Find all pairs $(x,y)$, $x,y\in \mathbb{N}$ for which $gcd(n(x!-xy-x-y+2)+2,n(x!-xy-x-y+3)+3)>1$ for $\forall$ $n\in \mathbb{N}$.

Find all positive integers $r$ with the property that there exists positive prime numbers $p$ and $q$ so that $$p^2 + pq + q^2 = r^2 .$$

Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.

$a$ and $b$ are positive integers. When written in binary, $a$ has $2004$ $1$'s, and $b$ has $2005$ $1$'s (not necessarily consecutive). What is the smallest number of $1$'s $a + b$ could possibly have?

Find all pairs of positive integers $(x,y)$, so that the number $x^3+y^3$ is a prime.

Let $A$ be a finite set of positive integers. Prove that there exists a finite set $B$ of positive integers such that $A \subseteq B$ and \[\prod_{x\in B} x = \sum_{x\in B} x^2.\]

[b]p1.[/b] Suppose that Yunseo wants to order a pizza that is cut into $4$ identical slices. For each slice, there are $2$ toppings to choose from: pineapples and apples. Each slice must have exactly one topping. How many distinct pizzas can Yunseo order? Pizzas that can be obtained by rotating one pizza are considered the same. [b]p2.[/b] How many triples of distinct positive integers $(E, M, C)$ are there such that $E = MC^2$ and $E \le 50$? [b]p3.[/b] Given that the cubic polynomial $p(x)$ has leading coefficient $1$ and satisfies $p(0) = 0$, $p(1) = 1$, and $p(2) = 2$. Find $p(3)$. [b]p4.[/b] Olaf asks Anna to guess a two-digit number and tells her that it’s a multiple of $7$ with two distinct digits. Anna makes her first guess. Olaf says one digit is right but in the wrong place. Anna adjusts her guess based on Olaf’s comment, but Olaf answers with the same comment again. Anna now knows what the number is. What is the sum of all the numbers that Olaf could have picked? [b]p5.[/b] Vincent the Bug draws all the diagonals of a regular hexagon with area $720$, splitting it into many pieces. Compute the area of the smallest piece. [b]p6.[/b] Given that $y - \frac{1}{y} = 7 + \frac{1}{7}$, compute the least integer greater than $y^4 + \frac{1}{y^4}$. [b]p7.[/b] At $9:00$ A.M., Joe sees three clouds in the sky. Each hour afterwards, a new cloud appears in the sky, while each old cloud has a $40\%$ chance of disappearing. Given that the expected number of clouds that Joe will see right after $1:00$ P.M. can be written in the form $p/q$ , where $p$ and $q$ are relatively prime positive integers, what is $p + q$? [b]p8.[/b] Compute the unique three-digit integer with the largest number of divisors. [b]p9.[/b] Jo has a collection of $101$ books, which she reads one each evening for $101$ evenings in a predetermined order. In the morning of each day that Jo reads a book, Amy chooses a random book from Jo’s collection and burns one page in it. What is the expected number of pages that Jo misses? [b]p10.[/b] Given that $x, y, z$ are positive real numbers satisfying $2x + y = 14 - xy$, $3y + 2z = 30 - yz$, and $z + 3x = 69 - zx$, the expression $x + y + z$ can be written as $p\sqrt{q} - r$, where $p, q, r$ are positive integers and $q$ is not divisible by the square of any prime. Compute $p + q + r$. [b]p11.[/b] In rectangle $TRIG$, points $A$ and $L$ lie on sides $TG$ and $TR$ respectively such that $TA = AG$ and $TL = 2LR$. Diagonal $GR$ intersects segments $IL$ and $IA$ at $B$ and $E$ respectively. Suppose that the area of the convex pentagon with vertices $TABLE$ is equal to $21$. What is the area of $TRIG$? [b]p12.[/b] Call a number nice if it can be written in the form $2^m \cdot 3^n$, where $m$ and $n$ are nonnegative integers. Vincent the Bug fills in a $3$ by $3$ grid with distinct nice numbers, such that the product of the numbers in each row and each column are the same. What is the smallest possible value of the largest number Vincent wrote? [b]p13.[/b] Let $s(n)$ denote the sum of digits of positive integer $n$ and define $f(n) = s(202n) - s(22n)$. Given that $M$ is the greatest possible value of $f(n)$ for $0 < n < 350$ and $N$ is the least value such that $f(N) = M$, compute $M + N$. [b]p14.[/b] In triangle $ABC$, let M be the midpoint of $BC$ and let $E, F$ be points on $AB, AC$, respectively, such that $\angle MEF = 30^o$ and $\angle MFE = 60^o$. Given that $\angle A = 60^o$, $AE = 10$, and $EB = 6$,compute $AB + AC$. [b]p15.[/b] A unit cube moves on top of a $6 \times 6$ checkerboard whose squares are unit squares. Beginning in the bottom left corner, the cube is allowed to roll up or right, rolling about its bottom edges to travel from square to square, until it reaches the top right corner. Given that the side of the cube facing upwards in the beginning is also facing upwards after the cube reaches the top right corner, how many total paths are possible? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$, \[a^n + a^{n-1} +
\cdots
+ a^1 + 1 \mid a^{n!} + a^{(n-1)!} +
\cdots

