Try these AMC 8 Combinatorics Questions and check your knowledge!
AMC 8, 2019, Problem 6
There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry for the square?
$\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20}\qquad\textbf{(E) }\frac{1}{2}$
AMC 8, 2014, Problem 18
The faces of each of two fair dice are numbered $1$, $2$, $3$, $5$, $7$, and $8$. When the two dice are tossed, what is the probability that their sum will be an even number?
$\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}$
AMC 8, 2019, Problem 10
The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?
$\textbf{(A) }$The mean increases by $1$ and the median does not change.
$\textbf{(B) }$ The mean increases by $1$ and the median increases by $1$.
$\textbf{(C) }$ The mean increases by $1$ and the median increases by $5$.
$\textbf{(D) }$ The mean increases by $5$ and the median increases by $1$.
$\textbf{(E) }$ The mean increases by $5$ and the median increases by $5$.
AMC 8, 2019, Problem 16
Qiang drives $15$ miles at an average speed of $30$ miles per hour. How many additional miles will he have to drive at $55$ miles per hour to average $50$ miles per hour for the entire trip?
AMC 8, 2019, Problem 19
In a tournament there are six teams that play each other twice. A team earns $3$ points for a win, $1$ point for a draw, and $0$ points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?
$\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30$
AMC 8, 2019, Problem 20
How many different real numbers $x$ satisfy the equation$(x^{2}-5)^{2}=16?$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8$
AMC 8, 2019, Problem 25
Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the people has at least $2$ apples?
$\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380$
AMC 8, 2019, Problem 14
Isabella has $6$ coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every $10$ days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the $6$ dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
$\textbf{(A) }\text{Monday}\qquad\textbf{(B) }\text{Tuesday}\qquad\textbf{(C) }\text{Wednesday}\qquad\textbf{(D) }\text{Thursday}\qquad\textbf{(E) }\text{Friday}$
AMC 8, 2019, Problem 18
The faces of each of two fair dice are numbered $1$, $2$, $3$, $5$, $7$, and $8$. When the two dice are tossed, what is the probability that their sum will be an even number?
$\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}$
AMC 8, 2019, Problem 25
Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the people has at least $2$ apples?
$\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380$
AMC 8, 2018, Problem 11
Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.
\begin{eqnarray} \text{X}&\quad\text{X}\quad&\text{X} \ \text{X}&\quad\text{X}\quad&\text{X} \end{eqnarray}
If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?
$\textbf{(A) } \frac{1}{3} \qquad \textbf{(B) } \frac{2}{5} \qquad \textbf{(C) } \frac{7}{15} \qquad \textbf{(D) } \frac{1}{2} \qquad \textbf{(E) } \frac{2}{3}$
AMC 8, 2018, Problem 16
Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?
$\textbf{(A) }1440\qquad\textbf{(B) }2880\qquad\textbf{(C) }5760\qquad\textbf{(D) }182,440\qquad \textbf{(E) }362,880$
AMC 8, 2018, Problem 19
n a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?
$\textbf{(A) } 2 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16$
AMC 8, 2018, Problem 23
From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?
$\textbf{(A) } \frac{2}{7} \qquad \textbf{(B) } \frac{5}{42} \qquad \textbf{(C) } \frac{11}{14} \qquad \textbf{(D) } \frac{5}{7} \qquad \textbf{(E) } \frac{6}{7}$
AMC 8, 2017, Problem 10
A box contains five cards, numbered $1, 2, 3, 4,$ and $5$. Three cards are selected randomly without replacement from the box. What is the probability that $4$ is the largest value selected?
$\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}\qquad\textbf{(D) }\frac{2}{5}\qquad\textbf{(E) }\frac{1}{2}$
AMC 8, 2017, Problem 11
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is $37$, how many tiles cover the floor?
$\textbf{(A) }148\qquad\textbf{(B) }324\qquad\textbf{(C) }361\qquad\textbf{(D) }1296\qquad\textbf{(E) }1369$
AMC 8, 2017, Problem 15
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.
(A) $8$
(B) $9$
(C) $12$
(D) $24$
(E) $36$
AMC 8, 2017, Problem 19
For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$ ?
$\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$
AMC 8, 2016, Problem 13
Two different numbers are randomly selected from the set ${ - 2, -1, 0, 3, 4, 5}$ and multiplied together. What is the probability that the product is $0$?
