Find the number of functions $f(x)$ from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1, 2, 3, 4, 5\}$.
Find the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\leq$ $k$ $\leq$ $5$, at least one of the first $k$ terms of the permutation is greater than $k$.
Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$.
The incircle $\omega$ of triangle $ABC$ is tangent to $\overline{BC}$ at $X$. Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega$. Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y$. Assume that $AP = 3$, $PB = 4$, $AC = 8$, and $AQ = \dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Find the number of functions $f$ from $\{0, 1, 2, 3, 4, 5, 6\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and $|x - y|$ $\leq$ $|f(x) - f(y)|$ $\leq$ $3|x - y|$ for all $x$ and $y$ in $\{0, 1, 2, 3, 4, 5, 6\}$.
Compute: $1\times 2\times 3 + 2\times 3\times 4 + \cdots + 18\times 19\times 20$.
Find the sum of all possible integer values of $a$ such that the equation $(a + 1)x^2-(a^2 + 1)x + (2a^2 − 6) = 0$ is solvable in integers.
(Hockey Sticker Identity) Show that for any positive integer $n \ge k$, the following relationship holds: $$\binom{k}{k} +\binom{k+1}{k} + \binom{k+2}{k} + \cdots + \binom{n}{k} = \binom{n+1}{k+1} $$
There are $100$ lights lined up in a long room. Each light has its own switch and is currently off. The room has an entry door and an exit door. There are $100$ people lined up outside the entry door. Each light is numbered consecutively from $1$ to $100$. So is each person.
Person No. $1$ enters the room, switches on every light, and exits. Person No. $2$ enters and flips the switch on every second light (i.e. turn off lights $2$, $4$, $6$...). Person No. $3$ enters and flips the switch on every third light (i.e. toggle lights $3$, $6$, $9$...). This continues until all $100$ people have passed through the room. How many of the lights are on at the end?
A person eats $X ( > 1)$ cookies in $N$ days in the following way:
What is the smallest possible value of $X$?
Compute $3^{2018} \mod{17}$.
Solve $15x\equiv 7\pmod{32}$.
Solve $x^{12}\equiv 3\pmod{11}$.
Show that $(2^{1194} + 1)$ is a multiple of $65$.
Compute $20!\pmod{23}$.
Let $p$ be a prime and integer $a$ is co-prime to $p$, show that $$a^{p(p-1)}\equiv 1\pmod{p^2}$$
Let $p$ and $q$ be two distinct primes, and integer $a$ is co-prime to both $p$ and $q$, show $$a^{(p-1)(q-1)}\equiv 1\pmod{pq}$$
Given $30!$ ends with some zeros, what is the digit that immediately precedes these zeros?
The two-digit integers from $19$ to $92$ are written consecutively to form the large integer $$N=192021\cdots 909192$$
Suppose that the $3^k$ is the highest power of $3$ that is a factor of $N$. What is $k$.
Show that for any positive integer $n$, $\varphi(2^n-1)$ is a multiple of $n$ where $\varphi(n)$ is Euler's totient function.
Let $p$ be an odd prime divisor of integer $(n^4 + 1)$. Show that $p\equiv 1\pmod{8}$.
Show that if there exist integer $x$, $y$, and $z$ such that $3^x + 4^y=5^z$, then both $x$ and $z$ must be even.
Let $a$ and $b$ be two positive integers such that both of them can be written as a sum of two squares. Show that their product can be written as a sum of two squares in two ways.
Show that two positive integers $m$ and $n$ are co-prime if and only if $\varphi(mn)=\varphi(m)\varphi(n)$.