I have $30$ coins consisting of nickels and quarters. The total value of the coins is $\$4.10$. How many of each kind do I have?
If we count by $3s$ starting with $1$, the following sequence is obtained. $1, 4, 7, 10, …$ What is the $100th$ number in the sequence?
A twelve hour clock looses $1$ minute every hour. Suppose it shows the correct time now. What is the least number of hours from now when it will again show the correct time?
The four-digit numeral 3AA1 is divisible by 9. What digit does A represent?
If I start by $2$ and count by $3s$ until I reach $449$, I will get: $2, 5, 8, 11, … , 449$, where $2$ is the first number, $5$ is the second number, $8$ is the third number and so forth. If $449$ is the $Nth$ number, what is the value of $N$?
An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, the ant does not return to its previous vertex, but chooses to move to one of the other two adjacent vertices. All choices are selected at random so that each of the possible moves is equally likely. The probability that after exactly $8$ moves that ant is at a vertex of the top face on the cube is $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Solution: Let $f(x,y,n)$ donate the probability that the ant eventually stays on the top after moving from vertex $x$ to vertex $y$ with $n$ steps remaining, where $0\le n<8$, $x$ and $y$ are either a vertex on the bottom surface of the cube, donate as $B$, or one on the top surface of the cube, donate as $T$. The probability that after exactly $8$ moves that ant is at a vertex of the top face on the cube is $$p = \dfrac{2}{3}f(B,B,7)+\dfrac{1}{3}f(B,T,7)=\dfrac{2}{3}f(B,B,7)+\dfrac{1}{3}f(T,T,6)$$
5 persons to be seated on 5 chairs arranged in a row. Two of these persons cannot sit next to each other. How many seating arrangements are possible?
How many four-digit integers for which the thousands digit equals the sum of the other three digits?
How many different 5-digit numbers can be obtained by using any 5 of the digits: 1, 3, 5, 5, 5 and 5?
You are to form a sequence of three numbers from the list 0, 1, 2, 3, 4, 5 and 6. Repetition is allowed, such as 0, 0, 6. How many possible lists can be formed such that the sum of the elements in the list is equal to 6?
A garage door opener has a ten-digit keypad. Codes to open the door must consist of 6 digits with no adjacent digits the same. How many codes are possible?
A number is multy if one of its digits is the product of other digits, such as 263. How many three-digit numbers are multy?
The following sequence of letters: $$A,A,B,A,B,C,A,B,C,D,A,B,C,D,E,A,B,C,D,E,F,A …$$ is formed by writing the first letter of the alphabet, followed by writing the first two letters of the alphabet, and continuing the pattern by writing one more letter of the alphabet each time. Continuing this pattern, what letter is the 284th letter in this sequence?
Start at S in the middle and form a path by moving to an adjacent letter to the right, left, up or down. How many paths spell the word SCHOOL ?
L
L O L
L O O O L
L O O H O O L
L O O H C H O O L
L O O H C S C H O O L
L O O H C H O O L
L O O H O O L
L O O O L
L O L
L
A party is attended by 11 politicians and 5 lawyers. Each politician shook hands exactly once with everyone. Each lawyer shook hands exactly once with each politician, but no handshakes among lawyers. How many handshakes took place?
A hexagon is inscribed in a circle. What is the maximum number of non-overlapping regions inside the circle divided by the edges and diagonals of the hexagon? How
Let $n$ is the number of side of the polygon inscribed in the circle. For $n=0$, there is one region of the circle when no chords have been created. Any time we connect two points out of $n$ points, one region is added. In this way $n\choose 2$ regions are added by the edges and diagonals. However, the line segment which connects any point with another point may intersect previously-drawn line segments creating new regions; there are $n\choose 4$ added by these intersections, as four points determine one additional region. In total, the maximum number of regions is: $$1+\dfrac{n(n-1)}{2}+\dfrac{n(n-1)(n-2)(n-3)}{4!}$$ When $n=6$, the answer is $31$.
