Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

Knowing that $23$ October $1948$ was a Saturday, which is more frequent for New Year’s Day, Sunday or Monday?

Let numbers $1,2,3,...,2010$ stand in a row at random. Consider row, obtain by next rule: For any number we sum it and it's number in a row (For example for row $( 2,7,4)$ we consider a row $(2+1;7+2;4+3)=(3;9;7)$ ); Proved, that in resulting row we can found two equals numbers, or two numbers, which is differ by $2010$

Elisa adds the digits of her year of birth and observes that the result coincides with the last two digits of the year her grandfather was born. Furthermore, the last two digits of the year she was born are precisely the current age of her grandfather. Find the year Elisa was born and the year her grandfather was born.

Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$. Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$.

1) Cells of $8 \times 8$ table contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exists integer written in the same row or in the same column such that it is not relatively prime with $a$. Find maximum possible number of prime integers in the table. 2) Cells of $2n \times 2n$ table, $n \ge 2,$ contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exist integers written in the same row and in the same column such that they are not relatively prime with $a$. Find maximum possible number of prime integers in the table.

Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n \plus{} 1$?

Let $\mathbb{F}_p$ be the set of integers modulo $p$. Call a function $f : \mathbb{F}_p^2 \to \mathbb{F}_p$ [i]quasiperiodic[/i] if there exist $a,b \in \mathbb{F}_p$, not both zero, so that $f(x + a, y + b) = f(x, y)$ for all $x,y \in \mathbb{F}_p$. Find the number of functions $\mathbb{F}_p^2 \to \mathbb{F}_p$ that can be written as the sum of some number of quasiperiodic functions.

Compute the remainder when $2^{3^5}+ 3^{5^2}+ 5^{2^3}$ is divided by $30$.

Fix a set of integers $S$. An integer is [i]clean[/i] if it is the sum of distinct elements of $S$ in exactly one way, and [i]dirty[/i] otherwise. Prove that the set of dirty numbers is either empty or infinite. [i]Note:[/i] We consider the empty sum to equal \(0\). [i]Proposed by Tony Wang and Ethan Tan[/i]

Determine all sequences $a_1, a_2, a_3, \dots$ of nonnegative integers such that $a_1 < a_2 < a_3 < \dots$ and $a_n$ divides $a_{n - 1} + n$ for all $n \geq 2$.

A set consisting of at least two distinct positive integers is called [i]centenary [/i] if its greatest element is $100$. We will consider the average of all numbers in a centenary set, which we will call the average of the set. For example, the average of the centenary set $\{1, 2, 20, 100\}$ is $\frac{123}{4}$ and the average of the centenary set $\{74, 90, 100\}$ is $88$. Determine all integers that can occur as the average of a centenary set.

Let $n\geqslant 2$ and $a_1,a_2,\ldots,a_n$ be non-zero integers such that $a_1+a_2+\cdots+a_n=a_1a_2\cdots a_n.$ Prove that \[(a_1^2-1)(a_2^2-1)\cdots(a_n^2-1)\]is a perfect square.

Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.

Compute the exact value of \[ \sum_{a=1}^{\infty}\sum_{b=1}^{\infty} \frac{\gcd(a, b)}{(a + b)^3}. \] If necessary, you may express your answer in terms of the Riemann zeta function, $Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, for integers $s \geq 2$.

What is the largest even positive integer that cannot be expressed as the sum of two composite odd numbers?

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

p1. Circle $M$ is the incircle of ABC, while circle $N$ is the incircle of $ACD$. Circles $M$ and $N$ are tangent at point $E$. If side length $AD = x$ cm, $AB = y$ cm, $BC = z$ cm, find the length of side $DC$ (in terms of $x, y$, and $z$). [img]https://cdn.artofproblemsolving.com/attachments/d/5/66ddc8a27e20e5a3b27ab24ff1eba3abee49a6.png[/img] p2. The address of the house on Jalan Bahagia will be numbered with the following rules: $\bullet$ One side of the road is numbered with consecutive even numbers starting from number $2$. $\bullet$ The opposite side is numbered with an odd number starting from number $3$. $\bullet$ In a row of even numbered houses, there is some land vacant house that has not been built. $\bullet$ The first house numbered $2$ has a neighbor next door. When the RT management ordered the numbers of the house, it is known that the cost of making each digit is $12.000$ Rp. For that, the total cost to be incurred is $1.020.000$ Rp. It is also known that the cost of all even-sided house numbers is $132.000$ Rp. cheaper than the odd side. When the land is empty later a house has been built, the number of houses on the even and odd sides is the same. Determine the number of houses that are now on Jalan Bahagia . p3. Given the following problem: Each element in the set $A = \{10, 11, 12,...,2008\}$ multiplied by each element in the set $B = \{21, 22, 23,...,99\}$. The results are then added together to give value of $X$. Determine the value of $X$. Someone answers the question by multiplying $2016991$ with $4740$. How can you explain that how does that person make sense? p4. Let $P$ be the set of all positive integers between $0$ and $2008$ which can be expressed as the sum of two or more consecutive positive integers . (For example: $11 = 5 + 6$, $90 = 29 + 30 + 31$, $100 = 18 + 19 +20 + 21 + 22$. So $11, 90, 100$ are some members of $P$.) Find the sum of of all members of $P$. p5. A four-digit number will be formed from the numbers at $0, 1, 2, 3, 4, 5$ provided that the numbers in the number are not repeated, and the number formed is a multiple of $3$. What is the probability that the number formed has a value less than $3000$?

Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that: [i](i)[/i] $i$ and $n - i$ always receive the same color, and [i](ii)[/i] for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$ Prove that all numbers in $N$ must receive the same color.

$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$.

Let us consider a polynomial $P(x)$ with integers coefficients satisfying $$P(-1)=-4,\ P(-3)=-40,\text{ and } P(-5)=-156.$$ What is the largest possible number of integers $x$ satisfying $$P(P(x))=x^2?$$

A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$

Suppose that $n$ is a natural number and $n$ is not divisible by $3$. Prove that $(n^{2n}+n^n+n+1)^{2n}+(n^{2n}+n^n+n+1)^n+1$ has at least $2d(n)$ distinct prime factors where $d(n)$ is the number of positive divisors of $n$. [i]proposed by Mahyar Sefidgaran[/i]