6. A triple of integers $(a,b,c)$ is said to be $\gamma$[i]-special[/i] if $a\le \gamma(b+c)$, $b\le \gamma(c+a)$ and $c\le\gamma(a+b)$. For each integer triple $(a,b,c)$ such that $1\le a,b,c \le 20$, Kodvick writes down the smallest value of $\gamma$ such that $(a,b,c)$ is $\gamma$-special. How many distinct values does he write down? [i]Proposed by winnertakeover[/i]

An odd positive integer $n$ can be expressed as the sum of two or more consecutive integers in exactly $2023$ ways. Find the greatest possible nonnegative integer $k$ such that $3^k$ is a factor of the least possible value of $n$.

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

Show that there is a multiple of $2^{1998}$ whose decimal representation consists only of the digits $1$ and $2$.

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

Find all pairs $(n, p)$ of positive integers such that $p$ is prime and \[ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). \]

Evaluate the following $$\prod^{50}_{j=1} \left( 2 cos \left( \frac{4\pi j}{101} \right) + 1\right).$$

For each integer $n \geq 2$, let $F(n)$ denote the greatest prime factor of $n$. A [i]strange pair[/i] is a pair of distinct primes $p$ and $q$ such that there is no integer $n \geq 2$ for which $F(n)F(n+1)=pq$. Prove that there exist infinitely many strange pairs.

The Magician and his Assistant show trick. The Viewer writes on the board the sequence of $N$ digits. Then the Assistant covers some pair of adjacent digits so that they become invisible. Finally, the Magician enters the show, looks at the board and guesses the covered digits and their order. Find the minimal $N$ such that the Magician and his Assistant can agree in advance so that the Magician always guesses right

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Given functions $f,g:N^*\rightarrow N^*$, $g$ is surjective and $2f(n)^2=n^2+g(n)^2$, $\forall n>0$. Prove that if $|f(n)-n|\le2005\sqrt n$, $\forall n>0$, then $f(n)=n$ for infinitely many $n$.

Find the minimal $N$ such that any $N$-element subset of $\{1, 2, 3, 4,...,7\}$ has a subset $S$ such that the sum of elements of $S$ is divisible by $7$.

[b]p1.[/b] (a) Put the numbers $1$ to $6$ on the circle in such way that for any five consecutive numbers the sum of first three (clockwise) is larger than the sum of remaining two. (b) Can you arrange these numbers so it works both clockwise and counterclockwise. [b]p2.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p3.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p4.[/b] Two consecutive three‐digit positive integer numbers are written one after the other one. Show that the six‐digit number that is obtained is not divisible by $1001$. [b]p5.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $k\ge 14$ be an integer, and let $p_k$ be the largest prime number which is strictly less than $k$. You may assume that $p_k\ge 3k/4$. Let $n$ be a composite integer. Prove: (a) if $n=2p_k$, then $n$ does not divide $(n-k)!$; (b) if $n>2p_k$, then $n$ divides $(n-k)!$.

A positive integer $n$ is such that $n(n+2013)$ is a perfect square. a) Show that $n$ cannot be prime. b) Find a value of $n$ such that $n(n+2013)$ is a perfect square.

If three integers $p, q$ and $r$ apply that $$p + q^2 = r ^2.$$Show that $6$ adds up to $pqr$ .

Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

Solve the equation in integer numbers $$y^3-x^3=91$$

Prove that the Fibonacci sequence $\{F_n\}^\infty_{n=1}$ defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for all $n \geq 1$ is a divisibility sequence, that is, if $m\mid n$ then $F_m \mid F_n$ for all positive integers $m$ and $n$.

Find all prime numbers $p,q$ such that: $$p^4+p^3+p^2+p=q^2+q$$

Given $1000000000$ first natural numbers. We change each number with the sum of its digits and repeat this procedure until there will remain $1000000000$ one digit numbers. Is there more "$1$"-s or "$2$"-s?

Given a natural number $n$, call a divisor $d$ of $n$ to be $\textit{nontrivial}$ if $d>1$. A natural number $n$ is $\textit{good}$ if one or more distinct nontrivial divisors of $n$ sum up to $n-1$. Prove that every natural number $n$ has a multiple that is good.

Find all pairs of positive integers $(x, y)$ such that $(xy+1)(xy+x+2)$ be a perfect square .

Find all real numbers $ x$ such that $ 4x^5 \minus{} 7$ and $ 4x^{13} \minus{} 7$ are both perfect squares.

We are looking for numbers ending with the digit $5$ such that in their decimal expansion each digit beginning with the second digit is no less than the previous one. Moreover the squares of these numbers must also possess the same property. (a) Find four such numbers. (b) Prove that there are infinitely many. (A. Andjans, Riga)