(E)}\  
 
−
</math>
 
  
 
[[2010 AMC 10B Problems/Problem 18|Solution]]
 
[[2010 AMC 10B Problems/Problem 18|Solution]]
Line 272: Line 244:
 
== Problem 19 ==
 
== Problem 19 ==
  
−
 
+
A circle with center <math>O</math> has area <math>156\pi</math>. Triangle <math>ABC</math> is equilateral, <math>\overbar{BC}</math> is a chord on the circle, <math>OA = 4\sqrt{3}</math>, and point <math>O</math> is outside <math>\triangle ABC</math>. What is the side length of <math>\triangle ABC</math>?
  
 
<math>
 
<math>
−
\mathrm{(A)}\  
+
\mathrm{(A)}\ 2\sqrt{3}
 
\qquad
 
\qquad
−
\mathrm{(B)}\  
+
\mathrm{(B)}\ 6
 
\qquad
 
\qquad
−
\mathrm{(C)}\  
+
\mathrm{(C)}\ 4\sqrt{3}
 
\qquad
 
\qquad
−
\mathrm{(D)}\  
+
\mathrm{(D)}\ 12
 
\qquad
 
\qquad
−
\mathrm{(E)}\  
+
\mathrm{(E)}\ 18
 
</math>
 
</math>
  
Line 290: Line 262:
 
== Problem 20 ==
 
== Problem 20 ==
  
 +
Two circles lie outside regular hexagon <math>ABCDEF</math>. The first is tangent to <math>\overbar{AB}</math>, and the second is tangent to <math>\overbar{DE}</math>. Both are tangent to lines <math>BC</math> and <math>FA</math>. What is the ratio of the area of the second circle to that of the first circle?
  
 
<math>
 
<math>
−
\mathrm{(A)}\  
+
\mathrm{(A)}\ 18
 
\qquad
 
\qquad
−
\mathrm{(B)}\  
+
\mathrm{(B)}\ 27
 
\qquad
 
\qquad
−
\mathrm{(C)}\  
+
\mathrm{(C)}\ 36
 
\qquad
 
\qquad
−
\mathrm{(D)}\  
+
\mathrm{(D)}\ 81
 
\qquad
 
\qquad
−
\mathrm{(E)}\  
+
\mathrm{(E)}\ 108
 
</math>
 
</math>
  
Line 308: Line 281:
  
  
 +
A palindrome between <math>1000</math> and <math>10,000</math> is chosen at random. What is the probability that it is divisible by <math>7</math>?
  
−
<math>
+
<math>\textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6} \qquad \textbf{(E)}\ \dfrac{1}{5}</math>
−
\mathrm{(A)}\  
 
−
\qquad
 
−
\mathrm{(B)}\  
 
−
\qquad
 
−
\mathrm{(C)}\  
 
−
\qquad
 
−
\mathrm{(D)}\  
 
−
\qquad
 
−
\mathrm{(E)}\  
 
−
</math>
 
  
 
[[2010 AMC 10B Problems/Problem 21|Solution]]
 
[[2010 AMC 10B Problems/Problem 21|Solution]]
Line 325: Line 289:
 
== Problem 22 ==
 
== Problem 22 ==
  
 +
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
  
 
<math>
 
<math>
−
\mathrm{(A)}\  
+
\mathrm{(A)}\ 1930
 
\qquad
 
\qquad
−
\mathrm{(B)}\  
+
\mathrm{(B)}\ 1931
 
\qquad
 
\qquad
−
\mathrm{(C)}\  
+
\mathrm{(C)}\ 1932
 
\qquad
 
\qquad
−
\mathrm{(D)}\  
+
\mathrm{(D)}\ 1933
 
\qquad
 
\qquad
−
\mathrm{(E)}\  
+
\mathrm{(E)}\ 1934
 
</math>
 
</math>
  
Line 341: Line 306:
  
 
== Problem 23 ==
 
== Problem 23 ==
 +
The entries in a <math>3 \times 3</math> array include all the digits from <math>1</math> through <math>9</math>, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
  
