Difference between revisions of "2010 AMC 10B Problems"

Latest revision as of 17:33, 27 August 2024

2010 AMC 10B (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions 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. 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. 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). Figures are not necessarily drawn to scale. 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

What is $100(100-3)-(100\cdot100-3)$ ?

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

Solution

Problem 2

Makayla 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?

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

Solution

Problem 4

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

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

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 $O$ , $\overline{AB}$ is a diameter and $C$ is a point on the circle with $\angle COB = 50^\circ$ . What is the degree measure of $\angle CAB$ ?

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

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 triangle. What is the perimeter of this rectangle?

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

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 $\textdollar 48$ , and a group of 10th graders buys tickets costing a total of $\textdollar 64$ . 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 parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$ , $b$ , $c$ , and $d$ were $1$ , $2$ , $3$ , and $4$ , respectively. What number did Larry substitute 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 $\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\textdollar 30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds $\textdollar 100$ .
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 Coupon C. What is $y - x$ ?

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

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 $30\%$ answered "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 $x = \left|2x-|60-2x|\right|$ ?

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

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?

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

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?

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

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, $\overline{BC}$ is a chord on the circle, $OA = 4\sqrt{3}$ , and point $O$ is outside $\triangle ABC$ . What is the side length of $\triangle ABC$ ?

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

Solution

Problem 20

Two circles lie outside regular hexagon $ABCDEF$ . The first is tangent to $\overline{AB}$ , and the second is tangent to $\overline{DE}$ . 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?

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

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?

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

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

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders 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?

$\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

Solution

Problem 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 $a$ ?

$\textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

Solution

2010 AMC 10B (Problems • Answer Key • Resources)
Preceded by 2010 AMC 10A Problems	Followed by 2011 AMC 10A Problems
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Difference between revisions of "2010 AMC 10B Problems"

