Two courier companies offer services in the country of Old Aland. For any two towns, at least one of the companies offers a direct link in both directions between them. Additionally, each company is willing to chain together deliveries (so if they offer a direct link from $A$ to $B$, and $B$ to $C$, and $C$ to $D$ for instance, they will deliver from $A$ to $D$.) Show that at least one of the two companies must be able to deliver packages between any two towns in Old Aland.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

Let $I$ be the incenter of a triangle $ABC$ and let $A', B', C'$ be midpoints of sides $BC$, $CA$, $AB$, respectively. If $IA'= IB'= IC'$ , then prove that triangle $ABC$ is equilateral.

A small square $PQRS$ is contained in a big square. Produce $PQ$ to $A$, $QR$ to $B$, $RS$ to $C$ and $SP$ to $D$ so that $A$, $B$, $C$ and $D$ lie on the four sides of the large square in order, produced if necessary. Prove that $AC = BD$ and $AC \perp BD$.

Let $n$ be a positive integer greater than 1. Determine all the collections of real numbers $x_1,\ x_2,\dots,\ x_n\geq1\mbox{ and }x_{n+1}\leq0$ such that the next two conditions hold: (i) $x_1^{\frac12}+x_2^{\frac32}+\cdots+x_n^{n-\frac12}= nx_{n+1}^\frac12$ (ii) $\frac{x_1+x_2+\cdots+x_n}{n}=x_{n+1}$

A [b][u]word[/u][/b] is formed by a number of letters of the alphabet. We show words with capital letters. A [b][u]sentence[/u][/b] is formed by a number of words. For example if $A=aa$ and $B=ab$ then the sentence $AB$ is equivalent to $aaab$. In this language, $A^n$ indicates $\underbrace{AA \cdots A}_{n}$. We have an equation when two sentences are equal. For example $XYX=YZ^2$ and it means that if we write the alphabetic letters forming the words of each sentence, we get two equivalent sequences of alphabetic letters. An equation is [b][u]simplified[/u][/b], if the words of the left and the right side of the sentences of the both sides of the equation are different. Note that every word contains one alphabetic letter at least. $\text{a})$We have a simplified equation in terms of $X$ and $Y$. Prove that both $X$ and $Y$ can be written in form of a power of a word like $Z$.($Z$ can contain only one alphabetic letter). $\text{b})$ Words $W_1,W_2,\cdots , W_n$ are the answers of a simplified equation. Prove that we can produce these $n$ words with fewer words. $\text{c})$ $n$ words $W_1,W_2,\cdots , W_n$ are the answers of a simplified system of equations. Define graph $G$ with vertices ${1,2 \cdots ,n}$ such that $i$ and $j$ are connected if in one of the equations, $W_i$ and $W_j$ be the two words appearing in the right side of each side of the equation.($\cdots W_i = \cdots W_j$). If we denote by $c$ the number of connected components of $G$, prove that these $n$ words can be produced with at most $c$ words. [i]Proposed by Mostafa Einollah Zadeh Samadi[/i]

Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

Prove that for any positive integer $n \geq 3$ there exist positive integers $a_1,a_2,\cdots , a_n$ such that \[a_1a_2\cdots a_n \equiv a_i \pmod {a_i^2} \qquad \forall i \in \{1,2,\cdots ,n\}\]

Let $\triangle ABC$ be an acute triangle, $O$ be its circumcenter, and $D$ be a point different that $A$ and $C$ on the smaller $AC$ arc of its circumcircle. Let $P$ be a point on $[AB]$ satisfying $\widehat{ADP} = \widehat {OBC}$ and $Q$ be a point on $[BC]$ satisfying $\widehat{CDQ}=\widehat {OBA}$. Show that $\widehat {DPQ} = \widehat {DOC}$.

A particular coin can land on heads (H), on tails (T), or in the middle (M), each with probability $\frac{1}{3}$. Find the expected number of flips necessary to observe the contiguous sequence HMMTHMMT...HMMT, where the sequence HMMT is repeated 2016 times.

In his spare time, Thomas likes making rectangular windows. He builds windows by taking four $30\text{ cm}\times20\text{ cm}$ rectangles of glass and arranging them in a larger rectangle in wood. The window has an $x\text{ cm}$ wide strip of wood between adjacent glass pieces and an $x\text{ cm}$ wide strip of wood between each glass piece and the adjacent edge of the window. Given that the total area of the glass is equivalent to the total area of the wood, what is $x$? [center]<see attached>[/center]

Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.

Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians. $ \textbf{(A)}\ \frac{\pi}{2} \qquad\textbf{(B)}\ \pi \qquad\textbf{(C)}\ \frac{3\pi}{2} \qquad\textbf{(D)}\ 3\pi \qquad\textbf{(E)}\ 4\pi $

An $n\times n$ Latin square is a $n\times n$ grid that is filled with $n$ $1$'s, $n$ $2$'s, $\dots$, and $n$ $n$'s such that each column and row of the grid contains exactly one of each $1$, $2$, $\dots$, $n$. For example, the following is a valid $2\times2$ Latin square: $\textstyle\begin{bmatrix}2&1\\1&2\end{bmatrix}$, but this is not: $\textstyle\begin{bmatrix}2&1\\2&1\end{bmatrix}$. How many $4\times4$ Latin squares are there?

Let $$A =\frac12 +\frac13 +\frac15 +\frac19,$$ $$B =\frac{1}{2 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{2 \cdot 9}+\frac{1}{3 \cdot 5}+\frac{1}{3 \cdot 9} +\frac{1}{5 \cdot 9},$$ $$C =\frac{1}{2 \cdot 3 \cdot 5} + \frac{1}{2 \cdot 3 \cdot 9} + \frac{1}{2 \cdot 5 \cdot 9} +\frac{1}{3 \cdot 5 \cdot 9}.$$ Compute the value of $A + B + C$.

Let $x$ and $y$ be integers from $-10$ to $10$, inclusive, with $xy \ne1$. Compute the number of ordered pairs $(x, y) $ such that $$\left| \frac{x + y}{1 - xy} \right|\le 1.$$

Prove that from every set of $n+1$ natural numbers, whose prime factors are in a given set of $n$ prime numbers, one can select several distinct numbers whose product is a perfect square.

For every positive integer $n \geq 2$, Prove that there is no $n-$tuple of distinct complex numbers $(x_1,x_2,\dots,x_n)$ such that for each $1 \leq k \leq n$ following equality holds. $\prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k - x_i) = \prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k + x_i) $ (20 points)

Suppose the roots of $$x^4 - 3x^2 + 6x - 12 = 1$$ are $\alpha$, $\beta$, $\gamma$ , and $\delta$. What is the value of $$\frac{\alpha+ \beta+ \gamma }{\delta^2}+\frac{\alpha+ \delta+ \gamma}{\beta^2}+\frac{\alpha+ \beta+ \delta}{\gamma^2}+\frac{\delta+ \beta+ \gamma }{\alpha^2}?$$

Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done? $\textbf{(A)} \ 96 \qquad \textbf{(B)} \ 108 \qquad \textbf{(C)} \ 156 \qquad \textbf{(D)} \ 204 \qquad \textbf{(E)} \ 372 $

Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.

John found all real numbers $p$ such that in the polynomial $g(x) = (x -1)^2(p + 2x)^2$ , the quadratic term has coefficient $2021$. What is the sum of all of these values $p$?

An ant walks around on the coordinate plane. It moves from the origin to $(3,4)$, then to $(-9, 9)$, then back to the origin. How many units did it walk? Express your answer as a decimal rounded to the nearest tenth.

Eight points are chosen on the circumference of a circle, labelled $P_1$, $P_2$, ..., $P_8$ in clockwise order. A route is a sequence of at least two points $P_{a_1}$, $P_{a_2}$, $...$, $P_{a_n}$ such that if an ant were to visit these points in their given order, starting at $P_{a_1}$ and ending at $P_{a_n}$, by following $n-1$ straight line segments (each connecting each $P_{a_i}$ and $P_{a_{i+1}}$), it would never visit a point twice or cross its own path. Find the number of routes.

Find out if there is a convex pentagon $A_1A_2A_3A_4A_5$ such that, for each $i = 1, \dots , 5$, the lines $A_iA_{i+3}$ and $A_{i+1}A_{i+2}$ intersect at a point $B_i$ and the points $B_1,B_2,B_3,B_4,B_5$ are collinear. (Here $A_{i+5} = A_i$.)