[b]p1.[/b] We say that integers $a$ and $b$ are [i]friends [/i] if their product is a perfect square. Prove that if $a$ is a friend of $b$, then $a$ is a friend of $gcd (a, b)$. [b]p2.[/b] On the island of knights and liars, a traveler visited his friend, a knight, and saw him sitting at a round table with five guests. "I wonder how many knights are among you?" he asked. " Ask everyone a question and find out yourself" advised him one of the guests. "Okay. Tell me one: Who are your neighbors?" asked the traveler. This question was answered the same way by all the guests. "This information is not enough!" said the traveler. "But today is my birthday, do not forget it!" said one of the guests. "Yes, today is his birthday!" said his neighbor. Now the traveler was able to find out how many knights were at the table. Indeed, how many of them were there if [i]knights always tell the truth and liars always lie[/i]? [b]p3.[/b] A rope is folded in half, then in half again, then in half yet again. Then all the layers of the rope were cut in the same place. What is the length of the rope if you know that one of the pieces obtained has length of $9$ meters and another has length $4$ meters? [b]p4.[/b] The floor plan of the palace of the Shah is a square of dimensions $6 \times 6$, divided into rooms of dimensions $1 \times 1$. In the middle of each wall between rooms is a door. The Shah orders his architect to eliminate some of the walls so that all rooms have dimensions $2 \times 1$, no new doors are created, and a path between any two rooms has no more than $N$ doors. What is the smallest value of $N$ such that the order could be executed? [b]p5.[/b] There are $10$ consecutive positive integers written on a blackboard. One number is erased. The sum of remaining nine integers is $2011$. Which number was erased? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.

There are $86400$ seconds in a day, which can be deduced from the conversions between seconds, minutes, hours, and days. However, the leading scientists decide that we should decide on $3$ new integers $x, y$, and $z$, such that there are $x$ seconds in a minute, $y$ minutes in an hour, and $z$ hours in a day, such that $xyz = 86400$ as before, but such that the sum $x + y + z$ is minimized. What is the smallest possible value of that sum?

A positive integer is called [i]pretty[/i] if it is equal to the sum of the fourth powers of five distinct divisors. [list=a] [*]Prove that every pretty number is divisible by $5$. [*]Determine if there are infinitely many beautiful numbers. [/list]

In $\triangle ABC$ with $AB=10$, $AC=13$, and $\measuredangle ABC = 30^\circ$, $M$ is the midpoint of $\overline{BC}$ and the circle with diameter $\overline{AM}$ meets $\overline{CB}$ and $\overline{CA}$ again at $D$ and $E$, respectively. The area of $\triangle DEM$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Compute $100m + n$. [i]Based on a proposal by Matthew Babbitt[/i]

[u]Set 6[/u] [b]p16.[/b] Let $x$ and $y$ be non-negative integers. We say point $(x, y)$ is square if $x^2 + y$ is a perfect square. Find the sum of the coordinates of all distinct square points which also satisfy $x^2 + y \le 64$. [b]p17.[/b] Two integers $a$ and $b$ are randomly chosen from the set $\{1, 2, 13, 17, 19, 87, 115, 121\}$, with $a > b$. What is the expected value of the number of factors of $ab$? [b]p18.[/b] Marnie the Magical Cello is jumping on nonnegative integers on number line. She starts at $0$ and jumps following two specific rules. For each jump she can either jump forward by $1$ or jump to the next multiple of $4$ (the next multiple must be strictly greater than the number she is currently on). How many ways are there for her to jump to $2022$? (Two ways are considered distinct only if the sequence of numbers she lands on is different.) PS. You should use hide for answers.