Latest revision as of 17:33, 27 August 2024

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

See also

@@ Line 1: / Line 1: @@
-Note: Excuse me for my notation, I hope someone can put these into correct format.
+{{AMC10 Problems|year=2010|ab=B}}
-. What is 100(100-3)-(100*100-3)?
-(A)-20,000 (B)-10,000 (C) -297 (D)-6 (E)0
-.Makayla 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?
-(A)15 (B)20 (C)25 (D)30 (E)35
-. 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?
-. For a real number x, define #(x) to be the average of x and x^2. What is #(1)+#(2)+#(3)?
-(A)3 (B)6 (C)10 (D)12 (E)20
-. 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?
-(A)2 (B)3 (C)4 (D)5 (E)6
-.A circle is centered at O, AB is a diameter and C is a point on the circle with angle COB=50 (degrees). What is the degree measure of CAB?
-(A)20 (B)25 (C)45 (D)50 (E)65
-.A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this triangle?
-(A)16 (B)24 (C)28(D)32 (E)36
-. A ticket to a school play costs x dollars, where x is a whole number. A group of 9th graders buys tickets costing a total of <math>48, and a group of 10th graders buys tickets costing a total of </math>64. How many values for x are possible?
-(A)1 (B)2 (C)3 (D)4 (E)5
-.Lucky Larry's teacher asked him to substitute numbers for a, b, c, d, and e in the expression a-(b-(c-(d+c))) and evaluate the result. Larry ignored the parenthesis but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for a,b,c, and d were 1,2,3, and 4, respectively. What number did Larry substitute for e?
-(A)-5 (B)-3 (C)0 (D)3 (E)5
-.Shelby drives her scooter at a speed of 30 miles per hour when it is not raining, and 20 miles per hour when 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?
-(A)18 (B)21 (C)24 (D) 27(E)30
-. A shopper plans to purchase an item that has a listed price greater than 100 dollars and can use any one of three coupons. Coupon A gives 15 percent of the listed price, B gives 30 dollars of the listed price, and C gives 25 percent of the amount by which the listed price exceeds 100 dollars.
-Let x and y by the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as B or C. What is y-x?
-(A)50 (B)60 (C)75 (D80 (E)100
-.At the beginning of the school year, 50 percent of all students in Mr. Well's class answered "yes" to the question "Do you like math?" and 50 percent answered "No". At the end of the school year, 70 percent answered "yes" and 30 percent answered "No". Altogether, x% of the students gave a different answer at the beginning and the end of the school year. What is the difference between the maximum and minimum values for x?
-(A)0 (B20(C)40 (D)60 (E)80
-. What is the sum of all the solutions of x=|2x-|60-2x||?
-(A)32 (B)60 (C)92 (D)120 (E)124
-.The average of the numbers 1, 2, 3, ...98, 99, and x is 100x.What is x?
-(A)49/101 (B)50/101 (C)1/2 (D)51/101 (E)50/99
-.On a 50-question multiple choice contest, students recieve 4 points for a correct answer, 0 points for 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 she could have answered correctly?
-(A)25 (B)27 (C)29 (D)31 (E)33
-. A square of side length 1 and a circle of radius sqrt(3)/3 share the same center. What is the area inside the circle, but outside the square?
-(A)(pi/3)-1 (B)(2pi/9)-sqrt(3)/3 (C)pi/18 (D)1/4 (E)2pi/9
-.Every high school in a city sent a team of 3 students to a math contest. Each participant in the contest recieved a different score. Andrea's score was the median, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city?
-(A)22 (B)23 (C)24 (D)25 (E)26
-.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
-. 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
-. 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
-.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
-.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
-.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
-.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
-. 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 ==
-. What is <math>100(100-3)-(100\times100-3)</math>?
+What is <math>100(100-3)-(100\cdot100-3)</math>?
 <math>
-\mathrm{(A)}\ -20,000
+\textbf{(A)}\ -20,000
 \qquad
-\mathrm{(B)}\ -10,000
+\textbf{(B)}\ -10,000
 \qquad
-\mathrm{(C)}\ -297
+\textbf{(C)}\ -297
 \qquad
-\mathrm{(D)}\ -6
+\textbf{(D)}\ -6
 \qquad
-\mathrm{(E)}\ 0
+\textbf{(E)}\ 0
 </math>
@@ Line 97: / Line 20: @@
 == Problem 2 ==
+Makayla attended two meetings during her <math>9</math>-hour work day. The first meeting took <math>45</math> minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
-<math>
+<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35</math>
-\mathrm{(A)}\
-\qquad
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 2|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?
 <math>
-\mathrm{(A)}\
+\textbf{(A)}\ 3
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 4
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 5
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 8
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 9
 </math>
@@ Line 129: / Line 46: @@
 == Problem 4 ==
+For a real number <math>x</math>, define <math>\heartsuit(x)</math> to be the average of <math>x</math> and <math>x^2</math>. What is <math>\heartsuit(1)+\heartsuit(2)+\heartsuit(3)</math>?
 <math>
-\mathrm{(A)}\
+\textbf{(A)}\ 3
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 6
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 10
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 12
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 20
 </math>
@@ Line 148: / Line 65: @@
-<math>
+A month with <math>31</math> 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?
-\mathrm{(A)}\
-\qquad
+<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math>
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 5|Solution]]
@@ Line 164: / Line 73: @@
 == Problem 6 ==
+A circle is centered at <math>O</math>, <math>\overline{AB}</math> is a diameter and <math>C</math> is a point on the circle with <math>\angle COB = 50^\circ</math>.
+What is the degree measure of <math>\angle CAB</math>?
 <math>
-\mathrm{(A)}\
+\textbf{(A)}\ 20
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 25
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 45
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 50
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 65
 </math>
@@ Line 181: / Line 92: @@
 == Problem 7 ==
+A triangle has side lengths <math>10</math>, <math>10</math>, and <math>12</math>. A rectangle has width <math>4</math> and area equal to the
+area of the triangle. What is the perimeter of this rectangle?
 <math>
-\mathrm{(A)}\
+\textbf{(A)}\ 16
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 24
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 28
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 32
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 36
 </math>
@@ Line 198: / Line 111: @@
 == Problem 8 ==
+A ticket to a school play cost <math>x</math> dollars, where <math>x</math> is a whole number. A group of 9th graders buys tickets costing a total of <math>\textdollar 48</math>, and a group of 10th graders buys tickets costing a total of <math>\textdollar 64</math>. How many values for <math>x</math> are possible?
-<math>
+<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math>
-\mathrm{(A)}\
-\qquad
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 8|Solution]]
@@ Line 215: / Line 119: @@
 == Problem 9 ==