$\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}$
AMC 8, 2016, Problem 17
]An ATM password at Fred's Bank is composed of four digits from $0$ to $9$, with repeated digits allowable. If no password may begin with the sequence $9,1,1,$ then how many passwords are possible?
$\textbf{(A)}\mbox{ }30\qquad\textbf{(B)}\mbox{ }7290\qquad\textbf{(C)}\mbox{ }9000\qquad\textbf{(D)}\mbox{ }9990\qquad\textbf{(E)}\mbox{ }9999$
AMC 8, 2016, Problem 18
In an All-Area track meet, $216$ sprinters enter a $100-$meter dash competition. The track has $6$ lanes, so only $6$ sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?
$\textbf{(A)}\mbox{ }36\qquad\textbf{(B)}\mbox{ }42\qquad\textbf{(C)}\mbox{ }43\qquad\textbf{(D)}\mbox{ }60\qquad\textbf{(E)}\mbox{ }72$
AMC 8, 2016, Problem 21
A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?
$\textbf{(A) }\dfrac{3}{10}\qquad\textbf{(B) }\dfrac{2}{5}\qquad\textbf{(C) }\dfrac{1}{2}\qquad\textbf{(D) }\dfrac{3}{5}\qquad \textbf{(E) }\dfrac{7}{10}$
AMC 8, 2015, Problem 4
The Dragonvale Middle School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible?
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }12$
AMC 8, 2015, Problem 7
Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
$\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{2}{9}\qquad\textbf{(C) }\frac{4}{9}\qquad\textbf{(D) }\frac{1}{2}\qquad \textbf{(E) }\frac{5}{9}$
AMC 8, 2015, Problem 11
In the small country of Icosahedrontopia, all automobile license plates have four symbols. The first must be a vowel ($A, E, I, O,$ or $U$), the second and third must be two different letters among the $21$ non-vowels, and the fourth must be a digit ($0$ through $9$). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "$AMC8$"?
$\textbf{(A) } \frac{1}{22,050} \qquad \textbf{(B) } \frac{1}{21,000}\qquad \textbf{(C) } \frac{1}{10,500}\qquad \textbf{(D) } \frac{1}{2,100} \qquad \textbf{(E) } \frac{1}{1,050}$
AMC 8, 2015, Problem 12
How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?
$\textbf{(A) }6 \quad\textbf{(B) }12 \quad\textbf{(C) } 18 \quad\textbf{(D) } 24 \quad \textbf{(E) } 36$
AMC 8, 2015, Problem 13
How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is $6$?
$\textbf{(A)}\text{ 1}\qquad\textbf{(B)}\text{ 2}\qquad\textbf{(C)}\text{ 3}\qquad\textbf{(D)}\text{ 5}\qquad\textbf{(E)}\text{ 6}$
AMC 8, 2015, Problem 1
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$?
$\textbf{(A) }21\qquad\textbf{(B) }31\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad \textbf{(E) }42$
AMC 8, 2015, Problem 23
Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. If a slip with $2$ goes into cup $E$ and a slip with $3$ goes into cup $B$, then the slip with $3.5$ must go into what cup?
$\textbf{(A) } A \qquad \textbf{(B) } B \qquad \textbf{(C) } C \qquad \textbf{(D) } D \qquad \textbf{(E) } E$
AMC 8, 2014, Problem 12
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?
$\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{1}{4}\qquad\textbf{(D) }\frac{1}{3}\qquad\textbf{(E) }\frac{1}{2}$
AMC 8, 2014, Problem 14
The "Middle School Eight" basketball conference has 8 teams. Every season, each team plays every other conference team twice (home and away), and each team also plays 4 games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
$\textbf{(A) }60\qquad\textbf{(B) }88\qquad\textbf{(C) }96\qquad\textbf{(D) }144\qquad\textbf{(E) }160$
AMC 8, 2013, Problem 1
Danica wants to arrange her model cars in rows with exactly $6$ cars in each row. She now has $23$ model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
AMC 8, 2013, Problem 8
A fair coin is tossed $3$ times. What is the probability of at least two consecutive heads?
$\textbf{(A)}\ \frac18 \qquad \textbf{(B)}\ \frac14 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac34$
AMC 8, 2013, Problem 14
Abe holds $1$ green and 1 red jelly bean in his hand. Bob holds $1$ green, $1$ yellow, and $2$ red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
$\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23$
AMC 8, 2013, Problem 22
Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether?