[u]Round 1[/u] [b]p1.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p2.[/b] A positive number is placed on each of the $10$ circles in this picture. It turns out that for each of the nine little equilateral triangles, the number on one of its corners is the sum of the numbers on the other two corners. Is it possible that all $10$ numbers are different? [img]https://cdn.artofproblemsolving.com/attachments/b/f/c501362211d1c2a577e718d2b1ed1f1eb77af1.png[/img] [b]p3.[/b] Pablo and Nina take turns entering integers into the cells of a $3 \times 3$ table. Pablo goes first. The person who fills the last empty cell in a row must make the numbers in that row add to $0$. Can Nina ensure at least two of the columns have a negative sum, no matter what Pablo does? [b]p4. [/b]All possible simplified fractions greater than $0$ and less than $1$ with denominators less than or equal to $100$ are written in a row with a space before each number (including the first). Zeke and Qing play a game, taking turns choosing a blank space and writing a “$+$” or “$-$” sign in it. Zeke goes first. After all the spaces have been filled, Zeke wins if the value of the resulting expression is an integer. Can Zeke win no matter what Qing does? [img]https://cdn.artofproblemsolving.com/attachments/3/6/15484835686fbc2aa092e8afc6f11cd1d1fb88.png[/img] [b]p5.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/0/c/d827cf26c8eaabfd5b0deb92612a6e6ebffb47.png[/img] [u]Round 2[/u] [b]p6.[/b] Prove that among any $3^{2022}$ integers, it is possible to find exactly $3^{2021}$ of them whose sum is divisible by $3^{2021}$. [b]p7.[/b] Given a list of three numbers, a zap consists of picking two of the numbers and decreasing each of them by their average. For example, if the list is $(5, 7, 10)$ and you zap $5$ and $10$, whose average is $7.5$, the new list is $(-2.5, 7, 2.5)$. Is it possible to start with the list $(3, 1, 4)$ and, through some sequence of zaps, end with a list in which the sum of the three numbers is $0$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

All the $5$-digit numbers from $11111$ to $99999$ are written on the cards. Those cards lies in a line in an arbitrary order. Prove that the resulting $444445$-digit number is not a power of two.

For two rational numbers $r,s$ we say:$$r\mid s$$whenever there exists $k\in\mathbb{Z}$ such that:$$s=kr$$ ${(a_n)}_{n\in\mathbb{N}}$ is an increasing sequence of pairwise coprime natural numbers and ${(b_n)}_{n\in\mathbb{N}}$ is a sequence of distinct natural numbers. Assume that for all $n\in\mathbb{N}$ we have: $$\sum_{i=1}^{n}\frac{1}{a_i}\mid\sum_{i=1}^{n}\frac{1}{b_i}$$ Prove that [b]for all[/b] $n\in\mathbb{N}$ we have: $a_n=b_n$.

The symbols $ (a,b,\ldots,g)$ and $ [a,b,\ldots,g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $ a,b,\ldots,g$. For example, $ (3,6,18)\equal{}3$ and $ [6,15]\equal{}30$. Prove that \[ \frac{[a,b,c]^2}{[a,b][b,c][c,a]}\equal{}\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.\]

Given positive integers $a,c$ and integer $b$, prove that there exists a positive integer $x$ such that \[ a^x + x \equiv b \pmod c, \] that is, there exists a positive integer $x$ such that $c$ is a divisor of $a^x + x - b$.

Find all integers $a, m, n, k,$ such that $(a^m+1)(a^n-1)=15^k.$

Show that there is precisely one sequence $ a_1,a_2,...$ of integers which satisfies $ a_1\equal{}1, a_2>1,$ and $ a_{n\plus{}1}^3\plus{}1\equal{}a_n a_{n\plus{}2}$ for $ n \ge 1$.

Given the natural numbers $a$ and $b$, with $1 \le a <b$, prove that there exist natural numbers $n_1<n_2< ...<n_k$, with $k \le a$ such that $$\frac{a}{b}=\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}$$

Positive integer $q{}$ is $m-additive$ $(m\in\mathbb{N}, m\geq2)$ if there exist pairwise distinct positive integers $a_1,a_2,\ldots,a_m$ such that $q=a_1+a_2+\ldots+a_m$ and $a_i | a_{i+1}$ for $i=1,2,\ldots,m-1$. [b]a)[/b] Prove that $1993$ is $8$-additive, but $9$-additive. [b]b)[/b] Determine the greatest integer $m$ for which $2102$ is $m$-additive.

Let $p\geq 5$ be a prime number, and set $M=\{1,2,\cdots,p-1\}.$ Define $$T=\{(n,x_n):p|nx_n-1\ \textup{and}\ n,x_n\in M\}.$$ If $\sum_{(n,x_n)\in T}n\left[\dfrac{nx_n}{p}\right]\equiv k \pmod {p},$ with $0\leq k\leq p-1,$ where $\left[\alpha\right]$ denotes the largest integer that does not exceed $\alpha,$ determine the value of $k.$

