Call a fraction $\frac{a}{b}$, not necessarily in the simplest form [i]special[/i] if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Let $M$ be a non-empty set of positive integers and let $S_M$ be the sum of all the elements of $M$. We define the [i]tlacoyo[/i] of $M$ as the sum of the digits of $S_M$. For example, if $M=\{2,7,34\}$, then $S_M=2+7+34=43$ and the tlacoyo of the set $M$ is $4+3=7$. \\ Prove that for every positive integer $n$, there exists a set $M$ of $n$ distinct positive integers, such that all its non-empty subsets have the same tlacoyo.

p1. If $x + y + z = 2$, show that $\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{xz+y-1}=\frac{-1}{(x-1)(y-1)(z-1)}$. p2. Determine the simplest form of $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\frac{5}{3!+4!+5!}+...+\frac{100}{98!+99!+100!}$ p3. It is known that $ABCD$ and $DEFG$ are two parallelograms. Point $E$ lies on $AB$ and point $C$ lie on $FG$. The area of $​​ABCD$ is $20$ units. $H$ is the point on $DG$ so that $EH$ is perpendicular to $DG$. If the length of $DG$ is $5$ units, determine the length of $EH$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/42453bf6768129ed84fbdc81ab7235e780b0e1.png[/img] p4. Each room in the following picture will be painted so that every two rooms which is directly connected to the door is given a different color. If $10$ different colors are provided and $4$ of them can not be used close together for two rooms that are directly connected with a door, determine how many different ways to color the $4$ rooms. [img]https://cdn.artofproblemsolving.com/attachments/4/a/e80a464a282b3fe3cdadde832b2fd38b51a41a.png[/img] 5. The floor of a hall is rectangular $ABCD$ with $AB = 30$ meters and $BC = 15$ meters. A cat is in position $A$. Seeing the cat, the mouse in the midpoint of $AB$ ran and tried to escape from cat. The mouse runs from its place to point $C$ at a speed of $3$ meters/second. The trajectory is a straight line. Watching the mice run away, at the same time from point $A$ the cat is chasing with a speed of $5$ meters/second. If the cat's path is also a straight line and the mouse caught before in $C$, determine an equation that can be used for determine the position and time the mouse was caught by the cat.

Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.

What is the smallest positive integer value of $x$ for which $x \equiv 4$ (mod $9$) and $x \equiv 7$ (mod $8$)?

$A$ -is $20$-digit number. We write $101$ numbers $A$ then erase last $11$ digits. Prove that this $2009$-digit number can not be degree of $2$

Let $q > 1$ be a power of $2$. Let $f: \mathbb{F}_{q^2} \to \mathbb{F}_{q^2}$ be an affine map over $\mathbb{F}_2$. Prove that the equation \[ f(x) = x^{q+1} \] has at most $2q - 1$ solutions.

Let $N=6+66+666+....+666..66$, where there are hundred $6's$ in the last term in the sum. How many times does the digit $7$ occur in the number $N$

Determine if there exist five consecutive positive integers such that their LCM is a perfect square.

Consider the polynomial $ P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38$. (a) Prove that there are at least nine integers $ a$ in the interval $ [1, 3^{2000}]$ for which $ P(a)$ is divisible by $ 3^{2000}$. (b) Find the number of integers $ a$ in $ [1, 3^{2000}]$ with the property from (a).

given $p_1,p_2,...$ be a sequence of integer and $p_1=2$, for positive integer $n$, $p_{n+1}$ is the least prime factor of $np_1^{1!}p_2^{2!}...p_n^{n!}+1 $ prove that all primes appear in the sequence (Proposed by Beatmania)

Given $2025$ positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining $2024$ numbers that are not hidden. What is the maximum number of the $2025$ written numbers that can be perfect squares? [i]Proposed by Oleksii Masalitin[/i]

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

Given $19$ red boxes and $200$ blue boxes filled with balls. None of which is empty. Suppose that every red boxes have a maximum of $200$ balls and every blue boxes have a maximum of $19$ balls. Suppose that the sum of all balls in the red boxes is less than the sum of all the balls in the blue boxes. Prove that there exists a subset of the red boxes and a subset of the blue boxes such that their sum is the same.

Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?

Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that: \[mp=x_1^2+x_2^2+x_3^2.\]

Let $P(x)$ and $Q(x)$ be integer polynomials of degree $p$ and $q$ respectively. Assume that $P(x)$ divides $Q(x)$ and all their coefficients are either $1$ or $2002$. Show that $p+1$ is a divisor of $q+1$. [i]Mihai Cipu[/i]

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for any two positive integers $a$ and $b$ we have $$ f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1)$$ Where $f^n(m)$ is defined in the standard iterative manner.

Find all perfect powers whose last $ 4$ digits are $ 2,0,0,8$, in that order.

Let $a$, $b$ and $c$ be given positive integers which are two by two coprime. A positive integer $n$ is called [i]sozopolian[/i], if it [u]can’t[/u] be written as $n=bcx+cay+abz$ where $x$, $y$, $z$ are also positive integers. Find the number of [i]sozopolian[/i] numbers as a function of $a$, $b$ and $c$.

Prove that there are do not exist natural numbers $k, m$ such that numbers $k^2+2m$, $m^2+2k$ to be squares of integers.

Does there exist a sequence $\{a_n\}$ of positive integers satisfying the following conditions: $a)$ every natural number occurs in this sequence and exactly once; $b)$ $a_1 + a_2 +... + a_n$ is divisible by $n^n$ for each $n = 1,2,3, ...$ ?

The positive integer $a$ is relatively prime with $10$. Prove that for any positive integer $n$, there exists a power of $a$ whose last $n$ digits are $\underbrace{0...0}_\text{n-1}1$.

Let $a, b$ and $c$ be pairwise relatively prime natural numbers. Find all possible values of $\frac{(a + b)(b + c)(c + a)}{abc}$ if known what it is integer.

$2 \leq d$ is a natural number. $B_{a,b}$={$a,a+b,a+2b,...,a+db$} $A_{c,q}$={$cq^n \vert n \in\mathbb{N}$} Prove that there are finite prime numbers like $p$ such exists $a,b,c,q$ from natural numbers : $i$ ) $ p \nmid abcq $ $ ii$ ) $A_{c,q} \equiv B_{a,b} (mod p ) $ (15 points )