Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

Let's call integer square-free if it's not divisible by $p^2$ for any prime $p$. You are given a square-free integer $n>1$, which has exactly $d$ positive divisors. Find the largest number of its divisors that you can choose, such that $a^2 + ab - n$ isn't a square of an integer for any $a, b$ among chosen divisors. [i](Proposed by Oleksii Masalitin)[/i]

Find all $n \in \mathbb N$ such that $n = \varphi (n) + 402$, where $\varphi$ denotes the Euler phi function.

Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime.

Let $ n$ be a positive integer. Show that \[ \left(\sqrt{2} \plus{} 1 \right)^n \equal{} \sqrt{m} \plus{} \sqrt{m\minus{}1}\] for some positive integer $ m.$

Determine the real numbers $ a,b, $ such that $$ [ax+by]+[bx+ay]=(a+b)\cdot [x+y],\quad\forall x,y\in\mathbb{R} , $$ where $ [t] $ is the greatest integer smaller than $ t. $

Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any positive integers $i, j$: $$d(a_i + a_j ) = i + j?$$ Here $d(n)$ is the number of positive divisors of a positive integer

Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.

The Fibonacci numbers $(F_n)_{n=1}^{\infty}$ are defined as follows: $$F_1 = F_2 = 1, F_n = F_{n-2} + F_{n-1}, n = 3, 4, ...$$ Assume $p$ is a prime greater than $3$. With $m$ being a natural number greater than $3$, find all $n$ numbers such that $F_n$ is divisible by $p^m$.

Denote by $a \bmod b$ the remainder of the euclidean division of $a$ by $b$. Determine all pairs of positive integers $(a,p)$ such that $p$ is prime and \[ a \bmod p + a\bmod 2p + a\bmod 3p + a\bmod 4p = a + p. \]

Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a [i]$p$-pod[/i] if for each $e \geq 0$, there is an $N \geq 0$ such that whenever $m \geq N$, $p^e$ divides the sum \[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\] Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$-pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$-pod.

Let $S$ be the set of positive integers in which the greatest and smallest elements are relatively prime. For natural $n$, let $S_n$ denote the set of natural numbers which can be represented as sum of at most $n$ elements (not necessarily different) from $S$. Let $a$ be greatest element from $S$. Prove that there are positive integer $k$ and integers $b$ such that $|S_n| = a \cdot n + b$ for all $ n > k $.

Find all integers $ a$, $ b$, $ n$ greater than $ 1$ which satisfy \[ \left(a^3 \plus{} b^3\right)^n \equal{} 4(ab)^{1995} \]

A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible? $ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$

Let $n = 2k + 1$ be an odd positive integer, and $m$ be an integer realtively prime to $n{}$. For each $j =1,2,\ldots,k$ we define $p_j$ as the unique integer from the interval $[-k, k]$ congruent to $m\cdot j$ modulo $n{}$. Prove that there are equally many pairs $(i,j)$ for which $1\leqslant i<j\leqslant k$ which satisfy $|p_i|>|p_j|$ as those which satisfy $p_ip_j<0$.

Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.

A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.

Let $\sum_{i=1}^n a_i x_i =0$, $a_i,x_i\in \mathbb{Z}$. It is known that however we color $\mathbb{Z}$ with finite number of colors, then the given equation has a monochromatic (of one color) solution. Prove that there is some non-empty sum of its coefficients equal to 0.

A sequence of integers $a_0, a_1 …$ is called [i]kawaii[/i] if $a_0 =0, a_1=1,$ and $$(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0$$ for all integers $n \geq 0$. An integer is called [i]kawaii[/i] if it belongs to some kawaii sequence. Suppose that two consecutive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that $m$ is divisible by $3,$ and that $m/3$ is also kawaii.

Suppose that $m$ and $n$ are positive integers with $m < n$ such that the interval $[m, n)$ contains more multiples of $2021$ than multiples of $2000$. Compute the maximum possible value of $n - m$.

Find the smallest and largest integers with decimal representation of the form $ababa$ ($a \neq 0$) that are divisible by $11$.

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$. [b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers. [b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$. [b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area? [img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$. Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?

Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer. [i]Note: A monic polynomial has a leading coefficient equal to 1.[/i] [i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]

Find all possible integer values of the sum: $$\frac{a}{b}+ \frac{2023 \times b}{4 \times a},$$ where $a$ and $b$ are positive integers with no prime factors in common.