Find all positive integers that can be written in the following way $\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}$ . Also, $a,b, c$ are positive integers that are pairwise relatively prime.

Let $a>1$ be an odd positive integer. Find the least positive integer $n$ such that $2^{2000}$ is a divisor of $a^n-1$. [i]Mircea Becheanu [/i]

Find all positive integers $n$ such that there exists an infinite set $A$ of positive integers with the following property: For all pairwise distinct numbers $a_1, a_2, \ldots , a_n \in A$, the numbers $$a_1 + a_2 + \ldots + a_n \text{ and } a_1\cdot a_2\cdot \ldots\cdot a_n$$ are coprime.

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 $(a,b,c)$.

Kevin and Yang are playing a game. Yang has $2017 + \tbinom{2017}{2}$ cards with their front sides face down on the table. The cards are constructed as follows: [list] [*] For each $1 \le n \le 2017$, there is a blue card with $n$ written on the back, and a fraction $\tfrac{a_n}{b_n}$ written on the front, where $\gcd(a_n, b_n) = 1$ and $a_n, b_n > 0$. [*] For each $1 \le i < j \le 2017$, there is a red card with $(i, j)$ written on the back, and a fraction $\tfrac{a_i+a_j}{b_i+b_j}$ written on the front. [/list] It is given no two cards have equal fractions. In a turn Kevin can pick any two cards and Yang tells Kevin which card has the larger fraction on the front. Show that, in fewer than $10000$ turns, Kevin can determine which red card has the largest fraction out of all of the red cards.

Let $a_n$ be a sequence defined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$. Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$.

Solve the equation \[ x^3+2y^3-4x-5y+z^2=2012, \] in the set of integers.

Call a set of 3 positive integers $\{a,b,c\}$ a [i]Pythagorean[/i] set if $a,b,c$ are the lengths of the 3 sides of a right-angled triangle. Prove that for any 2 Pythagorean sets $P,Q$, there exists a positive integer $m\ge 2$ and Pythagorean sets $P_1,P_2,\ldots ,P_m$ such that $P=P_1, Q=P_m$, and $\forall 1\le i\le m-1$, $P_i\cap P_{i+1}\neq \emptyset$.

Determine all positive integers equal to 13 times the sum of their digits.

When some number $a^2$ is written in base $b$, the result is $144_b$. $a$ and $b$ also happen to be integer side lengths of a right triangle. If $a$ and $b$ are both less than $20$, find the sum of all possible values of $a$.

Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then $$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$ is not an integer.

Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.

Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]

$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$

Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition \[a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,\] then $k-2$ is divisible by $3$. [i]Proposed by Okan Tekman, Turkey[/i]

Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that \[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \] and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$. [i]Proposed by Victor Wang[/i]

Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1$ (mod $2017$). ($2017$ is prime.)

Find all triplets of positive integers $(x, y, z)$ such that $2^x+1=7^y+2^z$.

Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying: (1)$\tau (n)=a$ (2)$n|\phi (n)+\sigma (n)$ Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

