Let $a_1$, $a_2$, ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the $2013$-th powers of the digits of $a_n$. Do there exist distinct positive integers $i$, $j$ such that $a_i=a_j$?

Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$, and there exists positive integer $x$ such that $x^k\equiv a \pmod p$. Suppose that $3\le |S| \le p-2$. Prove that the elements of $S$, when arranged in increasing order, does not form an arithmetic progression.

Exactly 1600 Coconuts are distributed on exactly 100 monkeys, where some monkeys also can have 0 coconuts. Prove that, no matter how you distribute the coconuts, at least 4 monkeys will always have the same amount of coconuts. (The original problem is written in German. So, I apologize when I've changed the original problem or something has become unclear while translating.)

Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]

Show that $1 + 1/2 + 1/3 + ... + 1/n$ is not an integer for $n > 1$.

[b]p1.[/b] What is the smallest possible value for the product of two real numbers that differ by ten? [b]p2.[/b] Determine the number of positive integers $n$ with $1 \le n \le 400$ that satisfy the following: $\bullet$ $n$ is a square number. $\bullet$ $n$ is one more than a multiple of $5$. $\bullet$ $n$ is even. [b]p3.[/b] How many positive integers less than $2019$ are either a perfect cube or a perfect square but not both? [b]p4.[/b] Felicia draws the heart-shaped figure $GOAT$ that is made of two semicircles of equal area and an equilateral triangle, as shown below. If $GO = 2$, what is the area of the figure? [img]https://cdn.artofproblemsolving.com/attachments/3/c/388daa657351100f408ab3f1185f9ab32fcca5.png[/img] [b]p5.[/b] For distinct digits $A, B$, and $ C$: $$\begin{tabular}{cccc} & A & A \\ & B & B \\ + & C & C \\ \hline A & B & C \\ \end{tabular}$$ Compute $A \cdot B \cdot C$. [b]p6 [/b] What is the difference between the largest and smallest value for $lcm(a,b,c)$, where $a,b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive? [b]p7.[/b] Let $A$ and $B$ be points on the circumference of a circle with center $O$ such that $\angle AOB = 100^o$. If $X$ is the midpoint of minor arc $AB$ and $Y$ is on the circumference of the circle such that $XY\perp AO$, find the measure of $\angle OBY$ . [b]p8. [/b]When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in $6$ hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in $4$ hours. How many hours does it take Ben and Sammy to finish that project if they each work together at their normal rates? [b][b]p9.[/b][/b] How many positive integer divisors $n$ of $20000$ are there such that when $20000$ is divided by $n$, the quotient is divisible by a square number greater than $ 1$? [b]p10.[/b] What’s the maximum number of Friday the $13$th’s that can occur in a year? [b]p11.[/b] Let circle $\omega$ pass through points $B$ and $C$ of triangle $ABC$. Suppose $\omega$ intersects segment $AB$ at a point $D \ne B$ and intersects segment $AC$ at a point $E \ne C$. If $AD = DC = 12$, $DB = 3$, and $EC = 8$, determine the length of $EB$. [b]p12.[/b] Let $a,b$ be integers that satisfy the equation $2a^2 - b^2 + ab = 18$. Find the ordered pair $(a,b)$. [b]p13.[/b] Let $a,b,c$ be nonzero complex numbers such that $a -\frac{1}{b}= 8, b -\frac{1}{c}= 10, c -\frac{1}{a}= 12.$ Find $abc -\frac{1}{abc}$ . [b]p14.[/b] Let $\vartriangle ABC$ be an equilateral triangle of side length $1$. Let $\omega_0$ be the incircle of $\vartriangle ABC$, and for $n > 0$, define the infinite progression of circles $\omega_n$ as follows: $\bullet$ $\omega_n$ is tangent to $AB$ and $AC$ and externally tangent to $\omega_{n-1}$. $\bullet$ The area of $\omega_n$ is strictly less than the area of $\omega_{n-1}$. Determine the total area enclosed by all $\omega_i$ for $i \ge 0$. [b]p15.[/b] Determine the remainder when $13^{2020} +11^{2020}$ is divided by $144$. [b]p16.[/b] Let $x$ be a solution to $x +\frac{1}{x}= 1$. Compute $x^{2019} +\frac{1}{x^{2019}}$ . [b]p17. [/b]The positive integers are colored black and white such that if $n$ is one color, then $2n$ is the other color. If all of the odd numbers are colored black, then how many numbers between $100$ and $200$ inclusive are colored white? [b]p18.[/b] What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once? [b]p19.[/b] Let $\vartriangle ABC$ have side lengths $AB = 19$, $BC = 2019$, and $AC = 2020$. Let $D,E$ be the feet of the angle bisectors drawn from $A$ and $B$, and let $X,Y$ to be the feet of the altitudes from $C$ to $AD$ and $C$ to $BE$, respectively. Determine the length of $XY$ . [b]p20.[/b] Suppose I have $5$ unit cubes of cheese that I want to divide evenly amongst $3$ hungry mice. I can cut the cheese into smaller blocks, but cannot combine blocks into a bigger block. Over all possible choices of cuts in the cheese, what’s the largest possible volume of the smallest block of cheese? PS. You had better use hide for answers.

