AMC 10 Algebra Previous Year Questions - Year wise

Get rolling on your preparation for AMC 10 with Cheenta. This post has all the AMC 10 Algebra previous year Questions, year-wise. Try out these problems:

AMC 10A, 2021, Problem 1

What is the value of

$\left(2^{2}-2\right)-\left(3^{2}-3\right)+\left(4^{2}-4\right)$

(A) 1
(B) 2
(C) 5
(D) 8
(E) 12

AMC 10A, 2021, Problem 2

Portia's high school has 3 times as many students as Lara's high school. The two high schools have a total of 2600 students. How many students does Portia's high school have?
(A) 600
(B) 650
(C) 1950
(D) 2000
(E) 2050

AMC 10A, 2021, Problem 3

The sum of two natural numbers is 17,402 . One of the two numbers is divisible by 10 . If the units digit of that number is erased, the other number is obtained.

What is the difference of these two numbers?
(A) 10,272
(B) 11,700
(C) 13,362
(D) 14,238
(E) 15,426

AMC 10A, 2021, Problem 4

A cart rolls down a hill, travelling 5 inches the first second and accelerating so that during each successive 1 -second time interval, it travels inches more than during the previous 1 -second interval.

The cart takes 30 seconds to reach the bottom of the hill. How far, in inches, does it travel?
(A) 215
(B) 360
(C) 2992
(D) 3195
(E) 3242

AMC 10A, 2021, Problem 5

The quiz scores of a class with $k>12$ students have a mean of 8 . The mean of a collection of 12 of these quiz scores is 14 .

What is the mean of the remaining quiz scores in terms of $k$ ?
(A) $\frac{14-8}{k-12}$
(B) $\frac{8 k-168}{k-12}$
(C) $\frac{14}{12}-\frac{8}{k}$
(D) $\frac{14(k-12)}{k^{2}}$
(E) $\frac{14(k-12)}{8 k}$

AMC 10A, 2021, Problem 6

Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking a 4 miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to 2 miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at 3 miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?

(A) $\frac{12}{13}$
(B) $1$
(C) $\frac{13}{12}$
(D) $\frac{24}{13}$
(E) $2$

AMC 10B, 2021, Problem 2

What is the value of $\sqrt{(3-2 \sqrt{3})^{2}}+\sqrt{(3+2 \sqrt{3})^{2}}$ ?
(A) 0
(B) $4 \sqrt{3}-6$
(C) 6
(D) $4 \sqrt{3}$
(E) $4 \sqrt{3}+6$

AMC 10B, 2021, Problem 15

The real number $x$ satisfies the equation $x+\frac{1}{x}=\sqrt{5}$ .

What is the value of $x^{11}-7 x^{7}+x^{3} ?$
(A) $-1$
(B) $0$
(C) $1$
(D) $2$
(E) $\sqrt{5}$

AMC 10A, 2020, Problem 22

Hiram's algebra notes are 50 pages long and are printed on 25 sheets of paper; the first sheet contains pages 1 and 2 , the second sheet contains pages 3 and 4 , and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly 19 . How many sheets were borrowed?
(A) 10
(B) 13
(C) 15
(D) 17
(E) 20

AMC 10A, 2020, Problem 1

What value of $x$ satisfies $x-\frac{3}{4}=\frac{5}{12}-\frac{1}{3} ?$

(A) $-\frac{2}{3}$
(B) $\frac{7}{36}$
(C) $\frac{7}{12}$
(D) $\frac{2}{3}$
(E) $\frac{5}{6}$

AMC 10A, 2020, Problem 2

The numbers $3,5,7, a$ , and $b$ have an average (arithmetic mean) of $15$ .

What is the average of $a$ and $b$ ?
(A) 0
(B) 15
(C) 30
(D) 45
(E) 60

AMC 10A, 2020, Problem 3

Assuming $a\neq3$ , $b\neq4$ , and $c\neq5$ , what is the value in simplest form of the following expression?

$\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}$

(A) $-1$
(B) 1
(C) $\frac{a b c}{60}$
(D) $\frac{1}{a b c}-\frac{1}{60}$
(E) $\frac{1}{60}-\frac{1}{a b c}$

AMC 10A, 2020, Problem 5

What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$

(A) $12$
(B) $15$
(C) $18$
(D) $21$
(E) $25$

AMC 10A, 2020, Problem 14

Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$ .