An ant is traveling along the side of a regular dodecagon with side length as $1$. What is the expected value of minimum distance the ant travels between two different vertices randomly chosen? Express your answer as a common fraction in simplest form.
A best-of-five series ends when one team wins three games. The probability of team A defeating team B in any game is $\frac{1}{5}$. The probability that team A will win the series can be expressed as a fraction in lowest terms as $\frac{m}{n}$. Find $m+n$.
A positive integer less than 100 is randomly chosen. The probability that at least one of its digits is 4 or that the number is divisible by 4 can be expressed as a common fraction as $\frac{a}{b}$. Find $a+b$.
A pair of dice is rolled until the sum of the two numbers obtained is a multiple of $4$. What is the probability that a multiple of $4$ is obtained at the $6th$ toss and not before? Express you answer as a decimal to the nearest hundredth.
A regular dodecagon is inscribed in a unit circle. What is the expected value of the distance between two different vertices of the dodecagon? Express you answer as a decimal to the nearest hundredth.
On a four-by-four grid, the points are $1$ unit apart horizontally and vertically. Two distinct points are randomly selected from this grid. What is the probability that the distance between those two points is less than $3.5$? Express your answer as a common fraction in simplest form.
What is the arithmetic mean of all possible 5-digit numbers which use the digits 1, 2, 3, 5 and 9?
Two standard dice are rolled. What is the probability that the product of the two numbers shown on the top of the dices exceeds $9$? Express your answer as a common fraction in simplest form.
A point E is randomly selected inside a unit square $ABCD$. What is the probability that $\angle{AEB}=90^\circ$?
A point E is randomly selected inside a unit square $ABCD$. What is the probability that the distance from $E$ to each of $A$, $B$, $C$ and $D$ is greater than $\frac{1}{2}$, and the distance from $E$ to each of $AB$, $BC$, $CD$ and $DA$ is greater than $\frac{1}{4}$?
Six cocker spaniels have a total weight of 192 pounds. Five golden retrievers have an average weight of 71 pounds. What is the average weight of all 11 dogs? Express your answer as a decimal to the nearest tenth.
How many different assortments of pennies, nickels, dimes and quarters can Ashley’s coin holder contain if it has 15 coins total?
Carla selects 5 fruit-flavored candies from a bowl containing 6 apple, 5 banana and 4 cherry candies. How many possible combinations of candies can Carla select?
Lynn has 3 cats and a row of 4 cat beds. Each cat bed can hold one or two cats, and each cat is in a bed. Listing the number of cats in each bed from left to right, how many unique sequences are there?
Sadako has a 6-inch by 8-inch rectangle of paper. She folds it in half from left to right, and then in half from top to bottom, so that the folds are on the top and left of the resulting rectangle, as shown in steps 1 and 2. Then she cuts the paper along a straight diagonal line from the bottom right corner to the top left corner, as shown in step 3. When she unfolds the paper, how many separate pieces are there?
Let $a↑b=a^2+b^2$, what is the value of $[(3↑1)↑2]-[(3↑(1↑2)]$?
A triangle has vertices $A(-1,2)$, $B(5,8)$ and $C(-1,-7)$. Point $D(x,y)$ is on side $BC$, and the area of triangle $ACD$ is half the area of triangle $ABC$. What is the value of $x+y$?
A pair of six-sided dice are rolled. What is the probability that the product of the numbers shown is a multiple of 3? Express your answer as a common fraction.
How many more quadrilaterals than triangles are in the figure shown?
What is the value of the expression $\dfrac{22^2-17^2}{20^2-19^2}$?
The graph of the line $y=6x-5$ intersects the graph of the parabola $y=x^2$ at two points $(x,y)$. What is the distance between those two points? Express your answer in simplest radical form.
A set of double-six dominoes consists of 28 dominoes. Each domino has two ends, with 0 to 6 dots on each end and every possible combination occurring on exactly one domino. For example, there is exactly one of the domino shown, which has 3 dots on one side and one dot on the other. How many total dots are there on all 28 dominoes?