$\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{0,1,2,\cdots,9\}$. There is an integer number $M$ such that $a_{n},b_{n}\neq 0$ for all $n\geq M$ and for each $n\geq 0$ $$(\overline{a_{n}\cdots a_{1}a_{0}})^{2}+999 \mid(\overline{b_{n}\cdots b_{1}b_{0}})^{2}+999 $$ prove that $a_{n}=b_{n}$ for $n\geq 0$.\\ (Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$.) [i]Proposed by Yahya Motevassel[/i]

All decimal digits of some natural number are $1,3,7$, and $9$. Prove that one can rearrange its digits so as to obtain a number divisible by $7$.

Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that: a) $d(f(x)) = x$ for all $x \in \mathbb{N}$ b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$

[b]p1[/b]. Prove that there is no interger $n$ such that $n^2 +1$ is divisible by $7$. [b]p2.[/b] Find all solutions of the system $$x^2-yz=1$$ $$y^2-xz=2$$ $$z^2-xy=3$$ [b]p3.[/b] A triangle with long legs is an isoceles triangle in which the length of the two equal sides is greater than or equal to the length of the remaining side. What is the maximum number, $n$ , of points in the plane with the property that every three of them form the vertices of a triangle with long legs? Prove all assertions. [b]p4.[/b] Prove that the area of a quadrilateral of sides $a, b, c, d$ which can be inscribed in a circle and circumscribed about another circle is given by $A=\sqrt{abcd}$ [b]p5.[/b] Prove that all of the squares of side length $$\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},...,\frac{1}{n},...$$ can fit inside a square of side length $1$ without overlap. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] (a) Suppose $101$ Dalmatians chase $2012$ squirrels. Each squirrel gets chased by at most one Dalmatian, and each Dalmatian chases at least one squirrel. Show that two Dalmatians chase the same number of squirrels. (b) What is the largest number of Dalmatians that can chase $2012$ squirrels in a way that each Dalmatian chases at least one squirrel and no two Dalmatians chase the same number of squirrels? [b]p2.[/b] Lucy and Linus play the following game. They start by putting the integers $1, 2, 3, ..., 2012$ in a hat. In each round of the game, Lucy and Linus each draw a number from the hat. If the two numbers are $a$ and $b$, they throw away these numbers and put the number $|a - b|$ back into the hat. After $2011$ rounds, there is only one number in the hat. If it is even, Lucy wins. If it is odd, Linus wins. (a) Prove that there is a sequence of drawings that makes Lucy win. (b) Prove that Lucy always wins. [b]p3.[/b] Suppose $x$ is a positive real number and $x^{1990}$, $x^{2001}$, and $x^{2012}$ differ by integers. Prove that $x$ is an integer. [b]p4.[/b] Suppose that each point in three-dimensional space is colored with one of five colors and suppose that each color is used at least once. Prove that there is some plane that contains at least four of the colors. [b]p5.[/b] Two circles, $C_1$ and $C_2$, are tangent at point $A$, with $C_1$ lying inside $C_2$ (and $C_1 \ne C_2$). The line through their centers intersects $C_1$ at $B_1$ and $C_2$ at $B_2$. A line $L$ is drawn through $A$ and it intersects $C_1$ at $P_1$ (with $P_1 \ne A$) and intersects $C_2$ at $P_2$ (with $P_2 \ne A$). The perpendicular from $P_2$ to the line $B_1B_2$ intersects the line $B_1B_2$ at $F$. Prove that if the line $P_1F$ is tangent to $C_1$ then $F$ is the midpoint of the line segment $B_1B_2$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/4db59be9fa764d3e910a828ed3296907ca5657.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.

How many integers $n$ (with $1 \le n \le 2021$) have the property that $8n + 1$ is a perfect square?

Consider the sequence $$1,1,2,1,2,4,1,2,4,8,1,2,4,8,16,1, . . .$$ formed by writing the first power of two, followed by the first two powers of two, followed by the first three powers of two, and so on. Find the smallest positive integer $N$ such that $N > 100$ and the sum of the first $N$ terms of this sequence is a power of two.

[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one. [b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$. [b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that (a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color; (b) there are no two neighboring blue chips. [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?