−
 
+
<math>\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60</math>
−
<math>
 
−
\mathrm{(A)}\  
 
−
\qquad
 
−
\mathrm{(B)}\  
 
−
\qquad
 
−
\mathrm{(C)}\  
 
−
\qquad
 
−
\mathrm{(D)}\  
 
−
\qquad
 
−
\mathrm{(E)}\  
 
−
 
 
−
</math>
 
  
 
[[2010 AMC 10B Problems/Problem 23|Solution]]
 
[[2010 AMC 10B Problems/Problem 23|Solution]]

Revision as of 12:28, 2 April 2010

24.A high school b-ball game between the R's and the W's was tied at the end of the first quarter. The number of point the R's scored in each of the four quarters formed an increasing geometric sequence, and the number of points the W's scored in each of the 4 quarters formed an increasing arithmetic sequence. At the end of the 4th quarter, the the R's had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half? (A)30 (B)31 (C)32 (D)33 (E)34

25. Let a>0, and let P(x) be a polynomial with integer coefficients such that: P(1)=P(3)=P(5)=P(7)=a, and P(2)=P(4)=P(6)=P(8)=-a. What is the smallest possible value of a? (A)105 (B)315 (C)845 (D)7! (E)8!


Problem 1

1. What is $100(100-3)-(100\times100-3)$?

$\mathrm{(A)}\ -20,000 \qquad \mathrm{(B)}\ -10,000 \qquad \mathrm{(C)}\ -297 \qquad \mathrm{(D)}\ -6 \qquad \mathrm{(E)}\ 0$


Solution

Problem 2

Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?

$\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$


Solution

Problem 3

A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?

$\mathrm{(A)}\ 3 \qquad \mathrm{(B)}\ 4 \qquad \mathrm{(C)}\ 5 \qquad \mathrm{(D)}\ 8 \qquad \mathrm{(E)}\ 9$

Solution

Problem 4

For a real number $x$, define $\heartsuit(x)$ to be the average of $x$ and $x^2$. What is $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$?

$\mathrm{(A)}\ 3 \qquad \mathrm{(B)}\ 6 \qquad \mathrm{(C)}\ 10 \qquad \mathrm{(D)}\ 12 \qquad \mathrm{(E)}\ 20$

Solution

Problem 5

A month with $31$ days has the same number of Mondays and Wednesdays.How many of the seven days of the week could be the first day of this month?

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

Solution

Problem 6

A circle is centered at $O$, $\overbar{AB}$ (Error compiling LaTeX. Unknown error_msg) is a diameter and $C$ is a point on the circle with $\angle COB = 50^\circ$. What is the degree measure of $\angle CAB$?

$\mathrm{(A)}\ 20 \qquad \mathrm{(B)}\ 25 \qquad \mathrm{(C)}\ 45 \qquad \mathrm{(D)}\ 50 \qquad \mathrm{(E)}\ 65$

Solution

Problem 7

A triangle has side lengths $10$, $10$, and $12$. A rectangle has width $4$ and area equal to the area of the rectangle. What is the perimeter of this rectangle?

$\mathrm{(A)}\ 16 \qquad \mathrm{(B)}\ 24 \qquad \mathrm{(C)}\ 28 \qquad \mathrm{(D)}\ 32 \qquad \mathrm{(E)}\ 36$

Solution

Problem 8

A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $$48$, and a group of 10th graders buys tickets costing a total of $$64$. How many values for $x$ are possible?

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

Solution

Problem 9

Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parenthese but added and subtracted correctly and obtained the correct result by coincidence. The number Larry sustitued for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitude for $e$?

$\textbf{(A)}\ -5 \qquad \textbf{(B)}\ -3 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 5$

Solution

Problem 10

Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain?

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 30$

Solution

Problem 11

A shopper plans to purchase an item that has a listed price greater than $$100$ and can use any one of the three coupns. Coupon A gives $15\%$ off the listed price, Coupon B gives $$30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds $$100$.
Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y$ − $x$?