What is the value of $x+\frac{x^{3}}{y^{2}}+\frac{y^{3}}{x^{2}}+y ?$

(A) $360$
(B) $400$
(C) $420$
(D) $440$
(E) $480$

AMC 10B, 2020, Problem 1

What is the value of

1 - (- 2) - 3 - (- 4) - 5 - (- 6) ?

$1-(-2)-3-(-4)-5-(-6) ?$

(A) $-20$
(B) $-3$
(C) $3$
(D) $5$
(E) $21$

AMC 10B, 2020, Problem 3

The ratio of $w$ to $x$ is $4: 3$ , the ratio of $y$ to $z$ is $3: 2$ ,

and the ratio of $z$ to $x$ is $1: 6$ . What is the ratio of $w$ to $y$ ?
(A) $4: 3$
(B) $3: 2$
(C) $8: 3$
(D) $4: 1$
(E) $16: 3$

AMC 10B, 2020, Problem 6

Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome-it reads the same forward and backward. Then 2 hours later, the odometer displayed the next higher palindrome.

What was her average speed, in miles per hour, during this 2 hour period?
(A) 50
(B) 55
(C) 60
(D) 65
(E) 70

AMC 10B, 2020, Problem 7

How many positive even multiples of 3 less than 2020 are perfect squares?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 12

AMC 10B, 2020, Problem 9

How many ordered pairs of integers $(x, y)$ satisfy the equation

x 2020 + y 2 = 2 y ?

$x^{2020}+y^{2}=2 y ?$

(A) 1
(B) 2
(C) 3
(D) 4
(E) infinitely many

AMC 10B, 2020, Problem 12

The decimal representation of

1 20 2

$\frac{1}{20^{2}}$

consists of a string of zeros after the decimal point, followed by a 9 and then several more digits.

How many zeros are in that initial string of zeros after the decimal point?
(A) 23
(B) 24
(C) 25
(D) 26
(E) 27

AMC 10B, 2020, Problem 22

What is the remainder when $2^{202}+202$ is divided by $2^{101}+2^{51}+1$ ?
(A) 100
(B) 101
(C) 200
(D) 201
(E) 202

AMC 10A, 2019, Problem 1

What is the value of

${ }_{2}^{\left(0^{\left(1^{9}\right)}\right)}+\left(\left(2^{0}\right)^{1}\right)^{9} ?$

(A) $0$
(B) $1$
(C) $2$
(D) $3$
(E) $4$

AMC 10A, 2019, Problem 2

What is the hundreds digit of $(20 !-15 !) ?$
(A) $0$
(B) $1$
(C) $2$
(D) $4$
(E) $5$

AMC 10A, 2019, Problem 3

Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was 5 times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n$ ?
(A) 3
(B) 5
(C) 9
(D) 12
(E) 15

AMC 10A, 2019, Problem 24

Let $p, q$ , and $r$ be the distinct roots of the polynomial $x^{3}-22 x^{2}+80 x-67$ . It is given that there exist real numbers $A, B$ , and $C$ such that

$\frac{1}{s^{3}-22 s^{2}+80 s-67}=\frac{A}{s-p}+\frac{B}{s-q}+\frac{C}{s-r}$

for all $s \notin{p, q, r}$ . What is $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$ ?

(A) $243$
(B) $244$
(C) $245$
(D) $246$
(E) $247$

AMC 10B, 2019, Problem 1

Alicia had two containers. The first was $\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?
(A) $\frac{5}{8}$
(B) $\frac{4}{5}$
(C) $\frac{7}{8}$
(D) $\frac{9}{10}$
(E) $\frac{11}{12}$

AMC 10B, 2019, Problem 2

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
(A) 11
(B) 15
(C) 19
(D) 21
(E) 27

AMC 10B, 2019, Problem 3

In a high school with 500 students, $40 \%$ of the seniors play a musical instrument, while $30 \%$ of the non-seniors do not play a musical instrument. In all, $46.8 \%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?
(A) 66
(B) 154
(C) 186
(D) 220
(E) 266

AMC 10B, 2019, Problem 4

All lines with equation $a x+b y=c$ such that $a, b, c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?
(A) $(-1,2)$
(B) $(0,1)$
(C) $(1,-2)$
(D) $(1,0)$
(E) $(1,2)$

AMC 10B, 2019, Problem 11

Two jars each contain the same number of marbles, and every marble is either blue or green.