Find all positive integers $(m,n)$ such that $$11^n+2^n+6=m^3$$

[u]Set 1[/u] [b]p1.[/b] Farmer John has $4000$ gallons of milk in a bucket. On the first day, he withdraws $10\%$ of the milk in the bucket for his cows. On each following day, he withdraws a percentage of the remaining milk that is $10\%$ more than the percentage he withdrew on the previous day. For example, he withdraws $20\%$ of the remaining milk on the second day. How much milk, in gallons, is left after the tenth day? [b]p2.[/b] Will multiplies the first four positive composite numbers to get an answer of $w$. Jeremy multiplies the first four positive prime numbers to get an answer of $j$. What is the positive difference between $w$ and $j$? [b]p3.[/b] In Nathan’s math class of $60$ students, $75\%$ of the students like dogs and $60\%$ of the students like cats. What is the positive difference between the maximum possible and minimum possible number of students who like both dogs and cats? [u]Set 2[/u] [b]p4.[/b] For how many integers $x$ is $x^4 - 1$ prime? [b]p5.[/b] Right triangle $\vartriangle ABC$ satisfies $\angle BAC = 90^o$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 60$ and $AB = 65$, find the area of $\vartriangle ABC$. [b]p6.[/b] Define $n! = n \times (n - 1) \times ... \times 1$. Given that $3! + 4! + 5! = a^2 + b^2 + c^2$ for distinct positive integers $a, b, c$, find $a + b + c$. [u]Set 3[/u] [b]p7.[/b] Max nails a unit square to the plane. Let M be the number of ways to place a regular hexagon (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Vincent, on the other hand, nails a regular unit hexagon to the plane. Let $V$ be the number of ways to place a square (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Find the nonnegative difference between $M$ and $V$ . [b]p8.[/b] Let a be the answer to this question, and suppose $a > 0$. Find $\sqrt{a +\sqrt{a +\sqrt{a +...}}}$ . [b]p9.[/b] How many ordered pairs of integers $(x, y)$ are there such that $x^2 - y^2 = 2019$? [u]Set 4[/u] [b]p10.[/b] Compute $\frac{p^3 + q^3 + r^3 - 3pqr}{p + q + r}$ where $p = 17$, $q = 7$, and $r = 8$. [b]p11.[/b] The unit squares of a $3 \times 3$ grid are colored black and white. Call a coloring good if in each of the four $2 \times 2$ squares in the $3 \times 3$ grid, there is either exactly one black square or exactly one white square. How many good colorings are there? Consider rotations and reflections of the same pattern distinct colorings. [b]p12.[/b] Define a $k$-[i]respecting [/i]string as a sequence of $k$ consecutive positive integers $a_1$, $a_2$, $...$ , $a_k$ such that $a_i$ is divisible by $i$ for each $1 \le i \le k$. For example, $7$, $8$, $9$ is a $3$-respecting string because $7$ is divisible by $1$, $8$ is divisible by $2$, and $9$ is divisible by $3$. Let $S_7$ be the set of the first terms of all $7$-respecting strings. Find the sum of the three smallest elements in $S_7$. [u]Set 5[/u] [b]p13.[/b] A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points $I$. Find the sum of all possible values of $I$. [b]p14.[/b] Mr. DoBa continuously chooses a positive integer at random such that he picks the positive integer $N$ with probability $2^{-N}$ , and he wins when he picks a multiple of 10. What is the expected number of times Mr. DoBa will pick a number in this game until he wins? [b]p15.[/b] If $a, b, c, d$ are all positive integers less than $5$, not necessarily distinct, find the number of ordered quadruples $(a, b, c, d)$ such that $a^b - c^d$ is divisible by $5$. PS. You had better use hide for answers. Last 4 sets have been posted [url=https://artofproblemsolving.com/community/c4h2777362p24370554]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $M$ and $N$ be two palindrome numbers, each having $9$ digits and the palindromes don't start with $0$. If $N>M$ and between $N$ and $M$ there aren't any palindromes, find all values of $N-M$.

Do there exist positive integers $a$ and $b$ such that $a^n+n^b$ and $b^n+n^a$ are relatively prime for all natural $n$?

Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers

Let $n$ and $k$ be given relatively prime natural numbers, $k<n.$ Each number in the set $M=\{1,2,...,n-1\}$ is colored either blue or white. It is given that [list] [*] for each $i\in M,$ both $i$ and $n-i$ have the same color, [*] for each $i\in M,i\ne k,$ both $i$ and $\left \vert i-k \right \vert $ have the same color. [/list] Prove that all numbers in $M$ have the same color.

Find all the pairs $a,b \in N$ such that $ab-1 |a^2 + 1$.

Let $\{x_i\}, 1\le i\le 6$ be a given set of six integers, none of which are divisible by $7$. $(a)$ Prove that at least one of the expressions of the form $x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6$ is divisible by $7$, where the $\pm$ signs are independent of each other. $(b)$ Generalize the result to every prime number.

Compute the smallest positive integer $n$ such that $\frac{n}{2}$ is a perfect square and $\frac{n}{3}$ is a perfect cube.