Practice With Solutions
back to index | new
Show that if $a$ and $b$ are relatively prime, then $\varphi(a)\varphi(b)=\varphi(ab)$ where $\varphi(n)$ is Euler's totient function.
Randomly select $3$ real numbers $x$, $y$, and $z$ between 0 and 1. What is the probability that $x^2 + y^2 + z^2 > 1$?
There are several equally spaced parallel lines on a table. The distance between two adjacent lines is $2a$. On the table, toss a coin with a radius of $r$, $(r < a)$. Find the probability that the coin does not touch any line.
Joe breaks a $10$-meter long stick into three shorter sticks. Find the probability that these three sticks can form a triangle.
In the following diagram, $\overline{AO}= 2$, $\overline{BO} = 5$, and $\angle{AOB} = 60^\circ$. Point $C$ is selected on $\overline{BO}$ randomly. Find the probability that $\triangle{AOC}$ is an acute triangle.
Show that when $x$ is an integer, $x^2 + 5x + 16$ is not divisible by $169$.
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: $$x_1+x_2+\cdots+x_5=14$$
Let $D_n$ be the derangement count, prove:
- $D_n =n\cdot D_{n−1} +(−1)^n$
- $D_n = (n−1)\cdot (D_{n−2} +D_{n−1})$
How many different ways are there to express $20$ as the sum of $1$, $2$, and $5$? (All numbers must appear.)
There are $2$ white balls, $3$ red balls, and $1$ yellow ball in a jar. How many different ways are there to retrieve $3$ balls?
There are $2$ white balls, $3$ red balls, and $1$ yellow balls in a jar. How many different ways are there to retrieve $3$ balls to form a line?
How many different $5$-digit numbers can be formed using $1$, $2$, $3$, and $4$ that satisfy the following conditions:
- the digit $1$ must appear either $2$ or $3$ times,
- the digit $2$ must appear even times,
- the digit $3$ must appear odd times, and
- the digit $4$ has no restriction
What is the last digit of $17^{17^{17^{17}}}$?
A girl and a guy are going to arrive at a train station. If they arrive within $10$ minutes of each other, they will instantly fall in love and live happily ever after. But after $10$ minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between $8$ AM and $9$ AM with equal probability. The guy will arrive between $7$ AM and $8:30$ AM, also with equal probability. Find the probability that the probability that they fall in love.
Let there be $320$ points arranged on a circle, labeled $1$, $2$, $3$, $\cdots$, $8$, $1$, $2$, $3$, $\cdots$, $8$, $\cdots$ in order. Line segments may only be drawn to connect points labeled with the same number. What is the largest number of non-intersecting line segments one can draw? (Two segments sharing the same endpoint are considered to be intersecting).
Consider an orange and black coloring of a $20\times 14$ square grid. Let $n$ be the number of coloring such that every row and column has an even number of orange square. Evaluate $\log_2 n$.
In $\bigtriangleup ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\bigtriangleup ABC$?
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?
Factorize: $f(a)=4a^4-3a^3-2a^2+3a-2$