In Jar 1 the ratio of blue to green marbles is $9: 1$ , and the ratio of blue to green marbles in Jar 2 is $8: 1$ . There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar $2 ?$
(A) 5
(B) 10
(C) 25
(D) 45
(E) 50

AMC 10B, 2019, Problem 13

What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17$ , and $x$ is equal to the mean of those five numbers?
(A) $-5$
(B) 0
(C) 5
(D) $\frac{15}{4}$
(E) $\frac{35}{4}$

AMC 10B, 2019, Problem 18

Henry decides one morning to do a workout, and he walks $\frac{3}{4}$ of the way from his home to his gym. The gym is 2 kilometers away from Henry's home. At that point, he changes his mind and walks $\frac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\frac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\frac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$ ?
(A) $\frac{2}{3}$
(B) 1
(C) $\frac{6}{5}$
(D) $\frac{5}{4}$
(E) $\frac{3}{2}$

AMC 10A, 2018, Problem 1

What is the value of

(((2 + 1) - 1 + 1) - 1 + 1) - 1 + 1 ?

$\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1 ?$

(A) $\frac{5}{8}$
(B) $\frac{11}{7}$
(C) $\frac{8}{5}$
(D) $\frac{18}{11}$
(E) $\frac{15}{8}$

AMC 10A, 2018, Problem 2

Liliane has $50 \%$ more soda than Jacqueline, and Alice has $25 \%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?
(A) Liliane has $20 \%$ more soda than Alice.
(B) Liliane has $25 \%$ more soda than Alice.
(C) Liliane has $45 \%$ more soda than Alice.
(D) Liliane has $75 \%$ more soda than Alice.
(E) Liliane has $100 \%$ more soda than Alice.

AMC 10A, 2018, Problem 3

A unit of blood expires after $10 !=10 \cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January $1 .$ On what day does his unit of blood expire?
(A) January 2
(B) January 12
(C) January 22
(D) February 11
(E) February 12

AMC 10A, 2018, Problem 6

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0 , and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90 , and that $65 \%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
(A) 200
(B) 300
(C) 400
(D) 500
(E) 600

AMC 10A, 2018, Problem 8

Joe has a collection of 23 coins, consisting of 5 -cent coins, 10 -cent coins, and 25 -cent coins. He has 3 more 10 -cent coins than 5 -cent coins and the total value of his collection is 320 cents. How many more 25 -cent coins does Joe have than 5 -cent coins?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

AMC 10A, 2018, Problem 10

Suppose that real number $x$ satisfies

49 - x 2 - - - - - - \sqrt - 25 - x 2 - - - - - - \sqrt = 3

$\sqrt{49-x^{2}}-\sqrt{25-x^{2}}=3$

What is the value of $\sqrt{49-x^{2}}+\sqrt{25-x^{2}}$ ?
(A) 8
(B) $\sqrt{33}+8$
(C) 9
(D) $2 \sqrt{10}+4$
(E) 12

AMC 10B, 2018, Problem 1

Kate bakes a 20-inch by 18 -inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?
(A) 90
(B) 100
(C) 180
(D) 200
(E) 360

AMC 10B, 2018, Problem 2

Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was $60 \mathrm{mph}$ (miles per hour), and his average speed during the second 30 minutes was $65 \mathrm{mph}$ . What was his average speed, in mph, during the last 30 minutes?
(A) 64
(B) 65
(C) 66
(D) 67
(E) 68

AMC 10A, 2018, Problem 4

A three-dimensional rectangular box with dimensions $X, Y$ , and $Z$ has faces whose surface areas are $24,24,48,48,72$ , and 72 square units. What is $X+Y+Z ?$
(A) 18
(B) 22
(C) 24
(D) 30
(E) 36

AMC 10A, 2018, Problem 19

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

AMC 10A, 2017, Problem 1

What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$ ?
(A) 70
(B) 97
(C) 127
(D) 159
(E) 729

AMC 10A, 2017, Problem 2

Pablo buys popsicles for his friends. The store sells single popsicles for $$ 1$ each, 3-popsicle boxes for $$ 2$ each, and 5-popsicle boxes for $$ 3$ What is the greatest number of popsicles that Pablo can buy with $$ 8$ ?
(A) $8$
(B) $11$
(C) $12$
(D) $13$
(E) $15$

