[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].

For two base-10 positive integers $a$ and $b$, we say $a \sim b$ if we can rearrange the digits of $a$ in some way to obtain $b$, where the leading digit of both $a$ and $b$ is nonzero. For instance, $463 \sim 463$ and $634 \sim 463$. Find the number of $11$-digit positive integers $K$ such that $K$ is divisible by $2$, $3$, and $5$, and there is some positive integer $K'$ such that $K' \sim K$ and $K'$ is divisible by $7$, $11$, $13$, $17$, $101$, and $9901$.

Define the [i]bigness [/i]of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and [i]bigness [/i]$N$ and another one with integer side lengths and [i]bigness [/i]$N + 1$.

Find all integer solutions to the equation $x^2=y^2(x+y^4+2y^2)$ .

For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that \[n = S(n) + U(n)^2.\]

Let $a,b,b',c,m,q$ be positive integers, where $m>1,q>1,|b-b'|\ge a$. It is given that there exist a positive integer $M$ such that $$S_q(an+b)\equiv S_q(an+b')+c\pmod{m}$$ holds for all integers $n\ge M$. Prove that the above equation is true for all positive integers $n$. (Here $S_q(x)$ is the sum of digits of $x$ taken in base $q$).

The distance between towns $A$ and $B$ is $999$ km. At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows: $\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$ How many signs are there, with both distances written with the help of only two distinct digits?

Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.

Find all primes $p, q, r$ such that $$p^3+p^2+p+1=qr.$$

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.

A set of integers is called [b]legendary[/b] if you can reach any integer from it by using the following action multiple times: If the numbers $x,y$ are in the set, we may add the number $xy-y^2-y+x$ to the set. Prove that any legendary set contains at least 8 numbers.

Chandler the Octopus is making a concoction to create the perfect ink. He adds $1.2$ grams of melanin, $4.2$ grams of enzymes, and $6.6$ grams of polysaccharides. But Chandler accidentally added n grams of an extra ingredient to the concoction, Chemical $X$, to create glue. Given that Chemical $X$ contains none of the three aforementioned ingredients, and the percentages of melanin, enzymes, and polysaccharides in the final concoction are all integers, find the sum of all possible positive integer values of $n$. [i]Proposed by Taiki Aiba[/i]

Call a positive whole number [i]rickety [/i] if it is three times the product of its digits. There are two $2$-digit numbers that are rickety. What is their sum?

[u]Part 1[/u] You might think this round is broken after solving some of these problems, but everything is intentional. [b]1.1.[/b] The number $n$ can be represented uniquely as the sum of $6$ distinct positive integers. Find $n$. [b]1.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = 5$ and $AC^2 = 1188$. Find $AB$. [b]1.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a \$45 dollar budget to divide between production and advertising. If it spent $k$ dollars on production, it could make $2k - 12$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $15 + 5k$ cents. The amount it made in sales for the previous month was $\$40.50$. Assuming the stand spent its entire budget on production and advertising, what was the absolute dierence between the amount spent on production and the amount spent on advertising? [b]1.4.[/b] Let $A$ be the number of dierent ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 1$, $1 \times 3$, and $1 \times 6$. Let B be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 2$ and $1 \times 5$, where there are 2 different colors available for the $1 \times 2$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.) [b]1.5.[/b] An integer $n \ge 0$ is such that $n$ when represented in base $2$ is written the same way as $2n$ is in base $5$. Find $n$. [b]1.6.[/b] Let $x$ be a positive integer such that $3$, $ \log_6(12x)$, $\log_6(18x)$ form an arithmetic progression in some order. Find $x$. [u]Part 2[/u] Oops, it looks like there were some [i]intentional [/i] printing errors and some of the numbers from these problems got removed. Any $\blacksquare$ that you see was originally some positive integer, but now its value is no longer readable. Still, if things behave like they did for Part 1, maybe you can piece the answers together. [b]2.1.[/b] The number $n$ can be represented uniquely as the sum of $\blacksquare$ distinct positive integers. Find $n$. [b]2.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = \blacksquare$ and $AC^2 = 1536$. Find $AB$. [b]2.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a $\$50$ dollar budget to divide between production and advertising. If it spent k dollars on production, it could make $2k - 2$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $25 + 5k$ cents. The amount it made in sales for the previous month was $\$\blacksquare$. Assuming the stand spent its entire budget on production and advertising, what was the absolute dierence between the amount spent on production and the amount spent on advertising? [b]2.4.[/b] Let $A$ be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times \blacksquare$, $1 \times \blacksquare$, and $1 \times \blacksquare$. Let $B$ be the number of different ways to tile a $1\times n$ rectangle with tiles of size $1 \times \blacksquare$ and $1 \times \blacksquare$, where there are $\blacksquare$ different colors available for the $1 \times \blacksquare$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.) [b]2.5.[/b] An integer $n \ge \blacksquare$ is such that $n$ when represented in base $9$ is written the same way as $2n$ is in base $\blacksquare$. Find $n$. [b]2.6.[/b] Let $x$ be a positive integer such that $1$, $\log_{96}(6x)$, $\log_{96}(\blacksquare x)$ form an arithmetic progression in some order. Find $x$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for any positive integer $k$ there exist infinitely many positive integers $m$ such that $3^k\mid m^3+10$.

