Practice (EndingDigits,TheDivideByNineMethod,MODBasic)
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Let $f(n)$ be the number of points of intersections of diagonals of a $n$-dimensional hypercube that is not the vertice of the cube. For example, $f(3) = 7$ because the intersection points of a cube's diagonals are at the centers of each face and the center of the cube. Find $f(5)$
Tom and Jerry are playing a game. In this game, they use pieces of paper with $2014$ positions, in which some permutation of the numbers $1, 2,\cdots, 2014$ are to be written. (Each number will be written exactly once). Tom fills in a piece of paper first. How many pieces of paper must Jerry fill in to ensure that at least one of her pieces of paper will have a permutation that has the same number as Tom's in at least one position?
Triangle $ABC$ is isosceles, and $\angle{ABC}=x^\circ$. If the sum of possible measurements of $\angle{BAC}=240^\circ$, find $x^\circ$.
Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when takes a blue pill, he will loose one pound. If Neo originally wights one pound, what is the minimum number of pills he must take to make his weight 2015 pound?
Consider all functions $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $$f(f(x)+2x+20)=15$$ Call an integer $n$ $\textit{good}$ if $f(n)$ can take any integer value. In other words, if we fix $n$, for any integer $m$, there exists a function $f$ such that $f(n)=m$. Find the sum of all good integers $x$.
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ that has center $I$. If $IA=5$, $IB=7$, $IC=4$, $ID=9$, find the value of $\frac{AB}{CD}$.
Find $\textit{any}$ quadruple of positive integers $(a, b, c, d)$ satisfying $a^3+b^4+c^5=d^{11}$ and $abc<10^5$.
Condier a $9\times 9$ grid of squares. Haraki fills each square in hthis grid with integer between 1 and 9, inclusive. The grid is called a $\textit{super-sudoku}$ if each of the following three conditions hold:
- Each column is this grid contains 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once
- Each row is this grid contains 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once
- Each $3\times 3$ sub-grid is this grid contains 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once
How any such super-sudokus are there?
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.
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?
