Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime. [i]Lucian Petrescu[/i]

A prime $p>3$ is given. Let $K$ be the number of such permutations $(a_1, a_2, \ldots, a_p)$ of $\{ 1, 2, \ldots, p\}$ such that $$a_1a_2+a_2a_3+\ldots + a_{p-1}a_p+a_pa_1$$ is divisible by $p$. Prove $K+p$ is divisible by $p^2$.

The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property: for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$. Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.

[u]Set 7[/u] [b]G19.[/b] How many ordered triples $(x, y, z)$ with $1 \le x, y, z \le 50$ are there such that both $x + y + z$ and $xy + yz + zx$ are divisible by$ 6$? [b]G20.[/b] Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. If $D$ is the foot of the perpendicular from $A$ to $BC$, then $AH = 8$ and $HD = 3$. If $\angle AOH = 90^o$, find $BC^2$. [b]G21.[/b] Nate flips a fair coin until he gets two heads in a row, immediately followed by a tails. The probability that he flips the coin exactly $12$ times is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 8[/u] [b]G22.[/b] Let $f$ be a function defined by $f(1) = 1$ and $$f(n) = \frac{1}{p}f\left(\frac{n}{p}\right)f(p) + 2p - 2,$$ where $p$ is the least prime dividing $n$, for all integers $n \ge 2$. Find $f(2022)$. [b]G23.[/b] Jessica has $15$ balls numbered $1$ through $15$. With her left hand, she scoops up $2$ of the balls. With her right hand, she scoops up $2$ of the remaining balls. The probability that the sum of the balls in her left hand is equal to the sum of the balls in her right hand can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]G24.[/b] Let $ABCD$ be a cyclic quadrilateral such that its diagonal $BD = 17$ is the diameter of its circumcircle. Given $AB = 8$, $BC = CD$, and that a line $\ell$ through A intersects the incircle of $ABD$ at two points $P$ and $Q$, the maximum area of $CP Q$ can be expressed as a fraction $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [u]Set 9[/u] [i]This set consists of three estimation problems, with scoring schemes described.[/i] [b]G25.[/b] Estimate $N$, the total number of participants (in person and online) at MOAA this year. An estimate of $e$ gets a total of max $ \left( 0, \lfloor 150 \left( 1- \frac{|N-e|}{N}\right) \rfloor -120 \right)$ points. [b]G26.[/b] If $A$ is the the total number of in person participants at MOAA this year, and $B$ is the total number of online participants at MOAA this year, estimate $N$, the product $AB$. An estimate of $e$ gets a total of max $(0, 30 - \lceil \log10(8|N - e| + 1)\rceil )$ points. [b]G27.[/b] Estimate $N$, the total number of letters in all the teams that signed up for MOAA this year, both in person and online. An estimate of e gets a total of max $(0, 30 - \lceil 7 log5(|N - E|)\rceil )$ points. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 4-6 [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Let $A\% B = BA - B - A + 1$. How many digits are in the number $1\%(3\%(3\%7))$ ? [b]p2. [/b]Three circles, of radii $1, 2$, and $3$ are all externally tangent to each other. A fourth circle is drawn which passes through the centers of those three circles. What is the radius of this larger circle? [b]p3.[/b] Express $\frac13$ in base $2$ as a binary number. (Which, similar to how demical numbers have a decimal point, has a “binary point”.) [b]p4. [/b] Isosceles trapezoid $ABCD$ with $AB$ parallel to $CD$ is constructed such that $DB = DC$. If $AD = 20$, $AB = 14$, and $P$ is the point on $AD$ such that $BP + CP$ is minimized, what is $AP/DP$? [b]p5.[/b] Let $f(x) = \frac{5x-6}{x-2}$ . Define an infinite sequence of numbers $a_0, a_1, a_2,....$ such that $a_{i+1} = f(a_i)$ and $a_i$ is always an integer. What are all the possible values for $a_{2014}$ ? [b]p6.[/b] $MATH$ and $TEAM$ are two parallelograms. If the lengths of $MH$ and $AE$ are $13$ and $15$, and distance from $AM$ to $T$ is $12$, find the perimeter of $AMHE$. [b]p7.[/b] How many integers less than $1000$ are there such that $n^n + n$ is divisible by $5$ ? [b]p8.[/b] $10$ coins with probabilities of $1, 1/2, 1/3 ,..., 1/10$ of coming up heads are flipped. What is the probability that an odd number of them come up heads? [b]p9.[/b] An infinite number of coins with probabilities of $1/4, 1/9, 1/16, ...$ of coming up heads are all flipped. What is the probability that exactly $ 1$ of them comes up heads? [b]p10.[/b] Quadrilateral $ABCD$ has side lengths $AB = 10$, $BC = 11$, and $CD = 13$. Circles $O_1$ and $O_2$ are inscribed in triangles $ABD$ and $BDC$. If they are both tangent to $BD$ at the same point $E$, what is the length of $DA$ ? PS. You had better use hide for answers.

