Let $S$ be a non-empty set of positive integers such that for any $n \in S$, all positive divisors of $2^n+1$ are also in $S$. Prove that $S$ contains an integer of the form $(p_1p_2 \ldots p_{2023})^{2023}$, where $p_1, p_2, \ldots, p_{2023}$ are distinct prime numbers, all greater than $2023$.

Let N be the greatest positive integer that can be expressed using all seven Roman numerals $I, V, X, L, C,D$, and $M$ exactly once each, and let n be the least positive integer that can be expressed using these numerals exactly once each. Find $N - n$. Note that the arrangement $CM$ is never used in a number along with the numeral $D$.

On the board is written a number with nine non-zero and distinct digits. Prove that we can delete at most seven digits so that the number formed by the digits left to be a perfect square.

Consider a sequence $(a_k)_{k \ge 1}$ of natural numbers defined as follows: $a_1=a$ and $a_2=b$ with $a,b>1$ and $\gcd(a,b)=1$ and for all $k>0$, $a_{k+2}=a_{k+1}+a_k$. Prove that for all natural numbers $n$ and $k$, $\gcd(a_n,a_{n+k}) <\frac{a_k}{2}$.

Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

Let $N$ be the fifth largest number that can be created by combining $2021$ $1$'s using addition, multiplication, and exponentiation, in any order (parentheses are allowed). If $f(x)=\log_2(x)$, and $k$ is the least positive integer such that $f^k(N)$ is not a power of $2$, what is the value of $f^k(N)$? (Note: $f^k(N)=f(f(\cdots(f(N))))$, where $f$ is applied $k$ times.) [i]Proposed by Adam Bertelli[/i]

Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_1,b_2,..., b_K$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_1,b_2,..., b_K$.

For a nonnegative integer $n$, let $s(n)$ be the sum of the digits of the binary representation of $n$. Prove that $$\sum_{n=1}^{2^{2022}-1} \frac{(-1)^{s(n)}}{n+2022}>0.$$

$a,b \in \mathbb Z$ and for every $n \in \mathbb{N}_0$, the number $2^na+b$ is a perfect square. Prove that $a=0$.

Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.

Find the number of positive integer pairs $1\leqslant a,b \leqslant 2027$ that satisfy \[ 2027 \mid a^6+b^5+b^2.\] (Note: For integers $a$ and $b$, the notation $a \mid b$ means that there is an integer $c$ such that $ac=b$.) [i]Proposed by Valentio Iverson, Indonesia[/i]

The systems of equations \[\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. \] has a solution in positive integers $x_i$. Show that $n$ must be even.

Let $X$ be the sum of all divisors of the number $(3\cdot 2009)^{((2\cdot 2009)^{2009}-1)}$ . Find the last digit of $X$.

Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.

Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$

Ted flips seven fair coins. there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that Ted flips at least two heads given that he flips at least three tails. Find $m+n$.

Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.

Delete $100$ digits from the number $1234567891011... 9899100$ so that the remaining number were as big as possible.

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) ${F_{p + 1}} \equiv 0(\bmod p).$ b) $k(p)|2p+2.$ c) $k(p)$ is divisible by $4.$

Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of $\frac{P^2}{A}$ [color = red]The official statement does not have the final period.[/color]

Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, c)=1$. Is it possible that $S$ has $2012$ elements? [i]Proposed by Ognjen Stipetić.[/i]

Find all pairs $(x,y)$ of positive integers such that $y^{x^2}=x^{y+2}$.

$72$ was added to the natural number $n$ and in the sum we got a number written in the same digits as the number $n$, but in the reverse order. Find all numbers $n$ that satisfy the given condition.

