Prove that for every $ n \in \mathbb{N}^*$ exists a multiple of $ n$, having sum of digits equal to $ n$.

Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?

Let $A$ be a set of positive integers satisfying the following properties: (i) if $m$ and $n$ belong to $A$, then $m+n$ belong to $A$; (ii) there is no prime number that divides all elements of $A$. (a) Suppose $n_1$ and $n_2$ are two integers belonging to $A$ such that $n_2-n_1 >1$. Show that you can find two integers $m_1$ and $m_2$ in $A$ such that $0< m_2-m_1 < n_2-n_1$ (b) Hence show that there are two consecutive integers belonging to $A$. (c) Let $n_0$ and $n_0+1$ be two consecutive integers belonging to $A$. Show that if $n\geq n_0^2$ then $n$ belongs to $A$.

Let $a,b $ be two relatively prime positive integers.Also let $m,n $ be positive integers with $n> m $.\\ Prove that\\ $lcm [am+b,a (m+1)+b,...,an+b]\ge (n+1)\cdot \binom {n}{m}$ [i]Proposed by Navid Safaei[/i]

Prove that there are infinitely many multiples of 2005 that contain all the digits 0, 1, 2,...,9 an equal number of times.

[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].

Let $n_1,\ldots,n_k$ be positive integers, and define $d_1=1$ and $d_i=\frac{(n_1,\ldots,n_{i-1})}{(n_1,\ldots,n_{i})}$, for $i\in \{2,\ldots,k\}$, where $(m_1,\ldots,m_{\ell})$ denotes the greatest common divisor of the integers $m_1,\ldots,m_{\ell}$. Prove that the sums \[\sum_{i=1}^k a_in_i\] with $a_i\in\{1,\ldots,d_i\}$ for $i\in\{1,\ldots,k\}$ are mutually distinct $\mod n_1$.

Find the largest integer $k$ such that $k$ divides $n^{55} - n$ for all integer $n$.

Find the positive integer $n{}$ if $$\left(1-\frac{1}{1+2}\right)\cdot\left(1-\frac{1}{1+2+3}\right)\cdot\ldots\cdot\left(1-\frac{1}{1+2+\ldots+n}\right)=\frac{2021}{6057}.$$

Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$.

The numbers from $1$ to $2000^2$ were written on a board. Vasya choose $2000$ of them whose sum of them equal to two thousandth of the sum of all numbers. Proof that his friend, Petya, will be able to color each of the remaining numbers by one of other $1999$ colors so that the sum of numbers with each of total $2000$ colors are the same.

Find the number of eight-digit positive integers that are multiples of $9$ and have all distinct digits.

Find the number of positive divisors of $20^{22}$ that are perfect squares or perfect cubes.

[u]Round 1[/u] [b]1.1.[/b] A classroom has $29$ students. A teacher needs to split up the students into groups of at most $4$. What is the minimum number of groups needed? [b]1.2.[/b] On his history map quiz, Eric recalls that Sweden, Norway and Finland are adjacent countries, but he has forgotten which is which, so he labels them in random order. The probability that he labels all three countries correctly can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]1.3.[/b] In a class of $40$ sixth graders, the class average for their final test comes out to be $90$ (out of a $100$). However, a student brings up an issue with problem $5$, and $10$ students receive credit for this question, bringing the class average to a $90.75$. How many points was problem $5$ worth? [u]Round 2[/u] [b]2.1.[/b] Compute $1 - 2 + 3 - 4 + ... - 2022 + 2023$. [b]2.2.[/b] In triangle $ABC$, $\angle ABC = 75^o$. Point $D$ lies on side $AC$ such that $BD = CD$ and $\angle BDC$ is a right angle. Compute the measure of $\angle A$. [b]2.3.[/b] Joe is rolling three four-sided dice each labeled with positive integers from $1$ to $4$. The probability the sum of the numbers on the top faces of the dice is $6$ can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime integers. Find $p + q$. [u]Round 3[/u] [b]3.1.[/b] For positive integers $a, b, c, d$ that satisfy $a + b + c + d = 23$, what is the maximum value of $abcd$? [b]3.2.[/b] A buckball league has twenty teams. Each of the twenty teams plays exactly five games with each of the other teams. If each game takes 1 hour and thirty minutes, then how many total hours are spent playing games? [b]3.3.[/b] For a triangle $\vartriangle ABC$, let $M, N, O$ be the midpoints of $AB$, $BC$, $AC$, respectively. Let $P, Q, R$ be points on $AB$, $BC$, $AC$ such that $AP =\frac13 AB$, $BQ =\frac13 BC$, and $CR =\frac13 AC$. The ratio of the areas of $\vartriangle MNO$ and $\vartriangle P QR$ can be expressed as $\frac{m}{n}$ , where $ m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Round 4[/u] [b]4.1.[/b] $2023$ has the special property that leaves a remainder of $1$ when divided by $2$, $21$ when divided by $22$, and $22$ when divided by $23$. Let $n$ equal the lowest integer greater than $2023$ with the above properties. What is $n$? [b]4.2.[/b] Ants $A, B$ are on points $(0, 0)$ and $(3, 3)$ respectively, and ant A is trying to get to $(3, 3)$ while ant $B$ is trying to get to $(0, 0)$. Every second, ant $A$ will either move up or right one with equal probability, and ant $B$ will move down or left one with equal probability. The probability that the ants will meet each other be $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]4.3.[/b] Find the number of trailing zeros of $100!$ in base $ 49$. PS. You should use hide for answers. Rounds 5-9 have been posted [url=https://artofproblemsolving.com/community/c3h3129723p28347714]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The numbers $25$ and $76$ have the property that when squared in base $10$, their squares also end in the same two digits. A positive integer is called [i]amazing [/i] if it has at most $3$ digits when expressed in base $21$ and also has the property that its square expressed in base $21$ ends in the same $3$ digits. (For this problem, the last three digits of a one-digit number b are 00b, and the last three digits of a two-digit number $\underline{ab}$ are $0\underline{ab}$.) Compute the sum of all amazing numbers. Express your answer in base $21$.

Let $C=\{4,6,8,9,10,\ldots\}$ be the set of composite positive integers. For each $n\in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$. Determine whether the following series converges: $$\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]

Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

Given are positive integers $m, n$ and a sequence $a_1, a_2, \ldots, $ such that $a_i=a_{i-n}$ for all $i>n$. For all $1 \leq j \leq n$, let $l_j$ be the smallest positive integer such that $m \mid a_j+a_{j+1}+\ldots+a_{j+l_j-1}$. Prove that $l_1+l_2+\ldots+l_n \leq mn$.

Find the number of subsets of $\{1, 2,... , 2100\}$ such that each has sum of the elements giving a remainder of $3$ when divided by $7$.

Prove that for all natural numbers $n$, \[ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. \] Here, $\{x\}$ denotes the fractional part of $x$.

Determine all polynomials $P(x)$ with integer coefficients such that $P(n)$ is divisible by $\sigma(n)$ for all positive integers $n$. (As usual, $\sigma(n)$ denotes the sum of all positive divisors of $n$.)

$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$

Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.

Prove that if $m=5^n+3^n+1$ is a prime, then $12$ divides $n$.