$\mathrm{(A)}\ 50 \qquad \mathrm{(B)}\ 60 \qquad \mathrm{(C)}\ 75 \qquad \mathrm{(D)}\ 80 \qquad \mathrm{(E)}\ 100$

Solution

Problem 12

At the beginning of the school year, $50\%$ of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answerws "No." Altogether, $x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x$?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80$

Solution

Problem 13

What is the sum of all the solutions of $x = \left|2x-|60-2x|\right|$?

$\mathrm{(A)}\ 32 \qquad \mathrm{(B)}\ 60 \qquad \mathrm{(C)}\ 92 \qquad \mathrm{(D)}\ 120 \qquad \mathrm{(E)}\ 124$

Solution

Problem 14

The average of the numbers $1, 2, 3,\cdots, 98, 99,$ and $x$ is $100x$. What is $x$?

$\textbf{(A)}\ \dfrac{49}{101} \qquad \textbf{(B)}\ \dfrac{50}{101} \qquad \textbf{(C)}\ \dfrac{1}{2} \qquad \textbf{(D)}\ \dfrac{51}{101} \qquad \textbf{(E)}\ \dfrac{50}{99}$

Solution

Problem 15

On a $50$-question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99$. What is the maximum number of questions that Jesse could have answered correctly?

$\mathrm{(A)}\ 25 \qquad \mathrm{(B)}\ 27 \qquad \mathrm{(C)}\ 29 \qquad \mathrm{(D)}\ 31 \qquad \mathrm{(E)}\ 33$

Solution

Problem 16

A square of side length $1$ and a circle of radius $\dfrac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?

$\mathrm{(A)}\ \dfrac{\pi}{3}-1 \qquad \mathrm{(B)}\ \dfrac{2\pi}{9}-\dfrac{\sqrt{3}}{3} \qquad \mathrm{(C)}\ \dfrac{\pi}{18} \qquad \mathrm{(D)}\ \dfrac{1}{4} \qquad \mathrm{(E)}\ \dfrac{2\pi}{9}$

Solution

Problem 17

Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city?

$\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 26$

Solution

Problem 18

Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. What is the probability that $abc + ab + a$ is divisible by $3$?

$\textbf{(A)}\ \dfrac{1}{3} \qquad \textbf{(B)}\ \dfrac{29}{81} \qquad \textbf{(C)}\ \dfrac{31}{81} \qquad \textbf{(D)}\ \dfrac{11}{27} \qquad \textbf{(E)}\ \dfrac{13}{27}$

Solution

Problem 19

A circle with center $O$ has area $156\pi$. Triangle $ABC$ is equilateral, $\overbar{BC}$ (Error compiling LaTeX. Unknown error_msg) is a chord on the circle, $OA = 4\sqrt{3}$, and point $O$ is outside $\triangle ABC$. What is the side length of $\triangle ABC$?

$\mathrm{(A)}\ 2\sqrt{3} \qquad \mathrm{(B)}\ 6 \qquad \mathrm{(C)}\ 4\sqrt{3} \qquad \mathrm{(D)}\ 12 \qquad \mathrm{(E)}\ 18$

Solution

Problem 20

Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\overbar{AB}$ (Error compiling LaTeX. Unknown error_msg), and the second is tangent to $\overbar{DE}$ (Error compiling LaTeX. Unknown error_msg). Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle?

$\mathrm{(A)}\ 18 \qquad \mathrm{(B)}\ 27 \qquad \mathrm{(C)}\ 36 \qquad \mathrm{(D)}\ 81 \qquad \mathrm{(E)}\ 108$

Solution

Problem 21

A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?

$\textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6} \qquad \textbf{(E)}\ \dfrac{1}{5}$

Solution

Problem 22

Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?

$\mathrm{(A)}\ 1930 \qquad \mathrm{(B)}\ 1931 \qquad \mathrm{(C)}\ 1932 \qquad \mathrm{(D)}\ 1933 \qquad \mathrm{(E)}\ 1934$

Solution

Problem 23

The entries in a $3 \times 3$ array include all the digits from $1$ through $9$, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60$

Solution

Problem 24

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 25

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_10B_Problems&oldid=34140"