AMC 10A, 2017, Problem 4

Mia is "helping" her mom pick up 30 toys that are strewn on the floor. Mia's mom manages to put 3 toys into the toy box every 30 seconds, but each time immediately after those 30 seconds have elapsed, Mia takes 2 toys out of the box. How much time, in minutes, will it take Mia and her mom to put all 30 toys into the box for the first time?
(A) $13.5$
(B) 14
(C) $14.5$
(D) 15
(E) $15.5$

AMC 10A, 2017, Problem 5

The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?
(A) 1
(B) 2
(C) 4
(D) 8
(E) 12

AMC 10A, 2017, Problem 9

Minnie rides on a flat road at 20 kilometers per hour (kph), downhill at $30 \mathrm{kph}$ , and uphill at $5 \mathrm{kph}$ . Penny rides on a flat road at $30 \mathrm{kph}$ , downhill at $40 \mathrm{kph}$ , and uphill at $10 \mathrm{kph}$ . Minnie goes from town $A$ to town $B$ , a distance of $10 \mathrm{~km}$ all uphill, then from town $B$ to town $C$ , a distance of 15 $\mathrm{km}$ all downhill, and then back to town $A$ , a distance of $20 \mathrm{~km}$ on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the 45 -km ride than it takes Penny?
(A) 45
(B) 60
(C) 65
(D) 90
(E) 95

AMC 10A, 2017, Problem 10

Joy has 30 thin rods, one each of every integer length from $1 \mathrm{~cm}$ through $30 \mathrm{~cm}$ . She places the rods with lengths $3 \mathrm{~cm}, 7 \mathrm{~cm}$ , and $15 \mathrm{~cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
(A) 16
(B) 17
(C) 18
(D) 19
(E) 20

AMC 10A, 2017, Problem 14

Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20 \%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5 \%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?
(A) $9 \%$
(B) $19 \%$
(C) $22 \%$
(D) $23 \%$
(E) $25 \%$

AMC 10A, 2017, Problem 16

There are 10 horses, named Horse 1, Horse $2, \ldots$ , Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S>0$ , in minutes, at which all 10 horses will again simultaneously be at the starting point is $S=2520$ . Let $T>0$ be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of $T$ ?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

AMC 10B, 2017, Problem 1

Mary thought of a positive two-digit number. She multiplied it by 3 and added 11 . Then she switched the digits of the result, obtaining a number between 71 and 75 , inclusive. What was Mary's number?
(A) 11
(B) 12
(C) 13
(D) 14
(E) 15

AMC 10B, 2017, Problem 2

Sofia ran 5 laps around the 400-meter track at her school. For each lap, she ran the first 100 meters at an average speed of 4 meters per second and the remaining 300 meters at an average speed of 5 meters per second. How much time did Sofia take running the 5 laps?
(A) 5 minutes and 35 seconds
(B) 6 minutes and 40 seconds
(C) 7 minutes and 5 seconds
(D) 7 minutes and 25 seconds
(E) 8 minutes and 10 seconds

AMC 10B, 2017, Problem 3

Real numbers $x, y$ , and $z$ satisfy the inequalities $0<x<1,-1<y<0$ , and $1<z<2$ . Which of the following numbers is necessarily positive?
(A) $y+x^{2}$
(B) $y+x z$
(C) $y+y^{2}$
(D) $y+2 y^{2}$
(E) $y+z$

AMC 10B, 2017, Problem 4

Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3 x+y}{x-3 y}=-2$ . What is the value of $\frac{x+3 y}{3 x-y}$ ?
(A) $-3$
(B) $-1$
(C) 1
(D) 2
(E) 3

AMC 10B, 2017, Problem 5

Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?
(A) 10
(B) 20
(C) 30
(D) 40
(E) 50

AMC 10B, 2017, Problem 7

Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
(A) $2.0$
(B) $2.2$
(C) $2.8$
(D) $3.4$
(E) $4.4$

AMC 10B, 2017, Problem 10

The lines with equations $a x-2 y=c$ and $2 x+b y=-c$ are perpendicular and intersect at $(1,-5)$ . What is $c ?$
$(\mathbf{A})-13$
(B) $-8$
(C) 2
(D) 8
(E) 13

AMC 10B, 2017, Problem 11

At Typico High School, $60 \%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80 \%$ say that they like it, and the res say that they dislike it. Of those who dislike dancing, $90 \%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
(A) $10 \%$
(B) $12 \%$
(C) $20 \%$
(D) $25 \%$
(E) $33 \frac{1}{3} \%$

