Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23. [i]Author: K. Kokhas[/i]

Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.

Let $m,n$ be integers. It is known that there are integers $a,b$ such that $am+bn=1$ if, and only if, the greatest common divisor of $m,n$ is 1. [i]You are not required to prove this[/i]. Now suppose that $p,q$ are different odd primes. In each case determine if there are integers $a,b$ such that $ap+bq=1$ so that the given condition is satisfied: [list] a. $p$ divides $b$ and $q$ divides $a$; b. $p$ divides $a$ and $q$ divides $b$; c. $p$ does not divide $a$ and $q$ does not divide $b$. [/list]

p1. A telephone number with $7$ digits is called a [i]Beautiful Number [/i]if the digits are which appears in the first three numbers (the three must be different) repeats on the next three digits or the last three digits. For example some beautiful numbers: $7133719$, $7131735$, $7130713$, $1739317$, $5433354$. If the numbers are taken from $0, 1, 2, 3, 4, 5, 6, 7, 8$ or $9$, but the number the first cannot be $0$, how many Beautiful Numbers can there be obtained? p2. Find the number of natural numbers $n$ such that $n^3 + 100$ is divisible by $n +10$ p3. A function $f$ is defined as in the following table. [img]https://cdn.artofproblemsolving.com/attachments/5/5/620d18d312c1709b00be74543b390bfb5a8edc.png[/img] Based on the definition of the function $f$ above, then a sequence is defined on the general formula for the terms is as follows: $U_1=2$ and $U_{n+1}=f(U_n)$ , for $n = 1, 2, 3, ...$ p4. In a triangle $ABC$, point $D$ lies on side $AB$ and point $E$ lies on side $AC$. Prove for the ratio of areas: $\frac{ADE }{ABC}=\frac{AD\times AE}{AB\times AC}$ p5. In a chess tournament, a player only plays once with another player. A player scores $1$ if he wins, $0$ if he loses, and $\frac12$ if it's a draw. After the competition ended, it was discovered that $\frac12$ of the total value that earned by each player is obtained from playing with 10 different players who got the lowest total points. Especially for those in rank bottom ten, $\frac12$ of the total score one gets is obtained from playing with $9$ other players. How many players are there in the competition?

Find the least positive integer ending in $7$ with exactly $12$ positive divisors.

Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.

Prove that $n^4 - n^2$ is divisible by $12$ for all integers $n > 1$.

You are given a balance and one copy of each of ten weights of $1, 2, 4, 8, 16, 32, 64, 128, 256$ and $512$ grams. An object weighing $M$ grams, where $M$ is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance. (a) Prove that no object may be balanced in more than $89$ ways. (b) Find a value of $M$ such that an object weighing $M$ grams can be balanced in $89$ ways. (A Shapovalov, A Kulakov)

For positive integers $a$ and $b$, consider the curve $x^a + y^b = 1$ over real numbers $x$, $y$ and let $S(a, b)$ be the sum $P$ of the number of $x$-intercepts and $y$-intercepts of this curve. Compute $\sum^{10}_{a=1}\sum^5_{b=1} S(a, b).$

Peter picked an arbitrary positive integer, multiplied it by 5, multiplied the result by 5, then multiplied the result by 5 again and so on. Is it true that from some moment all the numbers that Peter obtains contain 5 in their decimal representation?

For a positive integer $n$ with prime factorization $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ let's define $\lambda(n) = (-1)^{\alpha_1 + \alpha_2 + \dots + \alpha_k}$. Define $L(n)$ as sum of $\lambda(x)$ over all integers from $1$ to $n$. Define $K(n)$ as sum of $\lambda(x)$ over all [b]composite[/b] integers from $1$ to $n$. For some $N>1$, we know, that for every $2\le n \le N$, $L(n)\le 0$. Prove that for this $N$, for every $2\le n \le N$, $K(n)\ge 0$. [i]Mykhailo Shtandenko[/i]

Let $\mathbb{F}_{p}$ denote the $p$-element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$. Find all functions $D\colon S\to S$ satisfying \[D(f\circ g)=(D(f)\circ g)\cdot D(g)\] for all $f,g \in {S}$. Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$, and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$.

$33$ horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time ?

Examine if we can put the sixteen positive divisors of $2024$ on the cells of the table shown such that the sum of the four numbers of any line or row to be a multiple of $3$. $ \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular} $

Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime.

2. Find all natural $n>1$ for which value of the sum $2^2+3^2+...+n^2$ equals $p^k$ where p is prime and k is natural

Suppose that $a,b,c$ are positive integers such that $a^b$ divides $b^c$, and $a^c$ divides $c^b$. Prove that $a^2$ divides $bc$.

Consider a prime number $p\geqslant 11$. We call a triple $a,b,c$ of natural numbers [i]suitable[/i] if they give non-zero, pairwise distinct residues modulo $p{}$. Further, for any natural numbers $a,b,c,k$ we define \[f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.\]Prove that there exist suitable $a,b,c$ for which $p\mid f_2(a,b,c)$. Furthermore, for each such triple, prove that there exists $k\geqslant 3$ for which $p\nmid f_k(a,b,c)$ and determine the minimal $k{}$ with this property. [i]Călin Popescu and Marian Andronache[/i]

Let $n$ be a positive integer and let $d_1, d_2, \dots, d_k$ be its positive divisors. Consider the number $$f(n) = (-1)^{d_1}d_1 + (-1)^{d_2}d_2 + \dots + (-1)^{d_k}d_k$$ Assume $f(n)$ is a power of 2. Show if $m$ is an integer greater than 1, then $m^2$ does not divide $n$.

Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n \plus{} 1$, $ n \plus{} 2$, $ n \plus{} 3$.

A $\textit{cubic sequence}$ is a sequence of integers given by $a_n =n^3 + bn^2 + cn + d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers. $\textbf{(a)}$ Show that there exists a cubic sequence such that the only terms of the sequence which are squares of integers are $a_{2015}$ and $a_{2016}$. $\textbf{(b)}$ Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part $\textbf{(a)}$.

Determine the least positive integer $n{}$ for which the following statement is true: the product of any $n{}$ odd consecutive positive integers is divisible by $45$.

Let $ p $ be prime. Denote by $ N (p) $ the number of integers $ x $ for which $ 1 \leq x \leq p $ and $$ x ^ {x} \equiv 1 \quad (\bmod p) $$Prove that there exist numbers $ c <1/2 $ and $ p_ {0}> 0 $ such that $$ N (p) \leq p ^ {c} $$if $ p \ge p_ {0} $.

Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence \[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \] forms an arithmetic progression. Prove that the terms of the sequence are equal.