(Leo Moser) Show that the Diophantine equation \[\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}+\frac{1}{x_{1}x_{2}\cdots x_{n}}= 1\] has at least one solution for every positive integers $n$.

A sequence of real numbers $a_0, a_1, . . .$ is said to be good if the following three conditions hold. (i) The value of $a_0$ is a positive integer. (ii) For each non-negative integer $i$ we have $a_{i+1} = 2a_i + 1 $ or $a_{i+1} =\frac{a_i}{a_i + 2} $ (iii) There exists a positive integer $k$ such that $a_k = 2014$. Find the smallest positive integer $n$ such that there exists a good sequence $a_0, a_1, . . .$ of real numbers with the property that $a_n = 2014$. [i]Proposed by Wang Wei Hua, Hong Kong[/i]

Calculate the sum of digit of the number $$9 \times 99 \times 9999 \times ... \times \underbrace{ 99...99}_{2^n}$$ where the number of nines doubles in each factor.

$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$

Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$. Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$.

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

Given an even positive integer $m$, find all positive integers $n$ for which there exists a bijection $f:[n]\to [n]$ so that, for all $x,y\in [n]$ for which $n\mid mx-y$, $$(n+1)\mid f(x)^m-f(y).$$ Note: For a positive integer $n$, we let $[n] = \{1,2,\dots, n\}$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

Let $ M (x,y)=x^2+xy-2y $ , x,y are positive integers a) Solve in positive integers $ x^2+xy-2y=64 $ b) Prove that if M (x,y) is a perfect square, then x+y+2 is composite if x>2.

Prove that $$ (n+2)^n=\prod_{k=1}^{n+1} \sum_{l=1}^{n+1} le^{\frac{2i\pi k (n-l+1)}{n+2}} , $$ for any natural number $ n. $ [i]Mihai Piticari[/i]

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

How many positive divisors does number $20!$ have?

The positive integers $a, b$ and $c$ satisfy that the three fractions $\frac{b}{a}$, $\frac{c + 100}{b}$ and $\frac{a + b + 169}{2c + 200}$ are all integers. Determine all possible values of $a$.

Prove there are no such integers $ k, m $ which satisfy $ k \ge 0, m \ge 0 $ and $ k!+48=48(k+1)^m $.

The sequence $ (x_n)$ is defined by $ x_1\equal{}1,x_2\equal{}a$ and $ x_n\equal{}(2n\plus{}1)x_{n\minus{}1}\minus{}(n^2\minus{}1)x_{n\minus{}2}$ $ \forall n \geq 3$, where $ a \in N^*$.For which value of $ a$ does the sequence have the property that $ x_i|x_j$ whenever $ i<j$.

The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.

For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1, 2, ... , n\}$ such that $\tau (\tau (\tau (x))) = x$ for $x = 1, 2, ..., n$. The first few values are $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 9$. Prove that $3^{334}$ divides $a_{2001}$. (A permutation of $\{1, 2, ... , n\}$ is a rearrangement of the numbers $\{1, 2, ... , n\}$ or equivalently, a one-to-one and onto function from $\{1, 2, ... , n\}$ to $\{1, 2, ... , n\}$. For example, one permutation of $\{1, 2, 3\}$ is the rearrangement $\{2, 1, 3\}$, which is equivalent to the function $\sigma : \{1, 2, 3\} \to \{1, 2, 3\}$ defined by $\sigma (1) = 2, \sigma (2) = 1, \sigma (3) = 3$.)

We call an infinite set $S\subseteq\mathbb{N}$ good if for all parwise different integers $a,b,c\in S$, all positive divisors of $\frac{a^c-b^c}{a-b}$ are in $S$. for all positive integers $n>1$, prove that there exists a good set $S$ such that $n \not \in S$. Proposed by Seyed Reza Hosseini Dolatabadi

[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].

