Find the smallest natural number that is the sum of $9$ consecutive natural numbers, is the sum of $10$ consecutive natural numbers and is also the sum of $11$ consecutive natural numbers.

Prove that there are no positive integers $a_1, a_2, \ldots , a_{2021}$ (not necessarily distinct) such that for $k = 1, 2, 3, \ldots , 2021$ the number of elements in the set $$A_k = \{ j \in \mathbb{N} : 1 \le j \le 2021 \text{ and } a_j|k \}$$ be exactly $a_k$.

Prove that for $n>1$ , $n$ does not divide $2^{n-1}+1$

[b]p1.[/b] Find $$2^{\left(0^{\left(2^3\right)}\right)}$$ [b]p2.[/b] Amy likes to spin pencils. She has an $n\%$ probability of dropping the $n$th pencil. If she makes $100$ attempts, the expected number of pencils Amy will drop is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p3.[/b] Determine the units digit of $3 + 3^2 + 3^3 + 3^4 +....+ 3^{2022} + 3^{2023}$. [b]p4.[/b] Cyclic quadrilateral $ABCD$ is inscribed in circle $\omega$ with center $O$ and radius $20$. Let the intersection of $AC$ and $BD$ be $E$, and let the inradius of $\vartriangle AEB$ and $\vartriangle CED$ both be equal to $7$. Find $AE^2 - BE^2$. [b]p5.[/b] An isosceles right triangle is inscribed in a circle which is inscribed in an isosceles right triangle that is inscribed in another circle. This larger circle is inscribed in another isosceles right triangle. If the ratio of the area of the largest triangle to the area of the smallest triangle can be expressed as $a+b\sqrt{c}$, such that $a, b$ and $c$ are positive integers and no square divides $c$ except $1$, find $a + b + c$. [b]p6.[/b] Jonny has three days to solve as many ISL problems as he can. If the amount of problems he solves is equal to the maximum possible value of $gcd \left(f(x), f(x+1) \right)$ for $f(x) = x^3 +2$ over all positive integer values of $x$, then find the amount of problems Jonny solves. [b]p7.[/b] Three points $X$, $Y$, and $Z$ are randomly placed on the sides of a square such that $X$ and $Y$ are always on the same side of the square. The probability that non-degenerate triangle $\vartriangle XYZ$ contains the center of the square can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p8.[/b] Compute the largest integer less than $(\sqrt7 +\sqrt3)^6$. [b]p9.[/b] Find the minimum value of the expression $\frac{(x+y)^2}{x-y}$ given $x > y > 0$ are real numbers and $xy = 2209$. [b]p10.[/b] Find the number of nonnegative integers $n \le 6561$ such that the sum of the digits of $n$ in base $9$ is exactly $4$ greater than the sum of the digits of $n$ in base $3$. [b]p11.[/b] Estimation (Tiebreaker) Estimate the product of the number of people who took the December contest, the sum of all scores in the November contest, and the number of incorrect responses for Problem $1$ and Problem $2$ on the October Contest. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the diagram $ABCDEFG$ is a regular heptagon (a $7$ sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\dfrac{m}{n}$ where m and n are relatively prime positive integers. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); string lab[]={"A","B","C","D","E","F","G"}; real r = 360/7; pair A=dir(90-r),B=dir(90),C=dir(90+r),D=dir(90+2*r),E=dir(90+3*r),F=dir(90+4*r),G=dir(90+5*r); draw(A--E--B--F--C--G--D--cycle); for(int k = -1;k <= 5;++k) { label("$"+lab[k+1]+"$",dir(90+k*r),dir(90+k*r)); } [/asy]

Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.

Find all positive integers $n$ such that the sum of the squares of the five smallest divisors of $n$ is a square.

Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.

Let $g : \mathbb{N} \to \mathbb{N}$ be a function satisfying: [list] [*] $g(xy) = g(x)g(y)$ for all $x, y \in \mathbb{N}$, [*] $g(g(x)) = x$ for all $x \in \mathbb{N}$, and [*] $g(x) \neq x$ for $2 \leq x \leq 2018$. [/list] Find the minimum possible value of $g(2)$.

Let $n=\frac{2^{2018}-1}{3}$. Prove that $n$ divides $2^n-2$.

Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

Given a positive integer $N$, determine all positive integers $n$, satisfying the following condition: for any list $d_1,d_2,\ldots,d_k$ of (not necessarily distinct) divisors of $n$ such that $\frac{1}{d_1} + \frac{1}{d_2} + \ldots + \frac{1}{d_k} > N$, some of the fractions $\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_k}$ add up to exactly $N$.

Prove that for any positive integer $n \geq 3$ there exist positive integers $a_1,a_2,\cdots , a_n$ such that \[a_1a_2\cdots a_n \equiv a_i \pmod {a_i^2} \qquad \forall i \in \{1,2,\cdots ,n\}\]

Determine all nonnegative integer solutions of the equation $2^x-2^y = 1$

Suppose that $p,p+d,p+2d,p+3d,p+4d$, and $p+5d$ are six prime numbers, where $p$ and $d$ are positive integers. Show that $d$ must be divisible by $2,3,$ and $5$.

Let $p$ be a prime number. Find all integers $k$ for which $\sqrt{k^2 -pk}$ is a positive integer.

Determine all integers $n>1$ whose positive divisors add up to a power of $3.$ [i]Andrei Bâra[/i]

Given that $a, b$, and $c$ are distinct positive integers such that $a \cdot b \cdot c = 2020$, the minimum possible positive value of $\frac{1}{a}-\frac{1}{b}-\frac{1}{c}$, is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find all positive integers $k$ such that there exists a positive integer $n$, for which $2^n + 11$ is divisible by $2^k - 1$.

Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.

Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.

Determine all triplets of integers $(x, y, z)$ that validate the inequality $x^2 + y^2 + z^2 <xy + 3y + 2z$.

Does there exist a set $ M$ with the following properties? [i](i)[/i] The set $ M$ consists of 1992 natural numbers. [i](ii)[/i] Every element in $ M$ and the sum of any number of elements have the form $ m^k$ $ (m, k \in \mathbb{N}, k \geq 2).$

(a) Decide whether there exist two decimal digits $a$ and $b$, such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$. (b) Decide whether there exist two decimal digits $c$ and $d$, such that every integer with decimal representation $cd222... 231$ is divisible by $79$.