It is given the sequence defined by $$\{a_{n+2}=6a_{n+1}-a_n\}_{n \in \mathbb{Z}_{>0}},a_1=1, a_2=7 \text{.}$$ Find all $n$ such that there exists an integer $m$ for which $a_n=2m^2-1$.

Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer.

Proof that for every primes $p$, $q$ \[p^{q^2-q+1}+q^{p^2-p+1}-p-q\] is never a perfect square. [i]Proposed by chengbilly[/i]

Are there positive integers $m,n$ such that there exist at least $2012$ positive integers $x$ such that both $m-x^2$ and $n-x^2$ are perfect squares? [i]David Yang.[/i]

Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$ \frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$

There are five positive integers written in a row. Each one except for the first one is the minimal positive integer that is not a divisor of the previous one. Can all these five numbers be distinct? Boris Frenkin

Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$. ([i]Serbia[/i])

Prove that, if there exist natural numbers $a_1,a_2…a_{2017}$ for which the product $(a_1^{2017}+a_2 )(a_2^{2017}+a_3 )…(a_{2016}^{2017}+a_{2017})(a_{2017}^{2017}+a_1)$ is a $k$-th power of a prime number, then $k=2017$ or $k\geq 2017.2018$.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Given a convex polygon with vertices at lattice points on a plane containing origin $O$. Let $V_1$ be the set of vectors going from $O$ to the vertices of the polygon, and $V_2$ be the set of vectors going from $O$ to the lattice points that lie inside or on the boundary of the polygon (thus, $V_1$ is contained in $V_2$.) Two grasshoppers jump on the whole plane: each jump of the first grasshopper shift its position by a vector from the set $V_1$, and the second by the set $V_2$. Prove that there exists positive integer $c$ that the following statement is true: if both grasshoppers can jump from $O$ to some point $A$ and the second grasshopper needs $n$ jumps to do it, then the first grasshopper can use at most $n+c$ jumps to do so.

Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

Let $n$ be a composite positive integer. Let $1=d_1<d_2<\cdots<d_k=n$ be the positive divisors of $n.{}$ Assume that the equations $d_{i+2}x^2-2d_{i+1}x+d_i=0$ for $i=1,\ldots,k-2$ all have real solutions. Prove that $n=p^{k-1}$ for some prime number $p.{}$

Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.

Define the infinite products \[ A = \prod\limits_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and } B = \prod\limits_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If $\tfrac{A}{B} = \tfrac{m}{n}$ where $m,n$ are relatively prime positive integers, determine $100m+n$. [i]Proposed by Lewis Chen[/i]

The set of positive integers is partitioned into $n$ disjoint infinite arithmetic progressions $S_1, S_2, \ldots, S_n$ with common differences $d_1, d_2, \ldots, d_n$, respectively. Prove that there exists exactly one index $1\leq i \leq n$ such that\[ \frac{1}{d_i}\prod_{j=1}^n d_j \in S_i.\]

Natural numbers have been divided in groups as follow: $(1), (2, 4), (3, 5, 7), (6, 8, 10, 12), (9, 11, 13, 15, 17), \ldots$. Let $S_n$ be the sum of the elements of the $n$th group. Prove that $\frac{S_{2n+1}}{2n+1}-\frac{S_{2n}}{2n}$ is even.

Let $a, b, c$ be nonnegative integers such that $a \le b \le c, 2b \neq a + c$ and $\frac{a+b+c}{3}$ is an integer. Is it possible to find three nonnegative integers $d, e$, and $f$ such that $d \le e \le f, f \neq c$, and such that $a^2+b^2+c^2 = d^2 + e^2 + f^2$?

Let $ n $ be a natural number. Find all integer numbers that can be written as $$ \frac{1}{a_1} +\frac{2}{a_2} +\cdots +\frac{n}{a_n} , $$ where $ a_1,a_2,...,a_n $ are natural numbers.

Suppose that $m>2$, and let $P$ be the product of the positive integers less than $m$ that are relatively prime to $m$. Show that $P \equiv -1 \pmod{m}$ if $m=4$, $p^n$, or $2p^{n}$, where $p$ is an odd prime, and $P \equiv 1 \pmod{m}$ otherwise.

