[b]Q.[/b] Show that for any given positive integers $k, l$, there exists infinitely many positive integers $m$, such that $i) m \geqslant k$ $ii) \text{gcd}\left(\binom{m}{k}, l\right)=1$ [i]Suggested by pigeon_in_a_hole[/i]

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.

Are there such pairwise distinct natural numbers $ m, n, p, q$ satisfying $ m \plus{} n \equal{} p \plus{} q$ and $ \sqrt{m} \plus{} \sqrt[3]{n} \equal{} \sqrt{p} \plus{} \sqrt[3]{q} > 2004$ ?

At each of the given $n$ points on a circle, either $+1$ or $-1$ is written. The following operation is performed: between any two consecutive numbers on the circle their product is written, and the initial $n$ numbers are deleted. Suppose that, for any initial arrangement of $+1$ and $-1$ on the circle, after finitely many operations all the numbers on the circle will be equal to $+1$. Prove that $n$ is a power of two.

Does there exist natural numbers $a,b,c$ all greater than $10^{10}$ such that their product is divisible by each of these numbers increased by $2012$?

[b]p1.[/b] A Slavic dragon has three heads. A knight fights the dragon. If the knight cuts off one dragon’s head three new heads immediately grow. Is it possible that the dragon has $2018$ heads at some moment of the fight? [b]p2.[/b] Peter has two squares $3\times 3$ and $4\times 4$. He must cut one of them or both of them in no more than four parts in total. Is Peter able to assemble a square using all these parts? [b]p3.[/b] Usually, dad picks up Constantine after his music lessons and they drive home. However, today the lessons have ended earlier and Constantine started walking home. He met his dad $14$ minutes later and they drove home together. They arrived home $6$ minutes earlier than usually. Home many minutes earlier than usual have the lessons ended? Please, explain your answer. [b]p4.[/b] All positive integers from $1$ to $2018$ are written on a blackboard. First, Peter erased all numbers divisible by $7$. Then, Natalie erased all remaining numbers divisible by $11$. How many numbers did Natalie remove? Please, explain your answer. [b]p5.[/b] $30$ students took part in a mathematical competition consisting of four problems. $25$ students solved the first problem, $24$ students solved the second problem, $22$ students solved the third, and, finally, $21$ students solved the fourth. Show that there are at least two students who solved all four problems. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

We call a positive integer [i] good[/i] if the sum of the reciprocals of all its natural divisors are integers. Prove that if $ m $ is a [i]good [/i] number, and $ p> m $ is a prime number, then $ pm $ is not [i]good[/i].

Let $n > 10$ be an odd integer. Determine the number of ways to place the numbers $1, 2, \ldots , n$ around a circle so that each number in the circle divides the sum its two neighbors. (Two configurations such that one can be obtained from the other per rotation are to be counted only once.)

[b]p1.[/b] Find the minimum value of $x^4 +2x^3 +3x^2 +2x+2$, where x can be any real number. [b]p2.[/b] A type of digit-lock has $5$ digits, each digit chosen from $\{1,2, 3, 4, 5\}$. How many different passwords are there that have an odd number of $1$'s? [b]p3.[/b] Tony is a really good Ping Pong player, or at least that is what he claims. For him, ping pong balls are very important and he can feel very easily when a ping pong ball is good and when it is not. The Ping Pong club just ordered new balls. They usually order form either PPB company or MIO company. Tony knows that PPB balls have $80\%$ chance to be good balls and MIO balls have $50\%$ chance to be good balls. I know you are thinking why would anyone order MIO balls, but they are way cheaper than PPB balls. When the box full with balls arrives (huge number of balls), Tony tries the first ball in the box and realizes it is a good ball. Given that the Ping Pong club usually orders half of the time from PPB and half of the time from MIO, what is the probability that the second ball is a good ball? [b]p4.[/b] What is the smallest positive integer that is one-ninth of its reverse? [b]p5.[/b] When Michael wakes up in the morning he is usually late for class so he has to get dressed very quickly. He has to put on a short sleeved shirt, a sweater, pants, two socks and two shoes. People usually put the sweater on after they put the short sleeved shirt on, but Michael has a different style, so he can do it both ways. Given that he puts on a shoe on a foot after he put on a sock on that foot, in how many dierent orders can Michael get dressed? [b]p6.[/b] The numbers $1, 2,..., 2015$ are written on a blackboard. At each step we choose two numbers and replace them with their nonnegative difference. We stop when we have only one number. How many possibilities are there for this last number? [b]p7.[/b] Let $A = (a_1b_1a_2b_2... a_nb_n)_{34}$ and $B = (b_1b_2... b_n)_{34}$ be two numbers written in base $34$. If the sum of the base-$34$ digits of $A$ is congruent to $15$ (mod $77$) and the sum of the base $34$ digits of $B$ is congruent to $23$ (mod $77$). Then if $(a_1b_1a_2b_2... a_nb_n)_{34} \equiv x$ (mod $77$) and $0 \le x \le 76$, what is $x$? (you can write $x$ in base $10$) [b]p8.[/b] What is the sum of the medians of all nonempty subsets of $\{1, 2,..., 9\}$? [b]p9.[/b] Tony is moving on a straight line for $6$ minutes{classic Tony. Several finitely many observers are watching him because, let's face it, you can't really trust Tony. In fact, they must watch him very closely{ so closely that he must never remain unattended for any second. But since the observers are lazy, they only watch Tony uninterruptedly for exactly one minute, and during this minute, Tony covers exactly one meter. What is the sum of the minimal and maximal possible distance Tony can walk during the six minutes? [b]p10.[/b] Find the number of nonnegative integer triplets $a, b, c$ that satisfy $$2^a3^b + 9 = c^2.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For $k = 1, 2, \ldots$ consider the $k$-tuples $(a_1, a_2, \ldots, a_k)$ of positive integers such that \[a_1 + 2a_2 + \cdots + ka_k = 1979.\] Show that there are as many such $k$-tuples with odd $k$ as there are with even $k$.