+Lucky Larry's teacher asked him to substitute numbers for <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, and <math>e</math> in the expression <math>a-(b-(c-(d+e)))</math> and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> were <math>1</math>, <math>2</math>, <math>3</math>, and <math>4</math>, respectively. What number did Larry substitute for <math>e</math>?
-<math>
+<math>\textbf{(A)}\ -5 \qquad \textbf{(B)}\ -3 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 5</math>
-\mathrm{(A)}\
-\qquad
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 9|Solution]]
@@ Line 232: / Line 127: @@
 == Problem 10 ==
+Shelby drives her scooter at a speed of <math>30</math> miles per hour if it is not raining, and <math>20</math> 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 <math>16</math> miles in <math>40</math> minutes. How many minutes did she drive in the rain?
-<math>
+<math>\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 30</math>
-\mathrm{(A)}\
-\qquad
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 10|Solution]]
 == Problem 11 ==
+A shopper plans to purchase an item that has a listed price greater than <math>\textdollar 100</math> and can use any one of the three coupons. Coupon A gives <math>15\%</math> off the listed price, Coupon B gives <math>\textdollar 30</math> off the listed price, and Coupon C gives <math>25\%</math> off the amount by which the listed price exceeds
+<math>\textdollar 100</math>. <br/>
+Let <math>x</math> and <math>y</math> be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or Coupon C. What is <math>y - x</math>?
 <math>
-\mathrm{(A)}\
+\textbf{(A)}\ 50
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 60
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 75
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 80
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 100
 </math>
@@ Line 266: / Line 154: @@
 == Problem 12 ==
+At the beginning of the school year, <math>50\%</math> of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and <math>50\%</math> answered "No." At the end of the school year, <math>70\%</math> answered "Yes" and <math>30\%</math> answered "No." Altogether, <math>x\%</math> 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 <math>x</math>?
-<math>
+<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80</math>
-\mathrm{(A)}\
-\qquad
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 12|Solution]]
 == Problem 13 ==
+What is the sum of all the solutions of <math>x = \left|2x-|60-2x|\right|</math>?
 <math>
-\mathrm{(A)}\
+\textbf{(A)}\ 32
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 60
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 92
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 120
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 124
 </math>
@@ Line 300: / Line 179: @@
 == Problem 14 ==
+The average of the numbers <math>1, 2, 3,\cdots, 98, 99,</math> and <math>x</math> is <math>100x</math>. What is <math>x</math>?
-<math>
+<math>\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}</math>
-\mathrm{(A)}\
-\qquad
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 14|Solution]]
 == Problem 15 ==
+On a <math>50</math>-question multiple choice math contest, students receive <math>4</math> points for a correct answer, <math>0</math> points for an answer left blank, and <math>-1</math> point for an incorrect answer. Jesse’s total score on the contest was <math>99</math>. What is the maximum number of questions that Jesse could have answered correctly?
 <math>
-\mathrm{(A)}\
+\textbf{(A)}\ 25
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 27
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 29
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 31
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 33
 </math>
@@ Line 333: / Line 203: @@
 == Problem 16 ==
+A square of side length <math>1</math> and a circle of radius <math>\dfrac{\sqrt{3}}{3}</math> share the same center. What is the area inside the circle, but outside the square?
 <math>
-\mathrm{(A)}\
+\textbf{(A)}\ \dfrac{\pi}{3}-1
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ \dfrac{2\pi}{9}-\dfrac{\sqrt{3}}{3}
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ \dfrac{\pi}{18}
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ \dfrac{1}{4}
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ \dfrac{2\pi}{9}
 </math>
@@ Line 350: / Line 220: @@
 == Problem 17 ==
+Every high school in the city of Euclid sent a team of <math>3</math> 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 <math>37</math>th and <math>64</math>th, respectively. How many schools are in the city?
+<math>\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 26</math>
-<math>
-\mathrm{(A)}\
-\qquad
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 17|Solution]]
@@ Line 368: / Line 228: @@
 == 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]]
@@ Line 386: / Line 236: @@
 == Problem 19 ==
+A circle with center <math>O</math> has area <math>156\pi</math>. Triangle <math>ABC</math> is equilateral, <math>\overline{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>
-\mathrm{(A)}\
+\textbf{(A)}\ 2\sqrt{3}
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 6
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 4\sqrt{3}
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 12
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 18
 </math>
@@ Line 404: / Line 254: @@
 == Problem 20 ==
+Two circles lie outside regular hexagon <math>ABCDEF</math>. The first is tangent to <math>\overline{AB}</math>, and the second is tangent to <math>\overline{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>
-\mathrm{(A)}\
+\textbf{(A)}\ 18
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 27
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 36
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 81
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 108
 </math>
@@ Line 422: / Line 273: @@
+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]]
@@ Line 439: / Line 281: @@
 == 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>
-\mathrm{(A)}\
+\textbf{(A)}\ 1930
 \qquad
-\mathrm{(B)}\
+\textbf{(B)}\ 1931
 \qquad
-\mathrm{(C)}\
+\textbf{(C)}\ 1932
 \qquad
-\mathrm{(D)}\
+\textbf{(D)}\ 1933
 \qquad
-\mathrm{(E)}\
+\textbf{(E)}\ 1934
 </math>
@@ Line 455: / Line 298: @@
 == 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]]
@@ Line 474: / Line 306: @@
 == Problem 24 ==
+A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than <math>100</math> points. What was the total number of points scored by the two teams in the first half?
-<math>
+<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34</math>
-\mathrm{(A)}\
-\qquad
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 24|Solution]]
@@ Line 491: / Line 314: @@
 == Problem 25 ==
+Let <math>a > 0</math>, and let <math>P(x)</math> be a polynomial with integer coefficients such that
+<center>
+<math>P(1) = P(3) = P(5) = P(7) = a</math>, and<br/>
+<math>P(2) = P(4) = P(6) = P(8) = -a</math>.
+</center>
+What is the smallest possible value of <math>a</math>?
-<math>
+<math>\textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!</math>
-\mathrm{(A)}\
-\qquad
-\mathrm{(B)}\
-\qquad
-\mathrm{(C)}\
-\qquad
-\mathrm{(D)}\
-\qquad
-\mathrm{(E)}\
-</math>
 [[2010 AMC 10B Problems/Problem 25|Solution]]
+==See also==
+{{AMC10 box|year=2010|ab=B|before=[[2010 AMC 10A Problems]]|after=[[2011 AMC 10A Problems]]}}
+* [[AMC 10]]
+* [[AMC 10 Problems and Solutions]]
+* [[2010 AMC 10B]]
+* [[Mathematics competition resources]]
+{{MAA Notice}}