[b]p1.[/b] What is the sum of the first $12$ positive integers? [b]p2.[/b] How many positive integers less than or equal to $100$ are multiples of both $2$ and $5$? [b]p3. [/b]Alex has a bag with $4$ white marbles and $4$ black marbles. She takes $2$ marbles from the bag without replacement. What is the probability that both marbles she took are black? Express your answer as a decimal or a fraction in lowest terms. [b]p4.[/b] How many $5$-digit numbers are there where each digit is either $1$ or $2$? [b]p5.[/b] An integer $a$ with $1\le a \le 10$ is randomly selected. What is the probability that $\frac{100}{a}$ is an integer? Express your answer as decimal or a fraction in lowest terms. [b]p6.[/b] Two distinct non-tangent circles are drawn so that they intersect each other. A third circle, distinct from the previous two, is drawn. Let $P$ be the number of points of intersection between any two circles. How many possible values of $P$ are there? [b]p7.[/b] Let $x, y, z$ be nonzero real numbers such that $x + y + z = xyz$. Compute $$\frac{1 + yz}{yz}+\frac{1 + xz}{xz}+\frac{1 + xy}{xy}.$$ [b]p8.[/b] How many positive integers less than $106$ are simultaneously perfect squares, cubes, and fourth powers? [b]p9.[/b] Let $C_1$ and $C_2$ be two circles centered at point $O$ of radii $1$ and $2$, respectively. Let $A$ be a point on $C_2$. We draw the two lines tangent to $C_1$ that pass through $A$, and label their other intersections with $C_2$ as $B$ and $C$. Let x be the length of minor arc $BC$, as shown. Compute $x$. [img]https://cdn.artofproblemsolving.com/attachments/7/5/915216d4b7eba0650d63b26715113e79daa176.png[/img] [b]p10.[/b] A circle of area $\pi$ is inscribed in an equilateral triangle. Find the area of the triangle. [b]p11.[/b] Julie runs a $2$ mile route every morning. She notices that if she jogs the route $2$ miles per hour faster than normal, then she will finish the route $5$ minutes faster. How fast (in miles per hour) does she normally jog? [b]p12.[/b] Let $ABCD$ be a square of side length $10$. Let $EFGH$ be a square of side length $15$ such that $E$ is the center of $ABCD$, $EF$ intersects $BC$ at $X$, and $EH$ intersects $CD$ at $Y$ (shown below). If $BX = 7$, what is the area of quadrilateral $EXCY$ ? [img]https://cdn.artofproblemsolving.com/attachments/d/b/2b2d6de789310036bc42d1e8bcf3931316c922.png[/img] [b]p13.[/b] How many solutions are there to the system of equations $$a^2 + b^2 = c^2$$ $$(a + 1)^2 + (b + 1)^2 = (c + 1)^2$$ if $a, b$, and $c$ are positive integers? [b]p14.[/b] A square of side length $ s$ is inscribed in a semicircle of radius $ r$ as shown. Compute $\frac{s}{r}$. [img]https://cdn.artofproblemsolving.com/attachments/5/f/22d7516efa240d00d6a9743a4dc204d23d190d.png[/img] [b]p15.[/b] $S$ is a collection of integers n with $1 \le n \le 50$ so that each integer in $S$ is composite and relatively prime to every other integer in $S$. What is the largest possible number of integers in $S$? [b]p16.[/b] Let $ABCD$ be a regular tetrahedron and let $W, X, Y, Z$ denote the centers of faces $ABC$, $BCD$, $CDA$, and $DAB$, respectively. What is the ratio of the volumes of tetrahedrons $WXYZ$ and $WAYZ$? Express your answer as a decimal or a fraction in lowest terms. [b]p17.[/b] Consider a random permutation $\{s_1, s_2, ... , s_8\}$ of $\{1, 1, 1, 1, -1, -1, -1, -1\}$. Let $S$ be the largest of the numbers $s_1$, $s_1 + s_2$, $s_1 + s_2 + s_3$, $...$ , $s_1 + s_2 + ... + s_8$. What is the probability that $S$ is exactly $3$? Express your answer as a decimal or a fraction in lowest terms. [b]p18.[/b] A positive integer is called [i]almost-kinda-semi-prime[/i] if it has a prime number of positive integer divisors. Given that there $are 168$ primes less than $1000$, how many almost-kinda-semi-prime numbers are there less than $1000$? [b]p19.[/b] Let $ABCD$ be a unit square and let $X, Y, Z$ be points on sides $AB$, $BC$, $CD$, respectively, such that $AX = BY = CZ$. If the area of triangle $XYZ$ is $\frac13$ , what is the maximum value of the ratio $XB/AX$? [img]https://cdn.artofproblemsolving.com/attachments/5/6/cf77e40f8e9bb03dea8e7e728b21e7fb899d3e.png[/img] [b]p20.[/b] Positive integers $a \le b \le c$ have the property that each of $a + b$, $b + c$, and $c + a$ are prime. If $a + b + c$ has exactly $4$ positive divisors, find the fourth smallest possible value of the product $c(c + b)(c + b + a)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $(a_n)_{n = 0}^{\infty} $ be a sequence of positive integers such that for every $n \geq 0$ it is true that $$a_{n+2} = a_0 a_1 + a_1 a_2 + ... + a_n a_{n+1} - 1 $$ a) Prove that there exist a prime number which divides infinitely many $a_n$ b) Prove that there exist infinitely many such prime numbers

Let $n$ be a positive integer and let $k$ be an odd positive integer. Moreover, let $a,b$ and $c$ be integers (not necessarily positive) satisfying the equations \[a^n+kb=b^n+kc=c^n+ka \] Prove that $a=b=c$.

Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]