$\textbf{(A)}\ 1920 \qquad \textbf{(B)}\ 1952 \qquad \textbf{(C)}\ 1980 \qquad \textbf{(D)}\ 2013 \qquad \textbf{(E)}\ 3932$
AMC 8, 2012, Problem 10
How many $4$-digit numbers greater than $1000$ are there that use the four digits of $2012$?
$\textbf{(A)}\hspace{.05in}6\qquad\textbf{(B)}\hspace{.05in}7\qquad\textbf{(C)}\hspace{.05in}8\qquad\textbf{(D)}\hspace{.05in}9\qquad\textbf{(E)}\hspace{.05in}12$
AMC 8, 2012, Problem 16
Each of the digits $0, 1, 2, 3, 4, 5, 6, 7, 8,$ and $9$ is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?
$\textbf{(A)}\hspace{.05in}76531\qquad\textbf{(B)}\hspace{.05in}86724\qquad\textbf{(C)}\hspace{.05in}87431\qquad\textbf{(D)}\hspace{.05in}96240\qquad\textbf{(E)}\hspace{.05in}97403$
AMC 8, 2012, Problem 22
Let $R$ be a set of nine distinct integers. Six of the elements are $2, 3, 4, 6, 9$, and $14$. What is the number of possible values of the median of $R$ ?
$\textbf{(A)}\hspace{.05in}4\qquad\textbf{(B)}\hspace{.05in}5\qquad\textbf{(C)}\hspace{.05in}6\qquad\textbf{(D)}\hspace{.05in}7\qquad\textbf{(E)}\hspace{.05in}8$
AMC 8, 2011, Problem 8
Bag A has three chips labeled $1, 3$, and $5$. Bag B has three chips labeled $2, 4,$ and $6$. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips?
$\textbf{(A) }4 \qquad\textbf{(B) }5 \qquad\textbf{(C) }6 \qquad\textbf{(D) }7 \qquad\textbf{(E) }9$
AMC 8, 2011, Problem 12
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?
$\textbf{(A) } \frac14 \qquad\textbf{(B) } \frac13 \qquad\textbf{(C) } \frac12 \qquad\textbf{(D) } \frac23 \qquad\textbf{(E) } \frac34$
AMC 8, 2011, Problem 18
A fair 6-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?
$\textbf{(A) }\dfrac16\qquad\textbf{(B) }\dfrac5{12}\qquad\textbf{(C) }\dfrac12\qquad\textbf{(D) }\dfrac7{12}\qquad\textbf{(E) }\dfrac56$
AMC 8, 2009, Problem 4
The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?
AMC 8, 2009, Problem 10
On a checkerboard composed of $64$ unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board?
$\textbf{(A)}\ \frac{1}{16}\qquad\textbf{(B)}\ \frac{7}{16}\qquad\textbf{(C)}\ \frac{1}2\qquad\textbf{(D)}\ \frac{9}{16}\qquad\textbf{(E)}\ \frac{49}{64}$
AMC 8, 2009, Problem 12
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{3}{4}$
(D) $\frac{7}{9}$
(E) $\frac{5}{6}$
AMC 8, 2009, Problem 13
A three-digit integer contains one of each of the digits $1$, $3$, and $5$. What is the probability that the integer is divisible by $5$?
$\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}$
AMC 8, 2009, Problem 16
How many $3$ -digit positive integers have digits whose product equals $24 ?$
(A) $12$
(B) $15$
(C) $18$
(D) $21$
(E) $24$
AMC 8, 2009, Problem 19
Two angles of an isosceles triangle measure $70^{\circ}$ and $x^{\circ}$. What is the sum of the three possible values of $x$ ?
(A) $95$
(B) $125$
(C) $140$
(D) $165$
(E) $180$
AMC 8, 2008, Problem 14
Three A's, three $\mathrm{B}$ 's, and three C's are placed in the nine spaces so that each row and column contain one of each letter. If $\mathrm{A}$ is placed in the upper left corner, how many arrangements are possible?
