Given $n>1$, what is the smallest positive integer $n$ whose positive divisors have a product of $n^6$?
What is the largest integer value of $n$ for which $8^n$ evenly divides $100!$?
A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a prime number. How many $3$-primable positive integers are there that are less than 1000?
How many whole numbers n, such that $100\le n \le1000$, have the same number of odd factors as even factors?
A right triangle has a hypotenuse of $10m$ and a perimeter of $22m$. In square meters, what is the area of the triangle?
Circle $O$ has radius $10$ units. Point $P$ is on radius $OQ$ and $OP=6$ units. How many chords containing $P$, including the diameter, have integer lengths?
What is the total surface area of the largest regular tetrahedron that can be inscribed inside of a cube of edge length $1 cm$. Express your answer in simplest radical form.
The diameter, in inches, of a sphere with twice the volume of a spbere of radius 9 inches can be expressed in the form of $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$.
A circular garden is surrounded by a sidewalk with a uniform width of $25$ foot. The total area of the sidewalk equals the total area of the garden. How many feet are in the diameter of the garden? Round your answer to the nearest whole number.
Pedro stood at the center of a circular field that had a radius of 120 feet. He walked due north halfway to the circle. He then turned and walked due east halfway to the circle. He turned again and walked due south hallway to the circle. Finally he turned and walked due west halfway to the circle. When he stopped, how many feet was Pedro from the center of the circle? Express your answer to the nearest foot.
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 $\frac{1}{4}$?
