Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n \minus{} 1.$

Let us denote with $P(n)$ the product of all the digits of $n$. Consider the sequence $$n_{k+1} = n_k + P(n_k)$$ Can it be unbounded for some $n_1$?

Let $p$ be some odd prime number and let $k=\frac{p+1}{2}$. The natural numbers $a_1,a_2…a_k$ are such that $a_i\neq a_j$ and $a_i<p$ for $\forall i,j=1,2…k$. Prove that for each natural number $r<p$ there exist not necessarily different $a_i$ and $a_j$, for which $a_i a_j\equiv r\, (mod\, p)$.

Let $n \ge 1$. Prove that it is possible to select some of the integers $1,2,...,2^n$ so that for each $p = 0,1,...,n - 1$ the sum of the $p$-th powers of the selected numbers is equal to the sum of the $p$-th powers of the remaining numbers.

Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$

Determine the smallest positive prime $p$ which satisfies the congruence \[p+p^{-1}\equiv 25\pmod{143}.\] Here, $p^{-1}$ as usual denotes multiplicative inverse.

For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.

Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and \[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\] Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

Is there a positive integer with at most four digits whose value is increased by exactly $60\%$ when the first digit is moved to the end of the number? For example, when the first digit of $1234$ is moved to the end of the number, the result is the integer $2341$.

[b]p1.[/b] Twelve people, three of whom are in the Mafia and one of whom is a police inspector, randomly sit around a circular table. What is the probability that the inspector ends up sitting next to at least one of the Mafia? [b]p2.[/b] Of the positive integers between $1$ and $1000$, inclusive, how many of them contain neither the digit “$4$” nor the digit “$7$”? [b]p3.[/b] You are really bored one day and decide to invent a variation of chess. In your variation, you create a new piece called the “krook,” which, on any given turn, can move either one square up or down, or one square left or right. If you have a krook at the bottom-left corner of the chessboard, how many different ways can the krook reach the top-right corner of the chessboard in exactly $17$ moves? [b]p4.[/b] Let $p$ be a prime number. What is the smallest positive integer that has exactly $p$ different positive integer divisors? Write your answer as a formula in terms of $p$. [b]p5.[/b] You make the square $\{(x, y)| - 5 \le x \le 5, -5 \le y \le 5\}$ into a dartboard as follows: (i) If a player throws a dart and its distance from the origin is less than one unit, then the player gets $10$ points. (ii) If a player throws a dart and its distance from the origin is between one and three units, inclusive, then the player gets awarded a number of points equal to the number of the quadrant that the dart landed on. (The player receives no points for a dart that lands on the coordinate axes in this case.) (iii) If a player throws a dart and its distance from the origin is greater than three units, then the player gets $0$ points. If a person throws three darts and each hits the board randomly (i.e with uniform distribution), what is the expected value of the score that they will receive? [b]p6.[/b] Teddy works at Please Forget Meat, a contemporary vegetarian pizza chain in the city of Gridtown, as a deliveryman. Please Forget Meat (PFM) has two convenient locations, marked with “$X$” and “$Y$ ” on the street map of Gridtown shown below. Teddy, who is currently at $X$, needs to deliver an eggplant pizza to $\nabla$ en route to $Y$ , where he is urgently needed. There is currently construction taking place at $A$, $B$, and $C$, so those three intersections will be completely impassable. How many ways can Teddy get from $X$ to $Y$ while staying on the roads (Traffic tickets are expensive!), not taking paths that are longer than necessary (Gas is expensive!), and that let him pass through $\nabla$ (Losing a job is expensive!)? [img]https://cdn.artofproblemsolving.com/attachments/e/0/d4952e923dc97596ad354ed770e80f979740bc.png[/img] [b]p7.[/b] $x, y$, and $z$ are positive real numbers that satisfy the following three equations: $$x +\frac{1}{y}= 4 \,\,\,\,\, y +\frac{1}{z}= 1\,\,\,\,\, z +\frac{1}{x}=\frac73.$$ Compute $xyz$. [b]p8.[/b] Alan, Ben, and Catherine will all start working at the Duke University Math Department on January $1$st, $2009$. Alan’s work schedule is on a four-day cycle; he starts by working for three days and then takes one day off. Ben’s work schedule is on a seven-day cycle; he starts by working for five days and then takes two days off. Catherine’s work schedule is on a ten-day cycle; she starts by working for seven days and then takes three days off. On how many days in $2009$ will none of the three be working? [b]p9.[/b] $x$ and $y$ are complex numbers such that $x^3 + y^3 = -16$ and $(x + y)^2 = xy$. What is the value of $|x + y|$? [b]p10.[/b] Call a four-digit number “well-meaning” if (1) its second digit is the mean of its first and its third digits and (2) its third digit is the mean of its second and fourth digits. How many well-meaning four-digit numbers are there? (For a four-digit number, its first digit is its thousands [leftmost] digit and its fourth digit is its units [rightmost] digit. Also, four-digit numbers cannot have “$0$” as their first digit.) [b]p11.[/b] Suppose that $\theta$ is a real number such that $\sum^{\infty}{k=2} \sin \left(2^k\theta \right)$ is well-defined and equal to the real number $a$. Compute: $$\sum^{\infty}{k=0} \left(\cot^3 \left(2^k\theta \right)-\cot \left(2^k\theta \right) \right) \sin^4 \left(2^k\theta \right).$$ Write your answer as a formula in terms of $a$. [b]p12.[/b] You have $13$ loaded coins; the probability that they come up as heads are $\cos\left( \frac{0\pi}{24 }\right)$,$ \cos\left( \frac{1\pi}{24 }\right)$, $\cos\left( \frac{2\pi}{24 }\right)$, $...$, $\cos\left( \frac{11\pi}{24 }\right)$ and $\cos\left( \frac{12\pi}{24 }\right)$, respectively. You throw all $13$ of these coins in the air at once. What is the probability that an even number of them come up as heads? [b]p13.[/b] Three married couples sit down on a long bench together in random order. What is the probability that none of the husbands sit next to their respective wives? [b]p14.[/b] What is the smallest positive integer that has at least $25$ different positive divisors? [b]p15.[/b] Let $A_1$ be any three-element set, $A_2 = \{\emptyset\}$, and $A_3 = \emptyset$. For each $i \in \{1, 2, 3\}$, let: (i) $B_i = \{\emptyset,A_i\}$, (ii) $C_i$ be the set of all subsets of $B_i$, (iii) $D_i = B_i \cup C_i$, and (iv) $k_i$ be the number of different elements in $D_i$. Compute $k_1k_2k_3$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a prime number. Determine all triples $(a,b,c)$ of positive integers such that $a + b + c < 2p\sqrt{p}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{p}$

