How many sequences of integers $(a_1, ... , a_7)$ are there for which $-1 \le a_i \le 1$ for every $i$, and $$a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_6 + a_6a_7 = 4 ?$$

Is it possible to place a positive integer in every cell of a $10 \times 10$ array in such a way that both the following conditions are satisfied? $\bullet$ Each number (not in the top row) is a proper divisor of the number immediately above. $\bullet$ Each row consists of 1$0$ consecutive positive integers (but not necessarily in order).

A $12$-digit positive integer consisting only of digits $1,5$ and $9$ is divisible by $37$. Prove that the sum of its digits is not equal to $76$.

The positive integer $a$ is greater than $10$, and all its digits are equal. Prove that $a$ is not a perfect square. (A perfect square is a number which can be expressed as $n^2$ , where $n$ is an integer.)

Let $m$ and $n$ be integers such that $m + n$ and $m - n$ are prime numbers less than $100$. Find the maximal possible value of $mn$.

For a positive integer $k$, define the $k$-[i]pop[/i] of a positive integer $n$ as the infinite sequence of integers $a_1, a_2, ...$ such that $a_1 = n$ and $$a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..$$ where $ \lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$. Furthermore, define a positive integer $m$ to be $k$-[i]pop avoiding[/i] if $k$ does not divide any nonzero term in the $k$-pop of $m$. For example, $14$ is 3-pop avoiding because $3$ does not divide any nonzero term in the $3$-pop of $14$, which is $14, 4, 1, 0, 0, ....$ Suppose that the number of positive integers less than $13^{2018}$ which are $13$-pop avoiding is equal to N. What is the remainder when $N$ is divided by $1000$?

Determine the number of pairs of natural numbers $(a,b)$ that simultaneously verify that $4620$ is a multiple of $a$, $4620$ is a multiple of $b$ and $b$ is a multiple of $a$.

Let $N_{15}$ be the answer to problem 15. For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits (for example, if $x = 2024$, then $x' = 4022$). Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$ (note: $x'$ is allowed to have a leading digit of zero). Compute the sum of the digits of $C$ when $C$ is expressed in base ten.

Let $ a_1, \ldots, a_n$ be distinct positive integers that do not contain a $ 9$ in their decimal representations. Prove that the following inequality holds \[ \sum^n_{i\equal{}1} \frac{1}{a_i} \leq 30.\]

find all positive integer $n$, satisfying $$\frac{n(n+2016)(n+2\cdot 2016)(n+3\cdot 2016) . . . (n+2015\cdot 2016)}{1\cdot 2 \cdot 3 \cdot . . . . . \cdot 2016}$$ is positive integer.

Find all positive integers $a$ and $b$ such that $(a + b^2)(b + a^2) = 2^m$ for some integer $m.$ [i](6 points)[/i]

Find all natural numbers $n$ with the following property: Given the decimal writing of $n$, adding a few digits one can obtain the decimal writing of $1996n$.

Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.

An integer $n\ge 1$ is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers $n$ such that there are exactly two balanced numbers among $n,n+1,n+2$ and $n+3$.

Prove that there exist $C>0$, which satisfies the following conclusion: For any infinite positive arithmetic integer sequence $a_1, a_2, a_3,\cdots$, if the greatest common divisor of $a_1$ and $a_2$ is squarefree, then there exists a positive integer $m\le C\cdot {a_2}^2$, such that $a_m$ is squarefree. Note: A positive integer $N$ is squarefree if it is not divisible by any square number greater than $1$. [i]Proposed by Qu Zhenhua[/i]

Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$

Among all triples $(a,b,c)$ of natural numbers satisfying \[(a+14\sqrt{3})(b-14c\sqrt{3})=2024,\] determine the one with the maximal value of $a$.

Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is (A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above.

Find the unique real number $c$ such that the polynomial $x^3+cx+c$ has exactly two real roots.

Prove that: (a) if $y<\frac12$ and $n\ge3$ is a natural number then $(y+1)^n\ge y^n+(1+2y)^\frac n2$; (b) if $x,y,z$ and $n\ge3$ are natural numbers for which $x^2-1\le2y$ then $x^n+y^n\ne z^n$.

Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*} [i]Proposed by Isaac Jiménez[/i]

Given that $x, y,$ and $z$ are positive real numbers such that $$x^{\log_2(yz)}=2^8\cdot3^4, \quad y^{\log_2(zx)}=2^9\cdot3^6, \quad \text{and}\quad z^{\log_2(xy)}=2^5 \cdot 3^{10},$$ compute the smallest possible value of $xyz.$

Given $n$ integers $a_1 = 1, a_2,..., a_n$ such that $a_i \le a_{i+1} \le 2a_i$ ($i = 1, 2, 3,..., n - 1$) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.

[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]

[b]p1.[/b] In triangle $ABC$, the median $BM$ is drawn. The length $|BM| = |AB|/2$. The angle $\angle ABM = 50^o$. Find the angle $\angle ABC$. [b]p2.[/b] Is there a positive integer $n$ which is divisible by each of $1, 2,3,..., 2018$ except for two numbers whose difference is$ 7$? [b]p3.[/b] Twenty numbers are placed around the circle in such a way that any number is the average of its two neighbors. Prove that all of the numbers are equal. [b]p4.[/b] A finite number of frogs occupy distinct integer points on the real line. At each turn, a single frog jumps by $1$ to the right so that all frogs again occupy distinct points. For some initial configuration, the frogs can make $n$ moves in $m$ ways. Prove that if they jump by $1$ to the left (instead of right) then the number of ways to make $n$ moves is also $m$. [b]p5.[/b] A square box of chocolates is divided into $49$ equal square cells, each containing either dark or white chocolate. At each move Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of distribution of chocolates in the box? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].