AMC 10B, 2017, Problem 12

Elmer's new car gives $50 \%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel which is $20 \%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?
(A) $20 \%$
(B) $26 \frac{2}{3} \%$
(C) $27 \frac{7}{9} \%$
(D) $33 \frac{1}{3} \%$
(E) $66 \frac{2}{3} \%$

AMC 10B, 2017, Problem 13

There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at least two classes. How many students are taking all three classes?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10A, 2016, Problem 1

What is the value of $\frac{11 !-10 !}{9 !}$ ?
(A) $99$
(B) $100$
(C) $110$
(D) $121$
(E) $132$

AMC 10A, 2016, Problem 2

For what value of $x$ does $10^{x} \cdot 100^{2 x}=1000^{5}$ ?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10A, 2016, Problem 3

For every dollar Ben spent on bagels, David spent 25 cents less. Ben paid $$ 12.50$ more than David. How much did they spend in the bagel store together?
(A) $$ 37.50$
(B) $$ 50.00$
(C) $$ 87.50$
(D) $$ 90.00$
(E) $$ 92.50$

AMC 10A, 2016, Problem 5

A rectangular box has integer side lengths in the ratio $1: 3: 4$ . Which of the following could be the volume of the box?
(A) $48$
(B) $56$
(C) $64$
(D) $96$
(E) $144$

AMC 10A, 2016, Problem 6

Ximena lists the whole numbers $1$ through $30$ once. Emilio copies Ximena's numbers, replacing each occurrence of the digit $2$ by the digit 1

X=10+12+14+\cdots+100 \
Y=12+14+16+\cdots+102

$630$

. How old was Tony at the end of the six month period?
(A) $9$

(B) $11$

(D) $13$

(E) $14$

AMC 10A, 2010, Problem 9

A palindrome, such as 83438 , is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit anc four-digit palindromes, respectively. What is the sum of the digits of $x$ ?
(A) $20$
(B) $21$
(C) $22$
(D) $23$
(E) $24$

AMC 10A, 2010, Problem 10

Marvin had a birthday on Tuesday, May 27 in the leap year 2008 . In what year will his birthday next fall on a Saturday?
(A) $2011$
(B) $2012$
(C) $2013$
(D) $2015$
(E) $2017$

AMC 10A, 2010, Problem 11

The length of the interval of solutions of the inequality $a \leq 2 x+3 \leq b$ is 10 . What is $b-a$ ?
(A) $6$
(B) $10$
(C) $15$
(D) $20$
(E) $30$

AMC 10A, 2010, Problem 12

Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds $0.1$ liters. How tall, in meters, should Logan make his tower?
(A) $0.04$
(B) $\frac{0.4}{\pi}$
(C) $0.4$
(D) $\frac{4}{\pi}$
(E) $4$

AMC 10A, 2010, Problem 13

Angelina drove at an average rate of $80 \mathrm{kmh}$ and then stopped 20 minutes for gas. After the stop, she drove at an average rate of $100 \mathrm{kmh}$ . Altogether she drove $250 \mathrm{~km}$ in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time $t$ in hours that she drove before her stop?
(A) $80 t+100\left(\frac{8}{3}-t\right)=250$
(B) $80 t=250$
(C) $100 t=250$
(D) $90 t=250$
(E) $80\left(\frac{8}{3}-t\right)+100 t=250$

AMC 10A, 2010, Problem 21

The polynomial $x^{3}-a x^{2}+b x-2010$ has three positive integer roots. What is the smallest possible value of $a$ ?
(A) $78$
(B) $88$
(C) $98$
(D) $108$
(E) $118$

AMC 10B, 2010, Problem 1

What is $100(100-3)-(100 \cdot 100-3)$ ?
(A) $-20,000$
(B) $-10,000$
(C) $-297$
(D) $-6$
(E) $0$

AMC 10B, 2010, 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?
(A) $15$
(B) $20$
(C) $25$
(D) $30$
(E) $35$

AMC 10B, 2010, 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?
(A) $2$
(B) $3$
(C) $4$
(D) $5$
(E) $6$

AMC 10B, 2010, Problem 7

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?
(A) 18
(B) 21
(C) 24
(D) 27
(E) 30

AMC 10B, 2010, Problem 8

A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of $9_{\text {th }}$ graders buys tickets costing a total of $$ 48$ , and a group of 10 th graders buys tickets costing a total of $$ 64$ . How many values for $x$ are possible?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10B, 2010, Problem 10