Determine all intergers $n\geq 2$ such that $a+\sqrt{2}$ and $a^n+\sqrt{2}$ are both rational for some real number $a$ depending on $n$

Consider 2016 points arranged on a circle. We are allowed to jump ahead by 2 or 3 points in clockwise direction. What is the minimum number of jumps required to visit all points and return to the starting point? (Gerd Baron)

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

For each positive integer $n$, let $c(n)$ be the number of digits of $n$. Let $A$ be a set of positive integers with the following property: If $a$ and $b$ are two distinct elements in $A$, then $c(a +b)+2 > c(a)+c(b)$. Find the largest number of elements that $A$ can have. PS. In the original wording: c(n) = ''cantidad de dıgitos''

Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \] [b]OR[/b] Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \]

Let $d(n)$ denote the number of positive divisors of $n$. For any given integer $a \geq 3$, define a sequence $\{a_i\}_{i=0}^\infty$ satisfying [list] [*] $a_{0}=a$, and [*] $a_{n+1}=a_{n}+(-1)^{n} d(a_{n})$ for each integer $n \geq 0$. [/list] For example, if $a=275$, the sequence would be \[275, \overline{281,279,285,277,279,273}.\] Prove that for each positive integer $k$ there exists a positive integer $N$ such that if such a sequence has period $2k$ and all terms of the sequence are greater than $N$ then all terms of the sequence have the same parity. [i]Proposed by Navid[/i]

Let $a,b,c,d,e,d,e,f$ be positive integers such that \(\dfrac a b < \dfrac c d < \dfrac e f\). Suppose \(af-be=-1\). Show that \(d \geq b+f\).

[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].

A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

Prove that, if the terms of an infinite arithmetic progression of natural numbers are not all equal, they cannot all be primes.

Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.