(A) $2$ (B) $3$ (C) $4$
(D) $5$ (E) $6$
AMC 8, 2009, Problem 15
In Theresa's first $8$ basketball games, she scored $7,4,3,6,8,3,1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points per game average for the $10$ games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
(A) $35$
(B) $40$
(C) $48$
(D) $56$
(E) $72$
AMC 8, 2009, Problem 17
Ms. Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
(A) $76$
(B) $120$
(C) $128$
(D) $132$
(E) $136$
AMC 8, 2009, Problem 19
Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?
(A) $\frac{1}{4}$
(B) $\frac{2}{7}$
(C) $\frac{4}{11}$
(D) $\frac{1}{2}$
(E) $\frac{4}{7}$
AMC 8, 2009, Problem 20
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
(A) $12$
(B) $17$
(C) $24$
(D) $27$
(E) $36$
AMC 8, 2009, Problem 24
Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
(A) $\frac{1}{10}$
(B) $\frac{1}{6}$
(C) $\frac{11}{60}$
(D) $\frac{1}{5}$
(E) $\frac{7}{30}$
AMC 8, 2007, Problem 4
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?
(A) $12$
(B) $15$
(C) $18$
(D) $30$
(E) $36$
AMC 8, 2007, Problem 5
Chandler wants to buy a 
mountain bike. For his birthday, his grandparents send him 
, his aunt sends him 
and his cousin gives him 
. He earns 
per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike?
(A) $24$
(B) $25$
(C) $26$
(D) $27$
(E) $28$
AMC 8, 2009, Problem 9
To complete the grid below, each of the digits $1$ through $4$ must occur once in each row and once in each column. What number will occupy the lower right-hand square?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) cannot be determined
AMC 8, 2007, Problem 11
Tiles $I, I I, I I I$ and $I V$ are translated so one tile coincides with each of the rectangles $A, B, C$ and $D .$ In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C ?$
(A) $I$
(B) $I I$
(C) $I I I$
(D) $I V$
(E) cannot be determined
AMC 8, 2009, Problem 16
Amanda draws five circles with radii $1,2,3,4$ and $5$ . Then for each circle she plots the point $(C, A),$ where $C$ is its circumference and $A$ is its area. Which of the following could be her graph?
AMC 8, 2007, Problem 20
Before district play, the Unicorns had won $45 \%$ of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?
(A) $48$
(B) $50$
(C) $52$
(D) $54$
(E) $60$
AMC 8, 2007, Problem 21
Two cards are dealt from a deck of four red cards labeled $A, B, C, D$ and four green cards labeled $A, B, C, D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?
(A) $\frac{2}{7}$
(B) $\frac{3}{8}$
(C) $\frac{1}{2}$
(D) $\frac{4}{7}$
(E) $\frac{5}{8}$
AMC 8, 2007, Problem 24
A bag contains four pieces of paper, each labeled with one of the digits $"1, 2,3"$ or $"4"$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$ ?
(A) $\frac{1}{4}$
(B) $\frac{1}{3}$
$(C) \frac{1}{2}$
(D) $\frac{2}{3}$
(E) $\frac{3}{4}$
AMC 8, 2007, Problem 25
On the dart board shown in the Figure, the outer circle has radius $6$ and the inner circle has a radius of $3$ . Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?
(A) $\frac{17}{36}$
(B) $\frac{35}{72}$
(C) $\frac{1}{2}$
(D) $\frac{37}{72}$
(E) $\frac{19}{36}$
AMC 8, 2006, Problem 4
Initially, a spinner points west. Chenille moves it clockwise $2 \frac{1}{4}$ revolutions and then counterclockwise $3 \frac{3}{4}$ revolutions. In what direction does the spinner point after the two moves?
(A) north
(B) east
(C) south
(D) west
(E) northwest
AMC 8, 2006, Problem 10
Jorge's teacher asks him to plot all the ordered pairs $(w, l)$ of positive integers for which $w$ is the width and $l$ is the length of a rectangle with area $12 .$ What should his graph look like?
AMC 8, 2006, Problem 17
Jeff rotates spinners $P, Q$ and $R$ and adds the resulting numbers. What is the probability that his sum is an odd number?
(A) $\frac{1}{4}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) $\frac{2}{3}$
(E) $\frac{3}{4}$
AMC 8, 2006, Problem 20
A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won $4$ games, Ines won $3$ games, Janet won $2$ games, Kendra won $2$ games and Lara won $2$ games, how many games did Monica (the sixth player) win?