+ a^{1!} + 1.\] [i]Proposed by Milos Milosavljevic[/i]

In how many ways can you choose several numbers from the numbers $1,2,3,..., 11$ so that among the selected numbers there are not three consecutive numbers?

Prove that for any integers $a,b$, the equation $2abx^4 - a^2x^2 - b^2 - 1 = 0$ has no integer roots. [i](Dan Schwarz)[/i]

Prove that there exists infinitely many positive integers $n$ such that equation $(x+y+z)^3=n^2xyz$ has solution $(x,y,z)$ in set $\mathbb{N}^3$

What is the units digit of $13^{2012}$ ? $\textbf{(A)}\hspace{.05in}1 \qquad \textbf{(B)}\hspace{.05in}3 \qquad \textbf{(C)}\hspace{.05in}5 \qquad \textbf{(D)}\hspace{.05in}7 \qquad \textbf{(E)}\hspace{.05in}9 $

We say a number is irie if it can be written in the form $1+\dfrac{1}{k}$ for some positive integer $k$. Prove that every integer $n \geq 2$ can be written as the product of $r$ distinct irie numbers for every integer $r \geq n-1$.

For any integer $n\ge 4$, prove that there exists a $n$-degree polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ satisfying the two following properties: [b](1)[/b] $a_i$ is a positive integer for any $i=0,1,\ldots,n-1$, and [b](2)[/b] For any two positive integers $m$ and $k$ ($k\ge 2$) there exist distinct positive integers $r_1,r_2,...,r_k$, such that $f(m)\ne\prod_{i=1}^{k}f(r_i)$.

Working alone, the expert can paint a car in one day, the amateur can paint a car in two days, and the beginner can paint a car in three days. If the three painters work together at these speeds to paint three cars, it will take them $\frac{m}{n}$ days where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$.

The number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special because it is not relatively prime to the sum of the special numbers less than it, $1 + 2$. The number $4$ is special because it is relatively prime to the sum of the special numbers less than it. So, a number bigger than $1$ is special only if it is relatively prime to the sum of the special numbers less than it. Find the twentieth special number.

How many pairs of integers $(a,b)$ are there such that $0\leq a < b \leq 100$ and such that $\tfrac{2^b-2^a}{2016}$ is an integer?

Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.

Find all pairs $(a, b)$ of coprime positive integers, such that $a<b$ and $$b \mid (n+2)a^{n+1002}-(n+1)a^{n+1001}-na^{n+1000}$$ for all positive integers $n$.

A company of $n$ soldiers is such that (i) $n$ is a palindrome number (read equally in both directions); (ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively. Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$.

Show that the $120$ five digit numbers which are permutations of $12345$ can be divided into two sets with each set having the same sum of squares.

Find all ordered pairs $(a,b)$ of positive integers such that $2a + 1$ divides $3b - 1$ and $2b + 1$ divides $3a -1$.

Find the last three digits in the product $1 \cdot 3\cdot 5\cdot 7 \cdot . . . \cdot 2009 \cdot 2011$.