For a given positive integer $n$, it's known that there exist $2010$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$. Prove that there exist $2004$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$.

[b]p1.[/b] It’s currently $6:00$ on a $12$ hour clock. What time will be shown on the clock $100$ hours from now? Express your answer in the form hh : mm. [b]p2.[/b] A tub originally contains $10$ gallons of water. Alex adds some water, increasing the amount of water by 20%. Barbara, unhappy with Alex’s decision, decides to remove $20\%$ of the water currently in the tub. How much water, in gallons, is left in the tub? Express your answer as an exact decimal. [b]p3.[/b] There are $2000$ math students and $4000$ CS students at Berkeley. If $5580$ students are either math students or CS students, then how many of them are studying both math and CS? [b]p4.[/b] Determine the smallest integer $x$ greater than $1$ such that $x^2$ is one more than a multiple of $7$. [b]p5.[/b] Find two positive integers $x, y$ greater than $1$ whose product equals the following sum: $$9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29.$$ Express your answer as an ordered pair $(x, y)$ with $x \le y$. [b]p6.[/b] The average walking speed of a cow is $5$ meters per hour. If it takes the cow an entire day to walk around the edges of a perfect square, then determine the area (in square meters) of this square. [b]p7.[/b] Consider the cube below. If the length of the diagonal $AB$ is $3\sqrt3$, determine the volume of the cube. [img]https://cdn.artofproblemsolving.com/attachments/4/d/3a6fdf587c12f2e4637a029f38444914e161ac.png[/img] [b]p8.[/b] I have $18$ socks in my drawer, $6$ colored red, $8$ colored blue and $4$ colored green. If I close my eyes and grab a bunch of socks, how many socks must I grab to guarantee there will be two pairs of matching socks? [b]p9.[/b] Define the operation $a @ b$ to be $3 + ab + a + 2b$. There exists a number $x$ such that $x @ b = 1$ for all $b$. Find $x$. [b]p10.[/b] Compute the units digit of $2017^{(2017^2)}$. [b]p11.[/b] The distinct rational numbers $-\sqrt{-x}$, $x$, and $-x$ form an arithmetic sequence in that order. Determine the value of $x$. [b]p12.[/b] Let $y = x^2 + bx + c$ be a quadratic function that has only one root. If $b$ is positive, find $\frac{b+2}{\sqrt{c}+1}$. [b]p13.[/b] Alice, Bob, and four other people sit themselves around a circular table. What is the probability that Alice does not sit to the left or right of Bob? [b]p14.[/b] Let $f(x) = |x - 8|$. Let $p$ be the sum of all the values of $x$ such that $f(f(f(x))) = 2$ and $q$ be the minimum solution to $f(f(f(x))) = 2$. Compute $p \cdot q$. [b]p15.[/b] Determine the total number of rectangles ($1 \times 1$, $1 \times 2$, $2 \times 2$, etc.) formed by the lines in the figure below: $ \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular} $ [b]p16.[/b] Take a square $ABCD$ of side length $1$, and let $P$ be the midpoint of $AB$. Fold the square so that point $D$ touches $P$, and let the intersection of the bottom edge $DC$ with the right edge be $Q$. What is $BQ$? [img]https://cdn.artofproblemsolving.com/attachments/1/1/aeed2c501e34a40a8a786f6bb60922b614a36d.png[/img] [b]p17.[/b] Let $A$, $B$, and $k$ be integers, where $k$ is positive and the greatest common divisor of $A$, $B$, and $k$ is $1$. Define $x\# y$ by the formula $x\# y = \frac{Ax+By}{kxy}$ . If $8\# 4 = \frac12$ and $3\# 1 = \frac{13}{6}$ , determine the sum $A + B + k$. [b]p18.[/b] There are $20$ indistinguishable balls to be placed into bins $A$, $B$, $C$, $D$, and $E$. Each bin must have at least $2$ balls inside of it. How many ways can the balls be placed into the bins, if each ball must be placed in a bin? [b]p19.[/b] Let $T_i$ be a sequence of equilateral triangles such that (a) $T_1$ is an equilateral triangle with side length 1. (b) $T_{i+1}$ is inscribed in the circle inscribed in triangle $T_i$ for $i \ge 1$. Find $$\sum^{\infty}_{i=1} Area (T_i).$$ [b]p20.[/b] A [i]gorgeous [/i] sequence is a sequence of $1$’s and $0$’s such that there are no consecutive $1$’s. For instance, the set of all gorgeous sequences of length $3$ is $\{[1, 0, 0]$,$ [1, 0, 1]$, $[0, 1, 0]$, $[0, 0, 1]$, $[0, 0, 0]\}$. Determine the number of gorgeous sequences of length $7$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a board, the positive integer $N$ is written. In each round, Olive can realize any one of the following operations: I - Switch the current number by a positive multiple of the current number. II - Switch the current number by a number with the same digits of the current number, but the digits are written in another order(leading zeros are allowed). For instance, if the current number is $2022$, Olive can write any of the following numbers $222,2202,2220$. Determine all the positive integers $N$, such that, Olive can write the number $1$ after a finite quantity of rounds.