[b]p1.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p2.[/b] A $5\times 6$ rectangle is drawn on the piece of graph paper (see the figure below). The side of each square on the graph paper is $1$ cm long. Cut the rectangle along the sides of the graph squares in two parts whose areas are equal but perimeters are different by $2$ cm. $\begin{tabular}{|l|l|l|l|l|l|} \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline \end{tabular}$ [b]p3.[/b] Three runners started simultaneously on a $1$ km long track. Each of them runs the whole distance at a constant speed. Runner $A$ is the fastest. When he runs $400$ meters then the total distance run by runners $B$ and $C$ together is $680$ meters. What is the total combined distance remaining for runners $B$ and $C$ when runner $A$ has $100$ meters left? [b]p4.[/b] There are three people in a room. Each person is either a knight who always tells the truth or a liar who always tells lies. The first person said «We are all liars». The second replied «Only you are a liar». Is the third person a liar or a knight? [b]p5.[/b] A $5\times 8$ rectangle is divided into forty $1\times 1$ square boxes (see the figure below). Choose 24 such boxes and one diagonal in each chosen box so that these diagonals don't have common points. $\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a prime of the form $p=4n-1$ where $n$ is a positive integer. Prove that $$\prod_{k=1}^p(k^2+1)\equiv4\pmod p.$$

$p > 3$ is a prime number such that $p|2^{p-1} - 1$ and $p \nmid 2^x - 1$ for $x = 1, 2,...,p-2$. Let $p = 2k + 3$. Now we define sequence $\{a_n\}$ as $$a_i = a_{i+k} = 2^i \,\, (1 \le i \le k ), \,\,\,\, a_{j+2k} = a_ja_{j+k} \,\, (j \le 1)$$ Prove that there exist $2k$ consecutive terms of sequence $a_{x+1},a_{x+2},..., a_{x+2k}$ such that $a_{x+i } \not\equiv a_{x+j}$ (mod $p$) for all $1 \le i < j \le 2k$ .

In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\overline{AC}$ with $V$ on $\overline{AW}$, points $X$ and $Y$ are on $\overline{BC}$ with $X$ on $\overline{CY}$, and points $Z$ and $U$ are on $\overline{AB}$ with $Z$ on $\overline{BU}$. In addition, the points are positioned so that $\overline{UV} \parallel \overline{BC}$, $\overline{WX} \parallel \overline{AB}$, and $\overline{YZ} \parallel \overline{CA}$. Right angle folds are then made along $\overline{UV}$, $\overline{WX}$, and $\overline{YZ}$. The resulting figure is placed on a level floor to make a table with triangular legs. Let $h$ be the maximum possible height of a table constructed from triangle $ABC$ whose top is parallel to the floor. Then $h$ can be written in the form $\tfrac{k \sqrt{m}}{n}$, where $k$ and $n$ are relatively prime positive integers and $m$ is a positive integer that is not divisible by the square of any prime. Find $k + m + n$. [asy] unitsize(1 cm); pair translate; pair[] A, B, C, U, V, W, X, Y, Z; A[0] = (1.5,2.8); B[0] = (3.2,0); C[0] = (0,0); U[0] = (0.69*A[0] + 0.31*B[0]); V[0] = (0.69*A[0] + 0.31*C[0]); W[0] = (0.69*C[0] + 0.31*A[0]); X[0] = (0.69*C[0] + 0.31*B[0]); Y[0] = (0.69*B[0] + 0.31*C[0]); Z[0] = (0.69*B[0] + 0.31*A[0]); translate = (7,0); A[1] = (1.3,1.1) + translate; B[1] = (2.4,-0.7) + translate; C[1] = (0.6,-0.7) + translate; U[1] = U[0] + translate; V[1] = V[0] + translate; W[1] = W[0] + translate; X[1] = X[0] + translate; Y[1] = Y[0] + translate; Z[1] = Z[0] + translate; draw (A[0]--B[0]--C[0]--cycle); draw (U[0]--V[0],dashed); draw (W[0]--X[0],dashed); draw (Y[0]--Z[0],dashed); draw (U[1]--V[1]--W[1]--X[1]--Y[1]--Z[1]--cycle); draw (U[1]--A[1]--V[1],dashed); draw (W[1]--C[1]--X[1]); draw (Y[1]--B[1]--Z[1]); dot("$A$",A[0],N); dot("$B$",B[0],SE); dot("$C$",C[0],SW); dot("$U$",U[0],NE); dot("$V$",V[0],NW); dot("$W$",W[0],NW); dot("$X$",X[0],S); dot("$Y$",Y[0],S); dot("$Z$",Z[0],NE); dot(A[1]); dot(B[1]); dot(C[1]); dot("$U$",U[1],NE); dot("$V$",V[1],NW); dot("$W$",W[1],NW); dot("$X$",X[1],dir(-70)); dot("$Y$",Y[1],dir(250)); dot("$Z$",Z[1],NE); [/asy]

For each positive integer $n$ let the set $A_n$ consist of all numbers $\pm 1 \pm 2 \pm ...\pm n$. For example, $$A_1 = \{-1,1\}, A_2 = \{ -3 ,-1 ,1 ,3 \} , A_3 = \{ -6 ,-4 ,-2 ,0 ,2 ,4 ,6 \}.$$ Find the number of elements in $A_n$ .