Find integral solutions to the equation \[(m^{2}-n^{2})^{2}=16n+1.\]

There's a special computer and it has a memory. At first, it's memory just contains $x$. We fill up the memory with the following rules. 1) If $f\neq 0$ is in the memory, then we can also put $\frac{1}{f}$ in it. 2) If $f,g$ are in the memory, then we can also put $ f+g$ and $f-g$ in it. Find all natural number $n$ such that we can have $x^n$ in the memory.

There are $n$ cities on each side of Hung river, with two-way ferry routes between some pairs of cities across the river. A city is “convenient” if and only if the city has ferry routes to all cities on the other side. The river is “clear” if we can find $n$ different routes so that the end points of all these routes include all $2n$ cities. It is known that Hung river is currently unclear, but if we add any new route, then the river becomes clear. Determine all possible values for the number of convenient cities. [i] Proposed by usjl[/i]

Let $ABCD$ be a parallelogram whose diagonals intersect in $M$. Suppose that the circumcircle of $ABM$ intersects the segment $AD$ in a point $E \ne A$ and that the circumcircle of $EMD$ intersects the segment $BE$ in a point $F \ne E$. Show that $\angle ACB=\angle DCF$.

Let $A$ be a set contains $2000$ distinct integers and $B$ be a set contains $2016$ distinct integers. $K$ is the numbers of pairs $(m,n)$ satisfying \[ \begin{cases} m\in A, n\in B\\ |m-n|\leq 1000 \end{cases} \] Find the maximum value of $K$

If $ ABCDEFG$ is a regular $ 7$-gon with side $ 1$, show that: $ \frac{1}{AC}\plus{}\frac{1}{AD}\equal{}1$.

Let $k>1$ be the given natural number and $p\in \mathbb{P}$ such that $n=kp+1$ is composite number. Given that $n\mid 2^{n-1}-1.$ Prove that $n<2^k.$

For each integer $1\le j\le 2017$, let $S_j$ denote the set of integers $0\le i\le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that \[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1\le j\le 2017} \left(1 - \prod_{i\in S_j} x_i\right).\] Compute the remainder when \[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right)\in\{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\] is divided by $2017$. [i]Proposed by Ashwin Sah[/i]

Calculate the given expression $$\sum_{k=0}^{n} \frac{2^k}{3^{2^k}+1}$$

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

During a game of chess, at a certain point, in each row and column of the board there is an odd number of pieces. Prove that the number of pieces that are on black squares is even. (Note: a chessboard has $8$ rows and $8$ columns)

A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? $ \textbf{(A)} \ 8! \qquad \textbf{(B)} \ 2^8 \cdot 8! \qquad \textbf{(C)} \ (8!)^2 \qquad \textbf{(D)} \ \frac {16!}{2^8} \qquad \textbf{(E)} \ 16!$

Let $f(n)$ be defined on the positive integers and satisfy: $f(prime) = 1$, $f(ab) = a f(b) + f(a) b$. Show that $f$ is unique and find all $n$ such that $n = f(n)$.

A convex $n$-gon $A_0A_1\dots A_{n-1}$ has been partitioned into $n-2$ triangles by certain diagonals not intersecting inside the $n$-gon. Prove that these triangles can be labeled $\triangle_1,\triangle_2,\dots,\triangle_{n-2}$ in such a way that $A_i$ is a vertex of $\triangle_i$, for $i=1,2,\dots,n-2$. Find the number of all such labellings.

Consider trapezoid $[ABCD]$ which has $AB\parallel CD$ with $AB = 5$ and $CD = 9$. Moreover, $\angle C = 15^\circ$ and $\angle D = 75^\circ$. Let $M_1$ be the midpoint of $AB$ and $M_2$ be the midpoint of $CD$. What is the distance $M_1M_2$? [i]Proposed by Daniel Li[/i]

A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$? $ \textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8 \qquad \textbf{(E) }9 \qquad $

Let $n$ be a positive integer. A square $ABCD$ is partitioned into $n^2$ unit squares. Each of them is divided into two triangles by the diagonal parallel to $BD$. Some of the vertices of the unit squares are colored red in such a way that each of these $2n^2$ triangles contains at least one red vertex. Find the least number of red vertices. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 4)[/i]

In a square field of side length $12$ there is a source that contains a system of straight irrigation ditches. This is laid out in such a way that for every point of the field the distance to the next ditch is at most $1$. Here, the source is as a point and are the ditches to be regarded as stretches. It must be verified that the total length of the irrigation ditches is greater than $70$ m. The sketch shows an example of a trench system of the type indicated. [img]https://cdn.artofproblemsolving.com/attachments/6/5/5b51511da468cf14b5823c6acda3c4d2fe8280.png[/img]

In an isosceles triangle $ABC$ ($AB = BC$), the midline parallel to side $BC$ intersects the incircle at a point $F$ that does not lie on the base $AC$. Prove that the tangent to the circle at point $F$ intersects the bisector of angle $C$ on side $AB$.

An [i]$n$-folding process[/i] on a rectangular piece of paper with sides aligned vertically and horizontally consists of repeating the following process $n$ times: [list] [*]Take the piece of paper and fold it in half vertically (choosing to either fold the right side over the left, or the left side over the right). [*]Rotate the paper $90^\circ$ clockwise. [/list] A $10$-folding process is performed on a piece of paper, resulting in a $1$-by-$1$ square base consisting of many stacked layers of paper. Let $d(i,j)$ be the Euclidean distance between the center of the $i$th square from the top and the center of the $j$th square from the top when the paper is unfolded. Determine the maximum possible value of $\sum_{i=1}^{1023} d(i, i+1).$

In each of the squares of an $n \times n$ board, with $n \ge 3$, a positive integer is written in such a way that the absolute value of the difference of the numbers written in any two neighboring cells is less than or equal to $2$ (two neighboring cells are those that have a common side). a) Show a $5 \times 5$ board on which $15$ integers have been written different following the indicated rule. b) Find, as a function of $n$, the maximum number of different numbers that can have the board of $n \times n$ squares.

For any non-empty subset $X$ of $M=\{1,2,3,...,2021\}$, let $a_X$ be the sum of the greatest and smallest elements of $X$. Determine the arithmetic mean of all the values of $a_X$, as $X$ covers all the non-empty subsets of $M$.

Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by $n.$ [b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$

Using a compass and a ruler, construct a point $K$ inside an acute-angled triangle $ABC$ so that $\angle KBA = 2\angle KAB$ and $ \angle KBC = 2\angle KCB$.

Let \(S\) be a set with 10 distinct elements. A set \(T\) of subsets of \(S\) (possibly containing the empty set) is called [i]union-closed[/i] if, for all \(A, B \in T\), it is true that \(A \cup B \in T\). Show that the number of union-closed sets \(T\) is less than \(2^{1023}\). [i]Proposed by Tony Wang[/i]