Let $m,n$ be integers greater than 1,$m \geq n$,$a_1,a_2,\dots,a_n$ are $n$ distinct numbers not exceed $m$,which are relatively primitive.Show that for any real $x$,there exists $i$ for which $||a_ix|| \geq \frac{2}{m(m+1)} ||x||$,where $||x||$ denotes the distance between $x$ and the nearest integer to $x$ .

Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

$p,q,r$ are real numbers satisfying \[\frac{(p+q)(q+r)(r+p)}{pqr} = 24\] \[\frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10.\] Given that $\frac{p}{q} + \frac{q}{r} + \frac{r}{p}$ can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers, compute $m+n$. [i]Author: Alex Zhu[/i]

Let $p$ be a prime number. At a school of $p^{2020}$ students it is required that each club consist of exactly $p$ students. Is it possible for each pair of students to have exactly one club in common?

Positive integers $m,n,k$ satisfy $1+2+3++...+n=mk$ and $m \ge n$. Show that we can partite $\{1,2,3,...,n \}$ into $k$ subsets (Every element belongs to exact one of these $k$ subsets), such that the sum of elements in each subset is equal to $m$.

A strange calculator has only two buttons with positive itegers, each of them consisting of two digits. It displays the number 1 at the beginning. Whenever a button with number $N$ is pressed, the calculator replaces the displayed number $X$ with the number $X\cdot N$ or $X+N$. Multiplication and addition alternate, multiplication is the first. (For example,if the number 10 is on the 1st button, the number 20 is on the 2nd button, and we consecutively press the 1st, 2nd, 1st and 1st button, we get the results $1\cdot 10=10$, $10+20=30$, $30\cdot 10=300$, and $300+10=310$.) Decide whether there exist particular values of the two-digit nubers on the buttons such that one can display infinitely many numbers (without cleaning the display, i.e. you must keep going and get infinitel many numbers) ending with (a) $2015$, (b) $5813$. [i]Proposed by Michal Rolínek and Peter Novotný[/i]

For which integers $n \ge 8$ is $n^{\frac{1}{n-7}}$ an integer?