(A) $0$
(B) $1$
(C) $2$
(D) $3$
(E) $4$
AMC 8, 2006, Problem 21
An aquarium has a rectangular base that measures $100 \mathrm{~cm}$ by $40 \mathrm{~cm}$ and has a height of $50 \mathrm{cm}$. The aquarium is filled with water to a depth of $37 \mathrm{~cm}$. A rock with volume $1000 \mathrm{~cm}^{3}$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise?
(A) $0.25$
(B) $0.5$
(C) $1$
(D) $1.25$
(E) $2.5$
AMC 8, 2006, Problem 22
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?
(A) $16$
(B) $24$
(C) $25$
(D) $26$
(E) $35$
AMC 8, 2006, Problem 23
A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
(A) $0$
(B) $1$
(C) $2$
(D) $3$
(E) $5$
AMC 8, 2006, Problem 24
In the multiplication problem below $A, B, C, D$ and are different digits. What is $A+B$ ?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $9$
AMC 8, 2005, Problem 20
Alice and Bob play a game involving a circle whose circumference is divided by $12$ equally spaced points. The points are numbered clockwise, from $1$ to $12 .$ Both start on point $12$ . Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves $5$ points clockwise and Bob moves $9$ points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
(A) $6$
(B) $8$
(C) $12$
(D) $14$
(E) $24$
AMC 8, 2005, Problem 21
How many distinct triangles can be drawn using three of the dots below as vertices?
(A) $9$
(B) $12$
(C) $18$
(D) $20$
(E) $24$
AMC 8, 2005, Problem 22
A company sells detergent in three different sized boxes: small (S), medium (M) and large
(L). The medium size costs $50 \%$ more than the small size and contains $20 \%$ less detergent than the large size. The large size contains twice as much detergent as the small size and costs $30 \%$ more than the medium size. Rank the three sizes from best to worst buy.
(A) SML
(B) LMS
(C) MSL
(D) LSM
(E) MLS
AMC 8, 2005, Problem 24
A certain calculator has only two keys $[+1]$ and $[x 2]$. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed $"9"$ and you pressed $[+1]$, it would display $"10."$ If you then pressed $[x2]$, it would display $"20."$ Starting with the display $"1,"$ what is the fewest number of keystrokes you would need to reach $"200"$?
(A) $8$
(B) $9$
(C) $10$
(D) $11$
(E) $12$
AMC 8, 2004, Problem 4
Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?
(A) $2$
(B) $4$
(C) $6$
(D) $8$
(E) $10$
AMC 8, 2004, Problem 5
Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?
(A) $4$
(B) $7$
(C) $8$
(D) $15$
(E) $16$
AMC 8, 2004, Problem 13
Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true.
I. Bill is the oldest.
II. Amy is not the oldest.
III. Celine is not the youngest.
Rank the friends from the oldest to the youngest.
(A) Bill, Amy, Celine
(B) Amy, Bill, Celine
(C) Celine, Amy, Bill
(D) Celine, Bill, Amy
(E) Amy, Celine, Bill
AMC 8, 2004, Problem 17
Three friends have a total of $6$ identical pencils, and each one has at least one pencil. In how many can this happen?
(A) $1$
(B) $3$
(C) $6$
(D) $10$
(E) $12$
AMC 8, 2004, Problem 18
Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers 1 through 10. Each throw hits the target in a region with a different value. The scores are Alice $16$ points, Ben $4$ points, Cindy $7$ points, Dave $11$ points, and Ellen $17$ points. Who hits the region worth $6$ points?
(A) Alice
(B) Ben
(C) Cindy
(D) Dave
(E) Ellen
AMC 8, 2004, Problem 20
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are $6$ empty chairs, how many people are in the room?
(A) $12$
(B) $18$
(C) $24$
(D) $27$
(E) $36$
AMC 8, 2004, Problem 21
Spinners $A$ and $B$ are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?
(A) $\frac{1}{4}$ (B) $\frac{1}{3}$ (C) $\frac{1}{2}$ (D) $\frac{2}{3}$
(E) $\frac{3}{4}$
AMC 8, 2004, Problem 23
Tess runs counterclockwise around rectangular block $JKLM$. She lives at corner $J$. Which graph could represent her straight-line distance from home?
