A computer program that works only with integer numbers reads the numbers on the screen, identifies the selected numbers and performs one of the following actions: • If button $A$ is pressed, the user selects $5$ numbers and then each selected number is changed to its successor; • If button $B$ is pressed, the user selects $5$ numbers and then each selected number is changed to its triple. Bento has this program on his computer with the numbers $1, 3, 3^2, · · ·, 3^{19}$ on the screen, each one appearing just once. a) By simply pressing button $A$ several times, is Bento able to make the sum of the numbers on the screen be $2024^{2025}$? b) What is the minimum number of times that Bento must press button $B$ to make all the numbers on the screen turn equal, without pressing button $A$?

Find the remainder of $$\prod_{n = 2}^{99} (1 - n^2 + n^4)(1 - 2n^2 + n^4)$$ when divided by $101$.

We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of$ n + 1$. Prove that $n$ is a prime number.

For some integers $u$,$ v$, and $w$, the equation $$26ab - 51bc + 74ca = 12(a^2 + b^2 + c^2)$$ holds for all real numbers a, b, and c that satisfy $$au + bv + cw = 0$$ Find the minimum possible value of $u^2 + v^2 + w^2$.

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

Let $p$ a prime number and $r$ the remainder of the division of $p$ by $210$. It is known that $r$ is a composite number and can be written as the sum of two non-zero perfect squares. Find all primes less than $2018$ that satisfy these conditions.

Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.

Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.

(a) Prove that from $2007$ given positive integers, one of them can be chosen so the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$. (2) (b) One of $2007$ given positive integers is $2006$. Prove that if there is a unique number among them such that the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$, then this unique number is $2006$. (2)

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that: a) ${5^{\frac{{p - 1}}{2}}} \equiv 1(\bmod p).$ b) ${F_{p - 1}} \equiv 0(\bmod p).$ c) $k(p)|p-1.$

Find the sum of all positive $30$-digit palindromes. The leading digit is not allowed to be $0$.

Find all integer triples $(a,m,n)$ such that $a^m+1|a^n+203$ where $a,m>1$. [i]Chen Yonggao[/i]

A positive integer $n$ is called [i]good[/i] if for all positive integers $a$ which can be written as $a=n^2 \sum_{i=1}^n {x_i}^2$ where $x_1, x_2, \ldots ,x_n$ are integers, it is possible to express $a$ as $a=\sum_{i=1}^n {y_i}^2$ where $y_1, y_2, \ldots, y_n$ are integers with none of them is divisible by $n.$ Find all good numbers.

Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$. Mary wants to prevent Pat from winning the game. Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.

[b]p1.[/b] To every face of a given cube a new cube of the same size is glued. The resulting solid has how many faces? [b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's? [b]p3.[/b] A radio transmitter has $4$ buttons. Each button controls its own switch: if the switch is OFF the button turns it ON and vice versa. The initial state of switches in unknown. The transmitter sends a signal if at least $3$ switches are ON. What is the minimal number of times you have to push the button to guarantee the signal is sent? [b]p4.[/b] $19$ matches are placed on a table to show the incorrect equation: $XXX + XIV = XV$. Move exactly one match to change this into a correct equation. [b]p5.[/b] Cut the grid shown into two parts of equal area by cutting along the lines of the grid. [img]https://cdn.artofproblemsolving.com/attachments/c/1/7f2f284acf3709c2f6b1bea08835d2fb409c44.png[/img] [b]p6.[/b] A family of funny dwarfs consists of a dad, a mom, and a child. Their names are: $A$, $R$, and $C$ (not in order). During lunch, $C$ made the statements: “$R$ and $A$ have different genders” and “$R$ and $A$ are my parents”, and $A$ made the statements “I am $C$'s dad” and “I am $R$'s daughter.” In fact, each dwarf told truth once and told a lie once. What is the name of the dad, what is the name of the child, and is the child a son or a daughter? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all natural numbers $n$ such that $$(n+3)^n=\sum_{k=3}^{n+2}k^n.$$

$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$ $(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$

Let $d_n(x)$ be the $n$-th decimal digit (after the decimal point) of $x$. For example, $d_3(\pi) = 1$ because $\pi = 3.14\underline{1}5...$ For a positive integer $k$, let $f(k) = p^4_k$, where $p_k$ is the $k$-th prime number. Compute the value of $$\sum^{2023}_{i=1} d_{f(i)} \left( \frac{1}{1275}\right).$$

Determine all pairs $(k, n)$ of positive integers that satisfy $$1! + 2! + ... + k! = 1 + 2 + ... + n.$$

Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.

The set $M$ consists of all $7$-digit positive integer numbers that contain (in decimal notation) each of the digits $1, 3, 4, 6, 7, 8$ and $9$ exactly once. (a) Find the smallest positive difference $d$ of two numbers from $M$. (b) How many pairs $(x, y)$ with $x$ and $y$ from M are there for which $x - y = d$? (Gerhard Kirchner)

Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.

Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

Javier multiplies four digits, not necessarily different, and obtains a number ending in $7$. Determine how much the sum of the four digits that Javier multiplies can be worth. Give all the possibilities.

Prove that there is positive integers $N$ such that the equation $$arctan(N)=\sum_{i=1}^{2020} a_i arctan(i),$$ does not hold for any integers $a_{i}.$