On the board there are written in a row, the integers from $1$ until $2030$ (included that) in an increasing order. We have the right of ''movement'' $K$: [i]We choose any two numbers $a,b$ that are written in consecutive positions and we replace the pair $(a,b)$ by the number $(a-b)^{2020}$.[/i] We repeat the movement $K$, many times until only one number remains written on the board. Determine whether it would be possible, that number to be: (i) $2020^{2020}$ (ii)$2021^{2020}$

[u]Part 1 [/u] [b]p1.[/b] Calculate $$(4!-5!+2^5 +2^6) \cdot \frac{12!}{7!}+(1-3)(4!-2^4).$$ [b]p2.[/b] The expression $\sqrt{9!+10!+11!}$ can be expressed as $a\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is squarefree. Find $a$. [b]p3.[/b] For real numbers $a$ and $b$, $f(x) = ax^{10}-bx^4+6x +10$ for all real $x$. Given that $f(42) = 11$, find $f (-42)$. [u]Part 2[/u] [b]p4.[/b] How many positive integers less than or equal to $2023$ are divisible by $20$, $23$, or both? [b]p5.[/b] Larry the ant crawls along the surface of a cylinder with height $48$ and base radius $\frac{14}{\pi}$ . He starts at point $A$ and crawls to point $B$, traveling the shortest distance possible. What is the maximum this distance could be? [b]p6.[/b] For a given positive integer $n$, Ben knows that $\lfloor 20x \rfloor = n$, where $x$ is real. With that information, Ben determines that there are $3$ distinct possible values for $\lfloor 23x \rfloor$. Find the least possible value of $n$. [u]Part 3 [/u] [b]p7.[/b] Let $ABC$ be a triangle with area $1$. Points $D$, $E$, and $F$ lie in the interior of $\vartriangle ABC$ in such a way that $D$ is the midpoint of $AE$, $E$ is the midpoint of $BF$, and $F$ is the midpoint of $CD$. Compute the area of $DEF$. [b]p8.[/b] Edwin and Amelia decide to settle an argument by running a race against each other. The starting line is at a given vertex of a regular octahedron and the finish line is at the opposite vertex. Edwin has the ability to run straight through the octahedron, while Amelia must stay on the surface of the octahedron. Given that they tie, what is the ratio of Edwin’s speed to Amelia’s speed? [b]p9.[/b] Jxu is rolling a fair three-sided die with faces labeled $0$, $1$, and $2$. He keeps going until he rolls a $1$, immediately followed by a $2$. What is the expected number of rolls Jxu makes? [u]Part 4 [/u] [b]p10.[/b] For real numbers $x$ and $y$, $x +x y = 10$ and $y +x y = 6$. Find the sum of all possible values of $\frac{x}{y}$. [b]p11.[/b] Derek is thinking of an odd two-digit integer $n$. He tells Aidan that $n$ is a perfect power and the product of the digits of $n$ is also a perfect power. Find the sum of all possible values of $n$. [b]p12.[/b] Let a three-digit positive integer $N = \overline{abc}$ (in base $10$) be stretchable with respect to $m$ if $N$ is divisible by $m$, and when $N$‘s middle digit is duplicated an arbitrary number of times, it‘s still divisible by $m$. How many three-digit positive integers are stretchable with respect to $11$? (For example, $432$ is stretchable with respect to $6$ because $433...32$ is divisible by $6$ for any positive integer number of $3$s.) [u]Part 5 [/u] [b]p13.[/b] How many trailing zeroes are in the base-$2023$ expansion of $2023!$ ? [b]p14.[/b] The three-digit positive integer $k = \overline{abc}$ (in base $10$, with a nonzero) satisfies $\overline{abc} = c^{2ab-1}$. Find the sum of all possible $k$. [b]p15.[/b] For any positive integer $k$, let $a_k$ be defined as the greatest nonnegative real number such that in an infinite grid of unit squares, no circle with radius less than or equal to $a_k$ can partially cover at least $k$ distinct unit squares. (A circle partially covers a unit square only if their intersection has positive area.) Find the sumof all positive integers $n \le 12$ such that $a_n \ne a_{n+1}$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3267915p30057005]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let \(S(n)\) denote the sum of the digits of the base-10 representation of an natural number \(n\). For example. \(S(2017) = 2+0+1+7 = 10\). Prove that for all primes \(p\), there exists infinitely many \(n\) which satisfy \(S(n) \equiv n \mod p\).

Let $p , q, r , s$ be four integers such that $s$ is not divisible by $5$. If there is an integer $a$ such that $pa^3 + qa^2+ ra +s$ is divisible be 5, prove that there is an integer $b$ such that $sb^3 + rb^2 + qb + p$ is also divisible by 5.

A two-digit integer is divided by the sum of its digits. Find the largest remainder that can occur.

Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties: 1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$; 2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$; Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.