[b]p1.[/b] If $a \diamond b = ab - a + b$, find $(3 \diamond 4) \diamond 5$ [b]p2.[/b] If $5$ chickens lay $5$ eggs in $5$ days, how many chickens are needed to lay $10$ eggs in $10$ days? [b]p3.[/b] As Alissa left her house to go to work one hour away, she noticed that her odometer read $16261$ miles. This number is a "special" number for Alissa because it is a palindrome and it contains exactly $1$ prime digit. When she got home that evening, it had changed to the next greatest "special" number. What was Alissa's average speed, in miles per hour, during her two hour trip? [b]p4.[/b] How many $1$ in by $3$ in by $8$ in blocks can be placed in a $4$ in by $4$ in by $9$ in box? [b]p5.[/b] Apple loves eating bananas, but she prefers unripe ones. There are $12$ bananas in each bunch sold. Given any bunch, if there is a $\frac13$ probability that there are $4$ ripe bananas, a $\frac16$ probability that there are $6$ ripe bananas, and a $\frac12$ probability that there are $10$ ripe bananas, what is the expected number of unripe bananas in $12$ bunches of bananas? [b]p6.[/b] The sum of the digits of a $3$-digit number $n$ is equal to the same number without the hundreds digit. What is the tens digit of $n$? [b]p7.[/b] How many ordered pairs of positive integers $(a, b)$ satisfy $a \le 20$, $b \le 20$, $ab > 15$? [b]p8.[/b] Let $z(n)$ represent the number of trailing zeroes of $n!$. What is $z(z(6!))?$ (Note: $n! = n\cdot (n-1) \cdot\cdot\cdot 2 \cdot 1$) [b]p9.[/b] On the Cartesian plane, points $A = (-1, 3)$, $B = (1, 8)$, and $C = (0, 10)$ are marked. $\vartriangle ABC$ is reflected over the line $y = 2x + 3$ to obtain $\vartriangle A'B'C'$. The sum of the $x$-coordinates of the vertices of $\vartriangle A'B'C'$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Compute $a + b$. [b]p10.[/b] How many ways can Bill pick three distinct points from the figure so that the points form a non-degenerate triangle? [img]https://cdn.artofproblemsolving.com/attachments/6/a/8b06f70d474a071b75556823f70a2535317944.png[/img] [b]p11.[/b] Say piece $A$ is attacking piece $B$ if the piece $B$ is on a square that piece $A$ can move to. How many ways are there to place a king and a rook on an $8\times 8$ chessboard such that the rook isn't attacking the king, and the king isn't attacking the rook? Consider rotations of the board to be indistinguishable. (Note: rooks move horizontally or vertically by any number of squares, while kings move $1$ square adjacent horizontally, vertically, or diagonally). [b]p12.[/b] Let the remainder when $P(x) = x^{2020} - x^{2017} - 1$ is divided by $S(x) = x^3 - 7$ be the polynomial $R(x) = ax^2 + bx + c$ for integers $a$, $b$, $c$. Find the remainder when $R(1)$ is divided by $1000$. [b]p13.[/b] Let $S(x) = \left \lfloor \frac{2020}{x} \right\rfloor + \left \lfloor \frac{2020}{x + 1} \right\rfloor$. Find the number of distinct values $S(x)$ achieves for integers $x$ in the interval $[1, 2020]$. [b]p14.[/b] Triangle $\vartriangle ABC$ is inscribed in a circle with center $O$ and has sides $AB = 24$, $BC = 25$, $CA = 26$. Let $M$ be the midpoint of $\overline{AB}$. Points $K$ and $L$ are chosen on sides $\overline{BC}$ and $\overline{CA}$, respectively such that $BK < KC$ and $CL < LA$. Given that $OM = OL = OK$, the area of triangle $\vartriangle MLK$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p15.[/b] Euler's totient function, $\phi (n)$, is defined as the number of positive integers less than $n$ that are relatively prime to $n$. Let $S(n)$ be the set of composite divisors of $n$. Evaluate $$\sum^{50}_{k=1}\left( k - \sum_{d\in S(k)} \phi (d) \right)$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs $(p,q)$ of prime numbers, such that $2p-1$, $2q-1$, $2pq-1$ are perfect square. F. Petrov, A. Smirnov

Let $ a$ and $b$ be positive integers. Prove that there exist positive integers $ x $ and $ y $ such that \[ \binom{x+y}{2} = ax + by . \]

What is the $101$st smallest integer which can represented in the form $3^a+3^b+3^c$, where $a,b,$ and $c$ are integers? [i]Proposed by Dilhan Salgado[/i]

Determine all pairs of positive integers $(a,b)$ such that \[ \dfrac{a^2}{2ab^2-b^3+1} \] is a positive integer.

If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we define $f_s(n) = d_1^s+d_2^s+..+d_k^s$. For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$. Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.

Prove that for every positive integer $n$ there exist integers $a$ and $b,$ both greater than $1,$ such that $a ^ 2 + 1 = 2b ^ 2$ and $a - b$ is a multiple of $n.$

Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$

Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.