#### Challenging

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For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) =\displaystyle\max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$.

There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \ldots, 30$ such that for $m \in \left\{{2, 3, 5}\right\}$, $m$ divides $(a_{n+m} - a_{n})$ for all integers $n$ with $1 \leq n < n+m \leq 30$. Find $N$.

Prove that there are infinitely many positive integers $n$ such that $(n^2+1)$ divides $n!$.

Show that the equation $x^2 + y^3 = z^4$ has infinitely many integer solutions.

Find all $n\in\mathbb{N}$ such that $$\binom{n}{k-1} = 2 \binom{n}{k} + \binom{n}{k+1}$$

for some natural number $k < n$.

Let $p$ be a prime. Prove that the equation $x^2-py^2 = -1$ has integral solution if and only if $p=2$ or $p\equiv 1\pmod{4}$.

If $p$ is a prime of the form $4k+3$, show that exactly one of the equations $x^2-py^2=\pm 2$ has an integral solution.

Show that $3^n-2$ is a square only for $n=1$ and $n=3$.

Let sequence $\{a_n\}$ satisfy $a_0=0, a_1=1$, and $a_n = 2a_{n-1}+a_{n-2}$. Show that $2^k\mid n$ if and only if $2^k\mid a_n$.

Let $\{a_n\}$ be a sequence defined as $a_n=\lfloor{n\sqrt{2}}\rfloor$ where $\lfloor{x}\rfloor$ indicates the largest integer not exceeding $x$. Show that this sequence has infinitely many square numbers.

Let sequence $g(n)$ satisfy $g(1)=0, g(2)=1, g(n+2)=g(n+1)+g(n)+1$ where $n\ge 1$. Show that if $n$ is a prime greater than 5, then $n\mid g(n)[g(n)+1]$.

Is it possible to arrange these numbers, $1, 1, 2, 2, 3, 3, \cdots, 1986, 1986$ to form a sequence for such there is $1$ number between two $1$'s, $2$ numbers between two $2$'s, $\cdots$, $1986$ numbers between two 1986's?

Solve in positive integers the equation $3^x + 4^y = 5^z$ .

Solve in positive integers the equation $8^x + 15^y = 17^z$.

How many terms with odd coefficients are there in the expanded form of $$((x+1)(x+2)\cdots(x+2015))^{2016}$$

Show that every integer $k > 1$ has a multiple which is less than $k^4$ and can be written in base 10 using at most 4 different digits.

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

Find all numbers $n \ge 3$ for which there exists real numbers $a_1, a_2, ..., a_{n+2}$ satisfying $a_{n+1} = a_1, a_{n+2} = a_2$ and$a_{i}a_{i+1} + 1 = a_{i+2}$for $i = 1, 2, ..., n.$

An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$

$4$$2\quad 6$$5\quad 7 \quad 1$$8\quad 3 \quad 10 \quad 9$

Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1 + 2 + 3 + \dots + 2018$?

A site is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD=BC \cdot DA.$ Point $X$ lies inside $ABCD$ so that $\angle XAB = \angle XCD$ and $\angle XBC = \angle XDA.$ Prove that $\angle BXA + \angle DXC = 180^{\circ}$ .

Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. Find the probability that she will roll the die an odd number of times.

Solve $x^{22} + x^{11}\equiv 2\pmod{11}$.

Let $p$ be an odd prime. Show that $$\sum_{j=0}^p\binom{p}{j}\binom{p+j}{j}\equiv 2^p +1 \pmod{p^2}$$

Show that if the equation $a^2 + 1\equiv 0\pmod{p}$ is solvable for some $a$, then $p$ can be represented as a sum of two squares.