Let $N$ be the smallest positive integer such that $N+2N+3N+\ldots +9N$ is a number all of whose digits are equal. What is the sum of digits of $N$?

Let $a, b, c$ be positive integers such that one of the values $$gcd(a,b) \cdot lcm(b,c), \,\,\,\, gcd(b,c)\cdot lcm(c,a), \,\,\,\, gcd(c,a)-\cdot lcm(a,b)$$ is equal to the product of the remaining two. Prove that one of the numbers $a, b, c$ is a multiple of another of them.

Find the smallest positive multiple of $77$ whose last four digits (from left to right) are $2020$.

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.

Prove that the equation $$\frac x{1+x^2}+\frac y{1+y^2}+\frac z{1+z^2}=\frac1{1996}$$has finitely many solutions in positive integers.

Prove that the product of $8$ consecutive natural numbers can never be a fourth power of natural number.

Let $m$ and $n$ be positive integers where $m$ has $d$ digits in base ten and $d\leq n$. Find the sum of all the digits (in base ten) of the product $(10^n-1)m$.

[b]p1.[/b] (a) Brian has a big job to do that will take him two hours to complete. He has six friends who can help him. They all work at the same rate, somewhat slower than Brian. All seven working together can finish the job in $45$ minutes. How long will it take to do the job if Brian worked with only three of his friends? (b) Brian could do his next job in $N$ hours, working alone. This time he has an unlimited list of friends who can help him, but as he moves down the list, each friend works more slowly than those above on the list. The first friend would take $kN$ ($k > 1$) hours to do the job alone, the second friend would take $k^2N$ hours alone, the third friend would take $k^3N$ hours alone, etc. Theoretically, if Brian could get all his infinite number of friends to help him, how long would it take to complete the job? [b]p2.[/b] (a) The centers of two circles of radius $1$ are two opposite vertices of a square of side $1$. Find the area of the intersection of the two circles. (b) The centers of two circles of radius $1$ are two consecutive vertices of a square of side $1$. Find the area of the intersection of the two circles and the square. (c) The centers of four circles of radius $1$ are the vertices of a square of side $1$. Find the area of the intersection of the four circles. [b]p3.[/b] For any real number$ x$, $[x]$ denotes the greatest integer that does not exceed $x$. For example, $[7.3] = 7$, $[10/3] = 3$, $[5] = 5$. Given natural number $N$, denote as $f(N)$ the following sum of $N$ integers: $$f(N) = [N/1] + [N/2] + [N/3] + ... + [N/n].$$ (a) Evaluate $f(7) - f(6)$. (b) Evaluate $f(35) - f(34)$. (c) Evaluate (with explanation) $f(1996) - f(1995)$. [b]p4.[/b] We will say that triangle $ABC$ is good if it satisfies the following conditions: $AB = 7$, the other two sides are integers, and $\cos A =\frac27$. (a) Find the sides of a good isosceles triangle. (b) Find the sides of a good scalene (i.e. non-isosceles) triangle. (c) Find the sides of a good scalene triangle other than the one you found in (b) and prove that any good triangle is congruent to one of the three triangles you have found. [b]p5.[/b] (a) A bag contains nine balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. How many of the nine balls were of one color and how many of the other color? (b) A bag contains $N$ balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. It is also found that $180 < N < 220$. Find the exact value of $N$ and determine how many of the $N$ balls were of one color and how many of the other color. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many ways are there to permute the first $n$ positive integers such that in the permutation, for each value of $k \le n$, the first $k$ elements of the permutation have distinct remainder mod $k$?

Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.

Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?

Let $N$ be the set of those positive integers $n$ for which $n\mid k^k-1$ implies $n\mid k-1$ for every positive integer $k$. Prove that if $n_1,n_2\in N$, then their greatest common divisor is also in $N$.

We say that a set $S$ of natural numbers is [i]synchronous [/i] provided that the digits of $a^2$ are the same (in occurence and numbers, if differently ordered) for all numbers $a$ in $S$. For example, $\{13, 14, 31\}$ is synchronous, since we find $\{13^2, 14^2, 31^2\} = \{169, 196, 961\}$. But $\{119, 121\}$ is not synchronous, for even though $119^2 = 14161$ and $121^2 = 14641$ have the same digits, they occur in different numbers. Show that there exists a synchronous set containing $2021$ different natural numbers.

Let $ a$, $ b$ and $ m$ be positive integers. Prove that there exists a positive integer $ n$ such that $ (a^n \minus{} 1)b$ is divisible by $ m$ if and only if $ \gcd (ab, m) \equal{} \gcd (b, m)$.

Find the sum (in base $10$) of the three greatest numbers less than $1000_{10}$ that are palindromes in both base $10$ and base $5$.

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

Let $a,b,c$ be given positive integers. Prove that there exists some positive integer $N$ such that \[ a\mid Nbc+b+c,\ b\mid Nca+c+a,\ c\mid Nab+a+b \] if and only if, denoting $d=\gcd(a,b,c)$ and $a=dx$, $b=dy$, $c=dz$, the positive integers $x,y,z$ are pairwise coprime, and also $\gcd(d,xyz) \mid x+y+z$. (Dan Schwarz)

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

Determine $ \gcd (n^2\plus{}2,n^3\plus{}1)$ for $ n\equal{}9^{753}$.

For each integer $n\geq 3$, find the least natural number $f(n)$ having the property $\star$ For every $A \subset \{1, 2, \ldots, n\}$ with $f(n)$ elements, there exist elements $x, y, z \in A$ that are pairwise coprime.

The natural number $a_n$ is obtained by writing together and ordered, in decimal notation , all natural numbers between $1$ and $n$. So we have for example that $a_1 = 1$,$a_2 = 12$, $a_3 = 123$, $. . .$ , $a_{11} = 1234567891011$, $...$ . Find all values of $n$ for which $a_n$ is not divisible by $3$.

Let \( p \geq 7 \) be a prime number and \( m \in \mathbb{N} \). Prove that \[\left| p^m - (p - 2)! \right| > p^2.\] [i]Proposed by Miloš Milićev[/i]

$a$ and $b$ are positive integers. When written in binary, $a$ has $2004$ $1$'s, and $b$ has $2005$ $1$'s (not necessarily consecutive). What is the smallest number of $1$'s $a + b$ could possibly have?

Denote by $x_n(p)$ the multiplicity of the prime $p$ in the canonical representation of the number $n!$ as a product of primes. Prove that $\frac{x_n(p)}{n}<\frac{1}{p-1}$ and $\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}$.

Prove that there exists and infinity of multiples of $1997$ that have $1998$ as first four digits and last four digits.