Practice (90/1000)
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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.
Let $a$ and $b$ be two randomly selected points on a line segment of unit length. What is the probability that their distance is not more than $\frac{1}{2}$?
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.
Break a stick into two parts. What is the probability that the length of one part is at least twice of that of the other?
Two people agree to meet at a place some time in the next 10 days. They have also agreed whoever arrives the place should wait for the other for 3 days and then leave. What is the probability that they will see each other?
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.
The sum of the three different positive unit fractions is $\frac{6}{7}$. What is the least number that could be the sum of the denominators of these fractions?
The sum of $n$ consecutive positive integers is 100. What is the greatest possible value of $n$?
Twin primes are prime numbers that differ by 2. Given that $a$ and $b$ are the greatest twin primes with $a < 100$, evaluate the value of $a + b$.
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: $$x_1+x_2+\cdots+x_5=14$$
Joe goes to a supermarket to buy 10 cakes. There are 6 different types of cakes, and each type has a sufficient quantity. How many different combinations of cakes can Joe have?
Library MAS has m bookshelves for books of m different categories. Each bookshelf has n books. Joe, the librarian, needs to re-arrange these books. Books of the same category still need to be put on the same bookshelf, but their order can change.
How many different arrangement plans are there that:
(a) no book is put on the same bookshelf as before?
(b) no book is put on it original position?
(c) no book is on the same bookshelf, and no book is at the relative position in the new bookshelf as it was before
Let $D_n$ be the derangement count, prove:
- $D_n =n\cdot D_{n−1} +(−1)^n$
- $D_n = (n−1)\cdot (D_{n−2} +D_{n−1})$
Joe has sufficient quantity of pennies, nickels, dimes, and quarters. He wants to pay a $\$1$ bill using these coins. How many different combinations does he have?
How many different weights can be measured using a set of 4 masses of 1, 2, 3, 4 grams each? For each measurable weight, how many different ways are possible?