An integer $n>1$ , whose positive divisors are $1=d_1<d_2< \cdots <d_k=n$, is called $\textit{southern}$ if all the numbers $d_2-d_1, d_3- d_2 , \cdots, d_k-d_{k-1}$ are divisors of $n$. a) Find a positive integer that is $\textit{not southern}$ and has exactly $2022$ positive divisors that are $\textit{southern}$. b) Show that there are infinitely many positive integers that are $\textit{not southern}$ and have exactly $2022$ positive divisors that are $\textit{southern}$.

Let $a,b,b',c,m,q$ be positive integers, where $m>1,q>1,|b-b'|\ge a$. It is given that there exist a positive integer $M$ such that $$S_q(an+b)\equiv S_q(an+b')+c\pmod{m}$$ holds for all integers $n\ge M$. Prove that the above equation is true for all positive integers $n$. (Here $S_q(x)$ is the sum of digits of $x$ taken in base $q$).

Let $a_n = 6^n + 8^n$. Determine the remainder on dividing $a_{83}$ by 49.

Call a $4$-digit number $\overline{a b c d}$ [i]unnoticeable [/i] if $a +c = b +d$ and $\overline{a b c d} +\overline{c d a b}$ is a multiple of $7$. Find the number of unnoticeable numbers. Note: $a$, $b$, $c$, and $d$ are nonzero distinct digits. [i]Proposed by Aditya Rao[/i]

In right triangle $ABC$ with right angle $C$, $CA=30$ and $CB=16$. Its legs $\overline{CA}$ and $\overline{CB}$ are extended beyond $A$ and $B$. Points $O_{1}$ and $O_{2}$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_{1}$ is tangent to the hypotenuse and to the extension of leg CA, the circle with center $O_{2}$ is tangent to the hypotenuse and to the extension of leg CB, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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

Compute the number of ordered triples $(a, b, c)$ of integers between $-100$ and $100$ inclusive satisfying the simultaneous equations $$a^3 - 2a = abc - b - c$$ $$b^3 - 2b = bca - c - a$$ $$c^3 - 2c = cab - a - b.$$

The sequence $(a_n)$ is determined by $a_1 = 0$ and $(n+1)^3a_{n+1} = 2n^2(2n+1)a_n+2(3n+1)$ for $n \geq 1$. Prove that infinitely many terms of the sequence are positive integers.

Prove that the sum of the squares of five consecutive integers cannot be a perfect square.

Let be $n$ a positive integer. Denote all its (positive) divisors as $1=d_1<d_2<\cdots<d_{k-1}<d_k=n$. Find all values of $n$ satisfying $d_5-d_3=50$ and $11d_5+8d_7=3n$. (Day 1, 1st problem author: Matúš Harminc)

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

Prova that any integer $ Z $ has a unique representation $$ a_0+a_12+a_22^2+a_32^3+\cdots +a_n2^n, $$ where $ n $ is natural, $ a_i\in\{ -1,0,+1\} $ for $ i=\overline{0,n} $ and $ a_ka_{k-1}=0 $ for $ k=\overline{1,n} . $

Let $A$ be the set of positive real numbers less than $1$ that have a periodic decimal expansion with a period of ten different digits. Find a positive integer $n$ greater than $1$ and less than $10^{10}$ such that $na-a$ is a positive integer for all $a$. of set $A$.

Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$. Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$. [i]N. Agakhanov[/i]

Consider the set $E$ consisting of pairs of integers $(a, b)$, with $a \geq 1$ and $b \geq 1$, that satisfy in the decimal system the following properties: [b](i)[/b] $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$, $\alpha_2 \neq 0$; [b](ii)[/b] $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$; [b](iii)[/b] $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$ Find the elements of $E$.

Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.

Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.

[b]p1.[/b] What is $20\times 19 + 20 \div (2 - 7)$? [b]p2.[/b] Will has three spinners. The first has three equally sized sections numbered $1$, $2$, $3$; the second has four equally sized sections numbered $1$, $2$, $3$, $4$; and the third has five equally sized sections numbered $1$, $2$, $3$, $4$, $5$. When Will spins all three spinners, the probability that the same number appears on all three spinners is $p$. Compute $\frac{1}{p}$. [b]p3.[/b] Three girls and five boys are seated randomly in a row of eight desks. Let $p$ be the probability that the students at the ends of the row are both boys. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p4.[/b] Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was $.300$. (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting $10$ home runs and striking out zero times in the last week, Jaron has now raised his batting average to $.310$. How many home runs has Jaron now hit? [b]p5.[/b] Suppose that the sum $$\frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100}$$ is expressible as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] Let $ABCD$ be a unit square with center $O$, and $\vartriangle OEF$ be an equilateral triangle with center $A$. Suppose that $M$ is the area of the region inside the square but outside the triangle and $N$ is the area of the region inside the triangle but outside the square, and let $x = |M -N|$ be the positive difference between $M$ and $N$. If $$x =\frac1 8(p -\sqrt{q})$$ for positive integers $p$ and $q$, find $p + q$. [b]p7.[/b] Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by $3$. For example, the number $1212121$ satisfies this property. [b]p8.[/b] There is a unique positive integer $x$ such that $x^x$ has $703$ positive factors. What is $x$? [b]p9.[/b] Let $x$ be the number of digits in $2^{2019}$ and let $y$ be the number of digits in $5^{2019}$. Compute $x + y$. [b]p10.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 13$ and $BC = 10$. Consider the set of all points $D$ in three-dimensional space such that $BCD$ is an equilateral triangle. This set of points forms a circle $\omega$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are tangent to $\omega$. If $EF^2$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, determine $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a prime number of the form $4k+3$. Prove that there are no integers $w,x,y,z$ whose product is not divisible by $p$, such that: \[ w^{2p}+x^{2p}+y^{2p}=z^{2p}. \]

Solve in natural numbers the equation $x^2-y^2=105$.

Determine the maximal size of the set $S$ such that: i) all elements of $S$ are natural numbers not exceeding $100$; ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$; iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$. [i]Yao Jiangang[/i]

Let $n,p>1$ be positive integers and $p$ be prime. We know that $n|p-1$ and $p|n^3-1$. Prove that $4p-3$ is a perfect square.

Does there exist an infinite non-constant arithmetic progression, each term of which is of the form $a^b$, where $a$ and $b$ are positive integers with $b\ge 2$?

What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.