Use digits $1$, $2$, $3$, $4$, and $5$ without repeating to create a number.
- How many 5-digit numbers can be formed?
- How many numbers will have the two even digits appearing between $1$ and $5$? (e.g.12345)
Joe plans to put a red stone, a blue stone, and a black stone on a $10 \times 10$ grid. The red stone and the blue stone cannot be in the same column. The blue stone and the black stone cannot be in the same row. How many different ways can Joe arrange these three stones?
How many different $6$-digit numbers can be formed by using digits $1$, $2$, and $3$, if no adjacent digits can be the same?
How many positive divisors does $20$ have?
Find the number of different rectangles that satisfy the following conditions:
- Its area is $2015$
- The lengths of all its sides are integers
How many integer solutions does the equation $(x+1)(y+1)=25$ have?
How many numbers between $1$ and $2020$ are multiples of $3$ or $4$ but not $5$?
How many positive integers, not exceeding $2019$, are relatively prime to $2019$?
Let $p$ be a prime number, computer $\varphi(p)$.
Let $p$ be a prime number and $n$ be a positive integer. Show that $\varphi(p^n)=p^n - p^{n-1}$ where $\varphi(n)$ is the Euler's totient function.
Show that if $a$ and $b$ are relatively prime, then $\varphi(a)\varphi(b)=\varphi(ab)$ where $\varphi(n)$ is Euler's totient function.
How many triangles are there in the following diagram?
How many rectangles or squares are there in the following diagram?
Six people form a line. $A$ must stand after $B$ (not necessarily immediately after $B$). How many different ways are there to form such a line?
Seven people form a line. If $A$ must stand next to $B$, and $C$ must stand next to $D$, how many possibilities are there?
Team MAS won a total of $10$ gold medals in a $6$-day tournament. It won at least one gold medal every day. How many different possibilities are there to count the number of gold medals won each day?
Find the number of positive integer solutions to the following equation: $$x_1+x_2+\cdots+x_5=14$$
How many integers between $1000$ and $9999$ have four distinct digits?
How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?
A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a 76 game schedule. How many games does a team play within its own division?
If one root of the equation $x^2 -6x+m^2-2m+5=0$ is $2$. Find the value of the other root and $m$.
Let $\alpha$ and $\beta$ be the two roots of $x^2 + 2x -5=0$. Evaluate $\alpha^2 + \alpha\beta + 2\alpha$.
If at least one real root of equation $x^2 - mx +5+m=0$ equals one root of $x^2 - (7m+1)x+13m+7=0$, compute the product of the four roots of these two equations.
If the two roots of $(a^2 -1)x^2 -(a+1)x+1=0$ are reciprocal, find the value of $a$.
Let $x_1$ and $x_2$ be the two roots of $2x^2 -7x -4=0$, compute the values of the following expressions using as many different ways as possible.
(1) $x_1^2 + x_2^2$
(2) $(x_1+1)(x_2+1)$
(3) $\mid x_1 - x_2 \mid$