Micah is at a donut shop and wants to choose his own dozen. There are ample glazed, chocolate and jelly donuts. If Micah wants at least two jelly donuts, how many different combinations of donuts can Micah use to create a box of a dozen donuts?
For a school project, Yun records the weather at his house every day for a 20-day period in September. On what percent of the days did it rain? Express your answer to the nearest percent.
Michelle charges \$45 per hour for tutoring. She is saving up to buy a \$14,000 car. How many full hours must she tutor to earn enough money to purchase the car?
A certain burger chain sells, on average, 75 burgers every second. How many burgers are sold each hour? Express your answer in scientific notation to two significant figures.
A specialty candy is packaged in a box holding only 1 candy, in a box holding 5 candies or in a box holding 16 candies. The company receives an order for 238 candies. What is the least number of boxes needed to exactly fill this order?
At Bruno’s shipping warehouse, any item weighing 5 pounds or less costs \$3.50 to ship. For items over 5 pounds, the item costs \$3.50 plus \$1.25 for each additional pound. If Rachel paid \$14.75 to ship a package, how many pounds did it weigh?
The bar graph shows the quiz scores received by the students in Ms. Novak’s Civics class. What was the mean quiz score? Express your answer as a decimal to the nearest hundredth.
The sum of four positive integers is 7. What is the least possible sum of their reciprocals? Express your answer as a mixed number.
The great pyramid of Giza is a pyramid with a square base that is 230 meters on each side, and it is 145 meters tall. It is built mostly of limestone that has a density of 2900 $kg/m^3$. Assume that the entire pyramid is solid. What is the pyramid’s mass, in kilograms? Express your answer in scientific notation to three significant digits.
The three-digit number ABC is 675 more than the three-digit number DEF. If the letters A through F represent distinct positive digits other than 6, 7 or 5, what is the value of ABC?
Roy juggles three balls—one red, one orange and one yellow. He starts by throwing the yellow ball into the air while the red ball is in his left hand and the orange ball is in his right hand. Every second, the yellow ball takes the place of the red ball, the red ball takes the place of the orange ball, and the orange ball takes the place of the yellow ball. How many seconds after starting will he take the red ball in his right hand for the fifth time?
A duck quacks every 3 minutes, a cow moos every 7 minutes, and a horse neighs every 12 minutes. At noon, the animals all sound off together. What is the next time that all three sound off together again?
There are 30 balls, each a different color, and 30 boxes in the same 30 colors. One ball is placed in each box. How many ways can the balls be placed so that exactly 3 of the balls do not match the color of the box they are in?
Nine fair coins are flipped. What is the probability that the total number of coins that land heads up is a multiple of 3? Express your answer as a common fraction.
A certain cube has volume $n\ ft^3$ and surface area $n\ in^2$. If the integer $n$ is written in the form $n=2^b\cdot 3^b$ for integers $a$ and $b$, what is the value of $a+b$?
How many rectangles of any size appear in the figure shown?
In the figure shown, lines AB and CD are parallel, line AB passes B through the center of circle O, AB = 12 cm, and the distance between lines AB and CD is 3 cm. What is the area of the shaded region? Express your answer as a decimal to the nearest tenth.
The numbers 1 through 10 are marked with points on the number line as shown. Two distinct points are chosen at random from these points. What is the expected value of the distance between the two points? Express your answer as a decimal to the nearest tenth.
Madison creates a batch of orange paint by mixing 1 pint of red paint with 1.5 quarts of yellow paint. How many more gallons of yellow paint does she need to add to the mixture to produce a 2-gallon batch of orange paint? Express your answer as a common fraction.
Convex quadrilateral ABCD has AB = 4, BC = 5, AC = 5, AD = 3 and CD = 4. What is the area of quadrilateral ABCD? Express your answer as a decimal to the nearest tenth.
At the end of Day 1, Larry has \$40 and Moe has \$9. Every day, Larry spends \$1.50 while Moe saves \$1.50. What is the first day Moe will end with more money than Larry?
A positive number $n$is $25\%$ of $\dfrac{6}{5}$ of its square. What is the value of $n$? Express your answer as a decimal to the nearest tenth.