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 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) $5$

AMC 10B, 2010, Problem 11

A shopper plans to purchase an item that has a listed price greater than $$ 100$ and can use any one of the three coupons. Coupon A gives $15 \%$ off the listed price, Coupon B gives $$ 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 $y-x$ ?
(A) $50$
(B) $60$
(C) $75$
(D) $80$
(E) $100$

AMC 10B, 2010, Problem 12

At the beginning of the school year, $50 \%$ of all students in Mr. Well's 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 $x$ ?
(A) $0$
(B) $20$
(C) $40$
(D) $60$
(E) $80$

AMC 10B, 2010, Problem 13

What is the sum of all the solutions of $x=|2 x-| 60-2 x||$ ?
(A) $32$
(B) $60$
(C) $92$
(D) $120$
(E) $124$

AMC 10B, 2010, Problem 14

The average of the numbers $1,2,3, \cdots, 98,99$ , and $x$ is $100 x$ . What is $x$ ?
(A) $\frac{49}{101}$
(B) $\frac{50}{101}$
(C) $\frac{1}{2}$
(D) $\frac{51}{101}$
(E) $\frac{50}{99}$

AMC 10A, 2010, 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?
(A) $25$
(B) $27$
(C) $29$
(D) $31$
(E) $33$

AMC 10A, 2009, Problem 1

One can holds $12$ ounces of soda, what is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
(A) $7$
(B) $8$
(C) $9$
(D) $10$
(E) $11$

AMC 10A, 2009, Problem 4

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?
(A) $15$
(B) $25$
(C) $35$
(D) $45$
(E) $55$

AMC 10A, 2009, Problem 3

Which of the following is equal to $1+\frac{1}{1+\frac{1}{1+1}}$ ?
(A) $\frac{5}{4}$
(B) $\frac{3}{2}$
(C) $\frac{5}{3}$
(D) $2$
(E) $3$

AMC 10A, 2009, Problem 4

Eric plans to compete in a triathlon. He can average 2 miles per hour in the $\frac{1}{4}$ mile swim and 6 miles per hour in the 3 -mile run. His goal is to finish the triathlon in 2 hours. To accomplish his goal what must his average speed in miles per hour, be for the 15 -mile bicycle ride?
(A) $\frac{120}{11}$
(B) $11$
(C) $\frac{56}{5}$
(D) $\frac{45}{4}$
(E) $12$

AMC 10A, 2009, Problem 5

What is the sum of the digits of the square of 111111111 ?
(A) $18$
(B) $27$
(C) $45$
(D) $63$
(E) $81$

AMC 10A, 2009, Problem 7

A carton contains milk that is $2 \%$ fat, an amount that is $40 \%$ less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
(A) $\frac{12}{5}$
(B) $3$
(C) $\frac{10}{3}$
(D) $38$
(E) $42$

AMC 10A, 2009, Problem 8

Three Generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50 \%$ discount as children. The two members of the oldest generation receive a $25 \%$ discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $$ 6.00$ , is paying for everyone. How many dollars must he pay?
(A) $34$
(B) $36$
(C) $42$
(D) $46$
(E) $48$

AMC 10A, 2009, Problem 9

Positive integers $a, b$ , and 2009, with $a<b<2009$ , form a geometric sequence with an integer ratio. What is $a$ ?
(A) $7$
(B) $41$
(C) $49$
(D) $289$
(E) $2009$

AMC 10A, 2009, Problem 16

Let $a, b, c$ , and $d$ be real numbers with $|a-b|=2,|b-c|=3$ , and $|c-d|=4$ . What is the sum of all possible values of $|a-d|$ ?
(A) $9$
(B) $12$
(C) $15$
(D) $18$
(E) $24$

AMC 10A, 2009, Problem 18

At Jefferson Summer Camp, $60 \%$ of the children play soccer, $30 \%$ of the children swim, and $40 \%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
(A) $30 \%$
(B) $40 \%$
(C) $49 \%$
(D) $51 \%$
(E) $70 \%$

AMC 10B, 2009, Problem 1

Each morning of her five-day workweek, Jane bought either a 50 -cent muffin or a 75 -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10B, 2009, Problem 3

Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms?
(A) $10$
(B) $12$
(C) $15$
(D) $18$
(E)