Let $z$ be a complex number with $|z| = 2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\tfrac{1}{z+w} = \tfrac{1}{z} + \tfrac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\sqrt{3},$ where $n$ is an integer. Find the remainder when $n$ is divided by $1000$.

Prove that for a natural number $ n > 2 $ the number $ n! $ is the sum of its $ n $ various divisors.

Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have \[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\] Prove that $\sqrt{(c-3)(c+1)}$ is rational.

Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$? $\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$

Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that \[ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. \]

Given that the expected amount of $1$s in a randomly selected $2021$-digit number is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Hannah Shen[/i]

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. The segment $AL$ intersects the inscribed circle for the second time at point $S$. Line $KL$ intersects the circumscribed circle of triangle $ASK$ for the second at point $M$. Prove that $PL = PM$.

Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and \[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \] $(a)$ Prove that $f$ has a fixed point different from $1$. $(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.

Let $ABC$ be a triangle with $AB = 5$, $BC = 6$, and $CA = 7$. Denote $\Gamma$ the incircle of $ABC$, let $I$ be the center of $\Gamma$ . The circumcircle of $BIC$ intersects $\Gamma$ at $X_1$ and $X_2$. The circumcircle of $CIA$ intersects $\Gamma$ at $Y_1$ and $Y_2$. The circumcircle of $AIB$ intersects $\Gamma$ at $Z_1$ and $Z_2$. The area of the triangle determined by $\overline{X_1X_2}$, $\overline{Y_1Y_2}$, and $\overline{Z_1Z_2}$ equals $\frac{m \sqrt{p}}{n}$ for positive integers $m, n$, and $p$, where $m$ and$ n$ are relatively prime and $p$ is squarefree. Compute $m+n+ p$.

Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$. Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$. Find a closed form for $a_n$. [i]Proposed by Bobby Shen[/i]

Arnulfo and Berenice play the following game: One of the two starts by writing a number from $ 1$ to $30$, the other chooses a number from $ 1$ to $30$ and adds it to the initial number, the first player chooses a number from $ 1$ to $30$ and adds it to the previous result, they continue doing the same until someone manages to add $2018$. When Arnulfo was about to start, Berenice told him that it was unfair, because whoever started had a winning strategy, so the numbers had better change. So they asked the following question: Adding chosen numbers from $1 $ to $a$, until reaching the number $ b$, what conditions must meet $a$ and $ b$ so that the first player does not have a winning strategy? Indicate if Arnulfo and Berenice are right and answer the question asked by them.

In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.

1007 distinct potatoes are chosen independently and randomly from a box of 2016 potatoes numbered $1, 2, \dots, 2016$, with $p$ being the smallest chosen potato. Then, potatoes are drawn one at a time from the remaining 1009 until the first one with value $q < p$ is drawn. If no such $q$ exists, let $S = 1$. Otherwise, let $S = pq$. Then given that the expected value of $S$ can be expressed as simplified fraction $\tfrac{m}{n}$, find $m+n$.

Annie and Bonnie are running laps around a 400-meter oval track. They started together, but Annie has pulled ahead because she is $25 \%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie? $\textbf{(A) }1 \frac{1}{4}\qquad\textbf{(B) }3 \frac{1}{3}\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }25$

A die is thrown six times. How many ways are there for the six rolls to sum to $21$?

How many solutions does the equation: $$[\frac{x}{20}]=[\frac{x}{17}]$$ have over the set of positve integers? $[a]$ denotes the largest integer that is less than or equal to $a$. [i]Proposed by Karl Czakler[/i]

Renata the robot packs boxes in a warehouse. Each box is a cube of side length $1$ foot. The warehouse floor is a square, $12$ feet on each side, and is divided into a $12$-by-$12$ grid of square tiles $1$ foot on a side. Each tile can either support one box or be empty. The warehouse has exactly one door, which opens onto one of the corner tiles. Renata fits on a tile and can roll between tiles that share a side. To access a box, Renata must be able to roll along a path of empty tiles starting at the door and ending at a tile sharing a side with that box. [list=a] [*]Show how Renata can pack $91$ boxes into the warehouse and still be able to access any box. [*]Show that Renata [b]cannot[/b] pack $95$ boxes into the warehouse and still be able to access any box.[/list]

Let $N$ be the number of functions $f$ from $\{1,2,\ldots, 8\}$ to $\{1,2,3,\ldots, 255\}$ with the property that: [list] [*] $f(k)=1$ for some $k \in \{1,2,3,4,5,6,7,8\}$ [*] If $f(a) =f(b)$, then $a=b$. [*] For all $n \in \{1,2,3,4,5,6,7,8\}$, if $f(n) \neq 1$, then $f(k)+1>\frac{f(n)}{2} \geq f(k)$ for some $k \in \{1,2,\ldots, 7,8\}$. [*] For all $k,n \in \{1,2,3,4,5,6,7,8\}$, if $f(n)=2f(k)+1$, then $k<n$. [/list] Compute the number of positive integer divisors of $N$. [i]2021 CCA Math Bonanza Individual Round #15[/i]

[b]7.[/b] Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that, with some positive number $C$, $\sum_{k=1}^{n}k\mid a_k \mid<n C$ ($n=1,2, \dots $) Putting $s_n= a_0 +a_1+\dots+a_n$, suppose that $\lim_{n \to \infty }(\frac{s_{0}+s_{1}+\dots+s_n}{n+1})= s$ exists. Prove that $\lim_{n \to \infty }(\frac{s_{0}^2+s_{1}^2+\dots+s_n^2}{n+1})= s^2$ [b](S. 7)[/b]

Let $ABC$ be a triangle and $AD$ be the angle bisector of $<BAC$, with $D$ on $BC$. Let $E$ be a point on segment $BC$ such that $BD = EC$. Through $E$ draw $l$ a parallel line to $AD$ and let $P$ be a point in $l$ inside the triangle. Let $G$ be the point where $BP$ intersects $AC$ and $F$ be the point where $CP$ intersects $AB$. Show $BF = CG$.

Let $v$ and $w$ be real numbers such that, for all real numbers $a$ and $b$, the inequality \[(2^{a+b}+8)(3^a+3^b) \leq v(12^{a-1}+12^{b-1}-2^{a+b-1})+w\] holds. Compute the smallest possible value of $128v^2+w^2$. [i]Proposed by Luke Robitaille[/i]

Let $ a,b,c,d$ be positive real numbers and $cd=1$. Prove that there exists a positive integer $n$ such that $ab\leq n^2\leq (a+c)(b+d)$

We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

In the exterior of an equilateral triangle $ABC$ of side $\alpha$ we construct an isosceles right-angled triangle $ACD$ with $\angle CAD=90^0.$The lines $DA$ and $CB$ meet at point $E$. (a) Find the angle $\angle DBC.$ (b) Express the area of triangle $CDE$ in terms of $\alpha.$ (c) Find the length of $BD.$

Let $H$ be the orthocenter of a nonisosceles triangle $ABC$. The bisector of angle $BHC$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. The perpendiculars to $AB$ and $AC$ from $P$ and $Q$ meet at $K$. Prove that $KH$ bisects the segment $BC$.