AMC 8, 2003, Problem 12
When a fair six-sided die is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the numbers on the five faces that can be seen is divisible by $6 ?$
(A) $\frac{1}{3}$
(B) $\frac{1}{2}$
(C) $\frac{2}{3}$
(D) $\frac{5}{6}$
(E) $1$
AMC 8, 2003, Problem 13
Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces?
(A) $4$
(B) $6 $
(C) $8$
(D) $10$
(E) $12$
AMC 8, 2003, Problem 14
In this addition problem, each letter stands for a different digit.
If $T=7$ and the letter $O$ represents an even number, what is the only possible value for $W ?$
(A) $0$
(B) $1$
(C) $2$
(D) $3$
(E) $4$
AMC 8, 2003, Problem 15
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?
(A) $3$
(B) $4$
(C) $5$
(D) $6$
(E) $7$
AMC 8, 2003, Problem 16
Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has $4$ seats: $1$ driver's seat, $1$ front passenger seat, and $2$ back passenger seats. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?
(A) $2$
(B) $4$
(C) $6$
(D) $12$
(E) $24$
AMC 8, 2003, Problem 17
The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings?
(A) Nadeen and Austin
(B) Benjamin and Sue
(C) Benjamin and Austin
(D) Nadeen and Tevy
(E) Austin and Sue.
AMC 8, 2003, Problem 18
Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party?
(A) $1$
(B) $4$
(C) $5$
(D) $6$
(E) $7$
AMC 8, 2003, Problem 23
In the pattern below, the cat moves clockwise through the four squares and the mouse moves counterclockwise through the eight exterior segments of the four squares.
If the pattern is continued, where would the cat and mouse be after the $247$ th move?
AMC 8, 2003, Problem 24
A ship travels from point $A$ to point $B$ along a semicircular path, centered at Island $X$. Then it travels along a straight path from $B$ to $C$. Which of these graphs best shows the ship's distance from Island $X$ as it moves along its course?
AMC 8, 2002, Problem 1
A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures?
(A) $2$
(B) $3$
(C) $4$
(D) $5$
(E) $6$
AMC 8, 2002, Problem 2
How many different combinations of $\$ 5$ bills and $\$ 2$ bills can be used to make a total of $\$ 17 ?$ order does not matter in this problem.
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
AMC 8, 2002, Problem 3
What is the smallest possible average of four distinct positive even integers?
(A) $3$
(B) $4$
(C) $5$
(D) $6$
(E) $7$
AMC 8, 2002, Problem 5
Carlos Montado was born on Saturday, November $9,2002 .$ On what day of the week will Carlos be 706 days old?
(A) Monday
(B) Wednesday
(C) Friday
(D) Saturday
(E) Sunday
AMC 8, 2002, Problem 6
A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of $20$ milliliters per minute and drains at the rate of $18$ milliliters per minute. One of these graphs shows the volume of water in the birdbath during the filling time and continuing into the overflow time. Which one is it?
(A) A
(B) B
(C) C
(D) D
(E) E
AMC 8, 2002, Problem 7
The students in Mrs. Sawyer's class were asked to do a taste test of five kinds of candy. Each student chose one kind of candy. A bar graph of their preferences is shown. What percent of her class chose candy E?
(A) $5$
(B) $12$
(C) $15$
(D) $16$
(E) $20$
AMC 8, 2002, Problem 8
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)
How many of his European stamps were issued in the '80s?
(A) $9$
(B) $15$
(C) $18$
(D) $24$
(E) $42$
AMC 8, 2002, Problem 9
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6$ cents each, Peru $4$ cents each, and Spain $5$ cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)
His South American stamps issued before the $'70$s cost him
(A) $\$ 0.40$
(B) $\$ 1.06$
(C) $\$ 1.80$
(D) $\$ 2.38$
(E) $\$ 2.64$
AMC 8, 2002, Problem 10
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6$ cents each, Peru $4$ cents each, and Spain $5$ cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)
The average price of his $'70$s stamps is closest to
(A) $3.5$ cents
(B) $4$ cents
(C) $4.5$ cents
(D) $5$ cents
(E) $5.4$ cents
AMC 8, 2002, Problem 11
A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth?
(A) $11$
(B) $12$
(C) $13$
(D) $14$
(E) $15$
AMC 8, 2002, Problem 12
A board game spinner is divided into three regions labeled $A, B$ and $C$. The probability of the arrow stopping on region $A$ is $\frac{1}{3}$ and on region $B$ is $\frac{1}{2}$. The probability of the arrow stopping on region $C$ is
(A) $\frac{1}{12}$
(B) $\frac{1}{6}$
(C) $\frac{1}{5}$
(D) $\frac{1}{3}$
(E) $\frac{2}{5}$
AMC 8, 2002, Problem 15
Which of the following polygons has the largest area?
