(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$, (2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$. Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?

For an integer $n \ge 0$, let $f(n)$ be the smallest possible value of $ |x + y|$, where $x$ and $y$ are integers such that $3x - 2y = n$. Evaluate $f(0) + f(1) + f(2) +...+ f(2013)$.

Let $p{}$ be a prime number. Prove that the equation has $x^2-x+3-ps=0$ with $x,s\in\mathbb{Z}$ has solutions if and only if the equation $y^2-y+25-pt=0$ with $y,t\in\mathbb{Z}$ has solutions.

Positive integers $a$, $b$ and $c$ satisfy the system of equations \begin{align*} (ab-1)^2&=c(a^2+b^2)+ab+1,\\ a^2+b^2&=c^2+ab. \end{align*} a) Prove that $c+1$ is a perfect square. b) Find all such triples $(a,b,c)$.

Let $\overline{a_1a_2a_3\ldots a_n}$ be the representation of a $n-$digits number in base $10.$ Prove that there exists a one-to-one function like $f : \{0, 1, 2, 3, \ldots, 9\} \to \{0, 1, 2, 3, \ldots, 9\}$ such that $f(a_1) \neq 0$ and the number $\overline{ f(a_1)f(a_2)f(a_3) \ldots f(a_n) }$ is divisible by $3.$

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

A sequence $(a_n)$ of positive integers is defined by $a_0=m$ and $a_{n+1}= a_n^5 +487$ for all $n\ge 0$. Find all positive integers $m$ such that the sequence contains the maximum possible number of perfect squares.

Is it possible for any finite group of people to choose a positive integer $N$ and assign a positive integer to each person in the group such that the product of two persons' number is divisible by $N$ if and only if they are friends?

[url=https://artofproblemsolving.com/community/c675547][b]Greece JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663730p10567608][b]Problem 1[/b][/url]. Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that $$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$ Also, find the values of $a,b,c$ for which the equality happens. [url=http://artofproblemsolving.com/community/c6h1663731p10567619][b]Problem 2[/b][/url]. Let $ABC$ be an acute-angled triangle inscribed in a circle $\mathcal C (O, R)$ and $F$ a point on the side $AB$ such that $AF < AB/2$. The circle $c_1(F, FA)$ intersects the line $OA$ at the point $A'$ and the circle $\mathcal C$ at $K$. Prove that the quadrilateral $BKFA'$ is cyclic and its circumcircle contains point $O$. [url=http://artofproblemsolving.com/community/c6h1663732p10567627][b]Problem 3[/b][/url]. Prove that for every positive integer $n$, the number $A_n = 7^{2n} -48n - 1$ is a multiple of $9$. [url=http://artofproblemsolving.com/community/c6h1663734p10567640][b]Problem 4[/b][/url]. Let $ABC$ be an equilateral triangle of side length $a$, and consider $D$, $E$ and $F$ the midpoints of the sides $(AB), (BC)$, and $(CA)$, respectively. Let $H$ be the the symmetrical of $D$ with respect to the line $BC$. Color the points $A, B, C, D, E, F, H$ with one of the two colors, red and blue. [list=1] [*] How many equilateral triangles with all the vertices in the set $\{A, B, C, D, E, F, H\}$ are there? [*] Prove that if points $B$ and $E$ are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set $\{A, B, C, D, E, F, H\}$ and having the same color. [*] Does the conclusion of the second part remain valid if $B$ is blue and $E$ is red? [/list]

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.

Let $p$ be a prime number and $h$ be a natural number smaller than $p$. We set $n = ph + 1$. Prove that if $2^{n-1}-1$, but not $2^h-1$, is divisible by $n$, then $n$ is a prime number.

Prove that if $m=5^n+3^n+1$ is a prime, then $12$ divides $n$.

In a mathematical version of baseball, the umpire chooses a positive integer $m$, $m \leq n$, and you guess positive integers to obtain information about $m$. If your guess is smaller than the umpire's $m$, he calls it a "ball"; if it is greater than or equal to $m$, he calls it a "strike." To "hit" it you must state the the correct value of $m$ after the 3rd strike or the 6th guess, whichever comes first. What is the largest $n$ so that there exists a strategy that will allow you to bat 1.000, i.e. always state $m$ correctly? Describe your strategy in detail.

Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]

Solve the equation for primes $p$ and $q$: $$p^3-q^3=pq^3-1.$$

Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k\geqslant 2$ such that $3^n=x^k+y^k$.

a) A positive integer is called [i]nice [/i] if it can be represented as an arithmetic mean of some (not necessarily distinct) positive integers each being a nonnegative power of $2$. Prove that all positive integers are nice. b) A positive integer is called [i]ugly [/i] if it can not be represented as an arithmetic mean of some pairwise distinct positive integers each being a nonnegative power of $2$. Prove that there exist infinitely many ugly positive integers. (A. Romanenko, D. Zmeikov)

For $n$ a positive integer, define $f_1(n)=n$ and then for $i$ a positive integer, define $f_{i+1}(n)=f_i(n)^{f_i(n)}$. Determine $f_{100}(75)\pmod{17}$. Justify your answer.

Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.

In how many ways can the numbers from $2$ to $2022$ be arranged so that the first number is a multiple of $1$, the second number is a multiple of $2$, the third number is a multiple of $3$, and so on untile the last number is a multiple of $2021$?

Determine all non-constant monic polynomials $f(x)$ with integer coefficients for which there exists a natural number $M$ such that for all $n \geq M$, $f(n)$ divides $f(2^n) - 2^{f(n)}$ [i] Proposed by Anant Mudgal [/i]

Find all integers $n$, such that the following number is a perfect square \[N= n^4 + 6n^3 + 11n^2 +3n+31. \]

Find all triples of positive integers $(a, b, c)$ satisfying $(a^3+b)(b^3+a)=2^c$.