Determine all positive integers $A= \overline{a_n a_{n-1} \ldots a_1 a_0}$ such that not all of its digits are equal and no digit is $0$, and $A$ divides all numbers of the following form: $A_1 = \overline{a_0 a_n a_{n-1} \ldots a_2 a_1}, A_2 = \overline{a_1 a_0 a_{n} \ldots a_3 a_2}, \ldots ,$ $ A_{n-1} = \overline{a_{n-2} a_{n-3} \ldots a_0 a_n a_{n-1}}, A_n = \overline{a_{n-1} a_{n-2} \ldots a_1 a_0 a_n}$.

$a_1, a_2,...,a_{2006}$ is a permutation of $1,2,...,2006$. Prove that $\prod_{i = 1}^{2006} (a_{i}^2-i) $ is a multiple of $3$. ($0$ is counted as a multiple of $3$)

$P(x)$ is polynomial with degree $n>5$ and integer coefficients have $n$ different integer roots. Prove that $P(x)+3$ have $n$ different real roots.

Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities: $$[a, b] =1000, [b, c] = 2000, [c, a] =2000.$$ ([x, y]represents the least common multiple of positive integers x,y)

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

Let $A=\{1,3,3^2,\ldots, 3^{2014}\}$. We obtain a partition of $A$ if $A$ is written as a disjoint union of nonempty subsets. [list=a] [*]Prove that there is no partition of $A$ such that the product of elements in each subset is a square. [*]Prove that there exists a partition of $A$ such that the sum of elements in each subset is a square.[/list]

Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$. [i]Proposed by Amirhossein Gorzi[/i]

a) Consider the positive integers $a, b, c$ so that $a < b < c$ and $a^2+b^2 = c^2$. If $a_1 = a^2$, $a_2 = ab$, $a_3 = bc$, $a_4 = c^2$, prove that $a_1^2+a_2^2+a_3^2=a_4^2$ and $a_1 < a_2 < a_3 < a_4$. b) Show that for any $n \in N$, $n\ge 3$, there exist the positive integers $a_1, a_2,..., a_n$ so that $a_1^2+a_2^2+...+ a_{n-1}^2=a_n^2$ and $a_1 < a_2 < ...< a_{n-1} < a_n$

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

Determine all numbers $ \overline{abcd}$ such that $ \overline{abcd}\equal{}11(a\plus{}b\plus{}c\plus{}d)^2$.

Find all triples $(p, q, r)$ of prime numbers for which \[(p+1)(q+2)(r+3)=4pqr. \]

The number $328$ is written on the board. Two players alternate writing positive divisors of $328$ on the board, subject to the following rules: $\bullet$ No divisor of a previously written number may be written. $\bullet$ The player who writes 328 loses. Who has a winning strategy, the first player or the second player?

Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$

Find two positive integers $a$ and $b$, when their sum and their least common multiple is given. Find the numbers when the sum is $3972$ and the least common multiple is $985928$.