(A) $A$
(B) $B$
(C) $C$
(D) $D$
(E) $E$
AMC 8, 2002, Problem 16
Right isosceles triangles are constructed on the sides of a $3-4-5$ right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true?
(A) $X+Z=W+Y$
(B) $W+X=Z$
(C) $3 X+4 Y=5 Z$
(D) $X+W=\frac{1}{2}(Y+Z)$
(E) $X+Y=Z$
AMC 8, 2002, Problem 17
In a mathematics contest with ten problems, a student gains $5$ points for a correct answer and loses $2$ points for an incorrect answer. If Olivia answered every problem and her score was $29,$ how many correct answers did she have?
(A) $5$
(B) $6$
(C) $7$
(D) $8$
(E) $9$
AMC 8, 2002, Problem 18
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time?
(A) $1 \mathrm{hr}$
(B) 1 hr $10 \mathrm{~min}$
(C) 1 hr $20 \mathrm{~min}$
(D) $1 \mathrm{hr} 40 \mathrm{~min}$
(E) $2 \mathrm{hr}$
AMC 8, 2002, Problem 21
Harold tosses a nickel four times. The probability that he gets at least as many heads as tails is
(A) $\frac{5}{16}$
(B) $\frac{3}{8}$
(C) $\frac{1}{2}$
(D) $\frac{5}{8}$
(E) $\frac{11}{16}$
AMC 8, 2002, Problem 22
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom and sides.
(A) $18$
(B) $24$
(C) $26$
(D) $30$
(E) $36$
AMC 8, 2001, Problem 14
Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose?
AMC 8, 2001, Problem 17
For the game show Who Wants To Be A Millionaire? the dollar values of each question are shown in the following table (where $\mathrm{K}=1000$ )
Between which two questions is the percent increase of the value the smallest?
(A) From $1$ to $2$
(B) From $2$ to $3$
(C) From $3$ to $4$
(D) From $11$ to $12$
(E) From $14$ to $15$
AMC 8, 2001, Problem 18
Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5 ?
(A) $\frac{1}{36}$
(B) $\frac{1}{18}$
(C) $\frac{1}{6}$
(D) $\frac{11}{36}$
(E) $\frac{1}{3}$
AMC 8, 2001, Problem 19
Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car $\mathrm{N}$ traveled at twice the speed for the same distance. If Car N's speed and time are shown as solid line, which graph illustrates this?
AMC 8, 2001, Problem 20
Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." List the scores from lowest to highest for Marty (M), Quay $(\mathrm{Q})$ and Shana (S).
(A) $\mathrm{S}, \mathrm{Q}, \mathrm{M}$
(B) $\mathrm{Q}, \mathrm{M}, \mathrm{S}$
(C) $\mathrm{Q}, \mathrm{S}, \mathrm{M}$
(D) $\mathrm{M}, \mathrm{S}, \mathrm{Q}$
(E) $\mathrm{S}, \mathrm{M}, \mathrm{Q}$
AMC 8, 2001, Problem 21
The mean of a set of five different positive integers is $15$ . The median is $18$ . The maximum possible value of the largest of these five integers is
(A) $19$
(B) $24$
(C) $32$
(D) $35$
(E) $40$
AMC 8, 2001, Problem 22
On a twenty-question test, each correct answer is worth $5$ points, each unanswered question is worth $1$ point and each incorrect answer is worth $0$ points. Which of the following scores is NOT possible?
(A) $90$
(B) $91$
(C) $92$
(D) $95$
(E) $97$
AMC 8, 2001, Problem 23
Points $R, S$ and $T$ are vertices of an equilateral triangle, and points $X, Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $20$
AMC 8, 2001, Problem 24
Each half of this figure is composed of $3$ red triangles, $5$ blue triangles and 8 white triangles. When the upper half is folded down over the centerline, $2$ pairs of red triangles coincide, as do $3$ pairs of blue triangles. There are $2$ red-white pairs. How many white pairs coincide?
(A) $4$
(B) $5$
(C) $6$
(D) $7$
(E) $9$