Find all the positive integers $N$ with an even number of digits with the property that if we multiply the two numbers formed by cutting the number in the middle we get a number that is a divisor of $N$ ( for example $12$ works because $1 \cdot 2$ divides $12$)

Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$. For instance $d(6)=12$ and $\phi(7)=6$. Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.

Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$. Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively prime to $i$.

Let $n$ be a positive integer. Prove that \[ 3^{\dfrac{5^{2^n}-1}{2^{n+2}}} \equiv (-5)^{\dfrac{3^{2^n}-1}{2^{n+2}}} \pmod{2^{n+4}}. \]

Show that there are infinitely many positive integers $c$, such that the following equations both have solutions in positive integers: $(x^2 - c)(y^2 -c) = z^2 -c$ and $(x^2 + c)(y^2 - c) = z^2 - c$.

Let $p$ be a primes ,$x,y,z $ be positive integers such that $x<y<z<p$ and $\{\frac{x^3}{p}\}=\{\frac{y^3}{p}\}=\{\frac{z^3}{p}\}$. Prove that $(x+y+z)|(x^5+y^5+z^5).$

[b]p1.[/b] Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$. Given that $\angle ABC$ is a right angle, determine the length of $AC$. [b]p2.[/b] Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$. Find the largest possible value of $m-n$. [b]p3.[/b] Six middle school students are sitting in a circle, facing inwards, and doing math problems. There is a stack of nine math problems. A random student picks up the stack and, beginning with himself and proceeding clockwise around the circle, gives one problem to each student in order until the pile is exhausted. Aditya falls asleep and is therefore not the student who picks up the pile, although he still receives problem(s) in turn. If every other student is equally likely to have picked up the stack of problems and Vishwesh is sitting directly to Aditya’s left, what is the probability that Vishwesh receives exactly two problems? [b]p4.[/b] Paul bakes a pizza in $15$ minutes if he places it $2$ feet from the fire. The time the pizza takes to bake is directly proportional to the distance it is from the fire and the rate at which the pizza bakes is constant whenever the distance isn’t changed. Paul puts a pizza $2$ feet from the fire at $10:30$. Later, he makes another pizza, puts it $2$ feet away from the fire, and moves the first pizza to a distance of $3$ feet away from the fire instantly. If both pizzas finish baking at the same time, at what time are they both done? [b]p5.[/b] You have $n$ coins that are each worth a distinct, positive integer amount of cents. To hitch a ride with Charon, you must pay some unspecified integer amount between $10$ and $20$ cents inclusive, and Charon wants exact change paid with exactly two coins. What is the least possible value of $n$ such that you can be certain of appeasing Charon? [b]p6.[/b] Let $a, b$, and $c$ be positive integers such that $gcd(a, b)$, $gcd(b, c)$ and $gcd(c, a)$ are all greater than $1$, but $gcd(a, b, c) = 1$. Find the minimum possible value of $a + b + c$. [b]p7.[/b] Let $ABC$ be a triangle inscribed in a circle with $AB = 7$, $AC = 9$, and $BC = 8$. Suppose $D$ is the midpoint of minor arc $BC$ and that $X$ is the intersection of $\overline{AD}$ and $\overline{BC}$. Find the length of $\overline{BX}$. [b]p8.[/b] What are the last two digits of the simplified value of $1! + 3! + 5! + · · · + 2009! + 2011!$ ? [b]p9.[/b] How many terms are in the simplified expansion of $(L + M + T)^{10}$ ? [b]p10.[/b] Ben draws a circle of radius five at the origin, and draws a circle with radius $5$ centered at $(15, 0)$. What are all possible slopes for a line tangent to both of the circles? PS. You had better use hide for answers.

Prove that no number of the form $ 2^n $, where $ n $ is a natural number, is the sum of two or more consecutive natural numbers.

$n\ge 2$ positive integers are written on the blackboard. A move consists of three steps: 1) choose an arbitrary number $a$ on the blackboard, 2) calculate the least common multiple $N$ of all numbers written on the blackboard, and 3) replace $a$ by $N/a$. Prove that using such moves it is always possible to make all the numbers on the blackboard equal to $1$. [i](A. Naradzetski)[/i]

Let $ P_1$ be a regular $ r$-gon and $ P_2$ be a regular $ s$-gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$. What's the largest possible value of $ s$?

(a) Is it true that, for arbitrary integer $n{}$ greater than $1$ and distinct positive integers $i{}$ and $j$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'}$ and $j^{'}$ whose product $i^{'}j^{'}$ is divisible by the product $ij$? (b) Is it true that, for arbitrary integer $n{}$ greater than $2$ and distinct positive integers $i, j, k$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'},j^{'},k^{'}$ whose product $i^{'}j^{'}k^{'}$ is divisible by the product $ijk$?

Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]

[b]p1.[/b] Let $X = 2022 + 022 + 22 + 2$. When $X$ is divided by $22$, there is a remainder of $R$. What is the value of $R$? [b]p2.[/b] When Amy makes paper airplanes, her airplanes fly $75\%$ of the time. If her airplane flies, there is a $\frac56$ chance that it won’t fly straight. Given that she makes $80$ airplanes, what is the expected number airplanes that will fly straight? [b]p3.[/b] It takes Joshua working alone $24$ minutes to build a birdhouse, and his son working alone takes $16$ minutes to build one. The effective rate at which they work together is the sum of their individual working rates. How long in seconds will it take them to make one birdhouse together? [b]p4.[/b] If Katherine’s school is located exactly $5$ miles southwest of her house, and her soccer tournament is located exactly $12$ miles northwest of her house, how long, in hours, will it take Katherine to bike to her tournament right after school given she bikes at $0.5$ miles per hour? Assume she takes the shortest path possible. [b]p5.[/b] What is the largest possible integer value of $n$ such that $\frac{4n+2022}{n+1}$ is an integer? [b]p6.[/b] A caterpillar wants to go from the park situated at $(8, 5)$ back home, located at $(4, 10)$. He wants to avoid routes through $(6, 7)$ and $(7, 10)$. How many possible routes are there if the caterpillar can move in the north and west directions, one unit at a time? [b]p7.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 2\sqrt{13}$, $BC = 6\sqrt2$. Construct square $BCDE$ such that $\vartriangle ABC$ is not contained in square $BCDE$. Given that $ACDB$ is a trapezoid with parallel bases $\overline{AC}$, $\overline{BD}$, find $AC$. [b]p8.[/b] How many integers $a$ with $1 \le a \le 1000$ satisfy $2^a \equiv 1$ (mod $25$) and $3^a \equiv 1$ (mod $29$)? [b]p9.[/b] Let $\vartriangle ABC$ be a right triangle with right angle at $B$ and $AB < BC$. Construct rectangle $ADEC$ such that $\overline{AC}$,$\overline{DE}$ are opposite sides of the rectangle, and $B$ lies on $\overline{DE}$. Let $\overline{DC}$ intersect $\overline{AB}$ at $M$ and let $\overline{AE}$ intersect $\overline{BC}$ at $N$. Given $CN = 6$, $BN = 4$, find the $m+n$ if $MN^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. [b]p10.[/b] An elimination-style rock-paper-scissors tournament occurs with $16$ players. The $16$ players are all ranked from $1$ to $16$ based on their rock-paper-scissor abilities where $1$ is the best and $16$ is the worst. When a higher ranked player and a lower ranked player play a round, the higher ranked player always beats the lower ranked player and moves on to the next round of the tournament. If the initial order of players are arranged randomly, and the expected value of the rank of the $2$nd place player of the tournament can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$ what is the value of $m+n$? [b]p11.[/b] Estimation (Tiebreaker) Estimate the number of twin primes (pairs of primes that differ by $2$) where both primes in the pair are less than $220022$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

Find the smallest possible positive integer n with the following property: For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$. (Gerhard J. Woeginger)

Find all integer numbers $x$ and $y$ such that: \[(y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y).\] [i]Proposed by Mahyar Sefidgaran[/i]

Show that there are infinitely many triples of distinct positive integers $a, b, c$ such that each divides the product of the other two and $a + b = c + 1$.

Does there exist a number $n=\overline{a_1a_2a_3a_4a_5a_6}$ such that $\overline{a_1a_2a_3}+4 = \overline{a_4a_5a_6}$ (all bases are $10$) and $n=a^k$ for some positive integers $a,k$ with $k \geq 3 \ ?$

Let \( \varphi \) be the Euler's totient function. 1. For any given integer \( a > 1 \), does there exist \( l \in \mathbb{N}_+ \) such that for any \( k \in \mathbb{N}_+ \), \( l \mid k \) and \( a^2 \nmid l \), \( \frac{\varphi(k)}{\varphi(l)} \) is a non-negative power of \( a \)? 2. For integer \( x > a \), are there integers \( k_1 \) and \( k_2 \) satisfying: \[ \varphi(k_i) \in \left ( \frac{x}{a} ,x \right ], i = 1,2; \quad \varphi(k_1) \neq \varphi(k_2). \] And these two different \( k_i \) correspond to the same \( l_1 \) and \( l_2 \) as described in (1), yet \( \varphi(l_1) = \varphi(l_2) \). 3. Define \( \#E \) as the number of elements in set \( E \). For integer \( x > a \), let \( V(x) = \#\{v \in \mathbb{N}_+ \mid v = \varphi(k) \leq x\} \) and \( W(x) = \#\{w \in \mathbb{N}_+ \mid w = \varphi(l) \leq x, a^2 \mid l\} \). Compare \( V\left( \frac{x}{a} \right) \) with \( W(x) \).

Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$