Find all $ f:N\rightarrow N$, such that $\forall m,n\in N $ $ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $

[u]Round 6 [/u] [b]p16.[/b] Triangle $ABC$ with $AB < AC$ is inscribed in a circle. Point $D$ lies on the circle and point $E$ lies on side $AC$ such that $ABDE$ is a rhombus. Given that $CD = 4$ and $CE = 3$, compute $AD^2$. [b]p17.[/b] Wam and Sang are walking on the coordinate plane. Both start at the origin. Sang walks to the right at a constant rate of $1$ m/s. At any positive time $t$ (in seconds),Wam walks with a speed of $1$ m/s with a direction of $t$ radians clockwise of the positive $x$-axis. Evaluate the square of the distance betweenWamand Sang in meters after exactly $5\pi$ seconds. [b]p18.[/b] Mawile is playing a game against Salamance. Every turn,Mawile chooses one of two moves: Sucker Punch or IronHead, and Salamance chooses one of two moves: Dragon Dance or Earthquake. Mawile wins if the moves used are Sucker Punch and Earthquake, or Iron Head and Dragon Dance. Salamance wins if the moves used are Iron Head and Earthquake. If the moves used are Sucker Punch and Dragon Dance, nothing happens and a new turn begins. Mawile can only use Sucker Punch up to $8$ times. All other moves can be used indefinitely. Assuming bothMawile and Salamance play optimally, find the probability thatMawile wins. [u]Round 7 [/u] [b]p19.[/b] Ephram is attempting to organize what rounds everyone is doing for the NEAML competition. There are $4$ rounds, of which everyone must attend exactly $2$. Additionally, of the 6 people on the team(Ephram,Wam, Billiam, Hacooba,Matata, and Derke), exactly $3$ must attend every round. In how many different ways can Ephram organize the teams like this? [b]p20.[/b] For some $4$th degree polynomial $f (x)$, the following is true: $\bullet$ $f (-1) = 1$. $\bullet$ $f (0) = 2$. $\bullet$ $f (1) = 4$. $\bullet$ $f (-2) = f (2) = f (3)$. Find $f (4)$. [b]p21.[/b] Find the minimum value of the expression $\sqrt{5x^2-16x +16}+\sqrt{5x^2-18x +29}$ over all real $x$. [u]Round 8 [/u] [b]p22.[/b] Let $O$ and $I$ be the circumcenter and incenter, respectively, of $\vartriangle ABC$ with $AB = 15$, $BC = 17$, and $C A = 16$. Let $X \ne A$ be the intersection of line $AI$ and the circumcircle of $\vartriangle ABC$. Find the area of $\vartriangle IOX$. [b]p23.[/b] Find the sum of all integers $x$ such that there exist integers $y$ and $z$ such that $$x^2 + y^2 = 3(2016^z )+77.$$ [b]p24.[/b] Evaluate $$ \left \lfloor \sum^{2022}_{i=1} \frac{1}{\sqrt{i}} \right \rfloor = \left \lfloor \frac{1}{\sqrt{1}} +\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+ \frac{1}{\sqrt{2022}}\right \rfloor$$ [u]Round 9[/u] [b]p25.[/b]Either: 1. Submit $-2$ as your answer and you’ll be rewarded with two points OR 2. Estimate the number of teams that choose the first option. If your answer is within $1$ of the correct answer, you’ll be rewarded with three points, and if you are correct, you’ll receive ten points. [b]p26.[/b] Jeff is playing a turn-based game that starts with a positive integer $n$. Each turn, if the current number is $n$, Jeff must choose one of the following: 1. The number becomes the nearest perfect square to $n$ 2. The number becomes $n-a$, where $a$ is the largest digit in $n$ Let $S(k)$ be the least number of turns Jeff needs to get from the starting number $k$ to $0$. Estimate $$\sum^{2023}_{k=1}S(k).$$ If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{6000} \right| \right \rfloor , 0 \right)$ points. [b]p27.[/b] Estimate the smallest positive integer n such that if $N$ is the area of the $n$-sided regular polygon with circumradius $100$, $10000\pi -N < 1$ is true. If your estimation is $E$ and the actual answer is $A$, you will receive $ \max \left \lfloor \left( 10 - \left| 10 \cdot \log_3 \left( \frac{A}{E}\right) \right|\right| ,0\right \rfloor.$ points. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167360p28825713]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].