Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$, such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube.

Find the remainder when \[\sum_{k=1}^{2^{16}}\binom{2k}{k}(3\cdot 2^{14}+1)^k (k-1)^{2^{16}-1}\]is divided by $2^{16}+1$. ([i]Note:[/i] It is well-known that $2^{16}+1=65537$ is prime.) [i]Victor Wang.[/i]

Consider the numbers from $1$ to $32$. A game is made by placing all the numbers in pairs and replacing each pair with the largest prime divisor of the sum of the numbers of that couple. For example, if we match the $32$ numbers as: $(1, 2), (3,4),(5, 6), (7, 8),..., (27, 28),(29, 30), (31,32)$, we get the following list of $16$ numbers: $3,7,11,5,...,11,59,7$. where there are repetitions. The game continues in a similar way until in the end only one number remains. Determine the highest possible value from the number that remains at the end.

Find all integers $a$, $b$, $c$ and $d$ such that $$a^2+5b^2-2c^2-2cd-3d^2=0$$

Find all three digit numbers $\overline{abc}$ such that $2 \cdot \overline{abc} = \overline{bca} + \overline{cab}$.

Find all $a,b \in \mathbb{N}$ such that $$2049^ba^{2048}-2048^ab^{2049}=1.$$ [i]Proposed by fattypiggy123 and 61plus[/i]

Let $N=2^a p_1^{b_1} p_2^{b_2} \ldots p_k^{b_k}$. Prove that there are $(b_1+1)(b_2+1)\ldots(b_k+1)$ number of $n$s which satisfies these two conditions. $\frac{n(n+1)}{2}\le N$, $N-\frac{n(n+1)}{2}$ is divided by $n$.

Let $n$ be a positive integer. We call $C_n$ the number of positive integers $x$ less than $10^n$ such that the sum of the digits of $2x$ is less than the sum of the digits of $x$. Show that $C_n\geq\frac{4}{9}(10^{n}-1)$.

Find all positive integers $n$ such that $3^n - 2^n - 1$ is a perfect square.

For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$. Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .

Positive integers $a$ and $b$ have $n$ digits each in their decimal representation. Assume that $m$ is a positive integer such that $\frac n2<m<n$ and assume that each of the leftmost $m$ digits of $a$ is equal to the corresponding digit of $b$. Prove that $$a^{\frac1n}-b^{\frac1n}<\frac1n.$$

Let $P = \{p_1,p_2,\ldots, p_{10}\}$ be a set of $10$ different prime numbers and let $A$ be the set of all the integers greater than $1$ so that their prime decomposition only contains primes of $P$. The elements of $A$ are colored in such a way that: [list] [*] each element of $P$ has a different color, [*] if $m,n \in A$, then $mn$ is the same color of $m$ or $n$, [*] for any pair of different colors $\mathcal{R}$ and $\mathcal{S}$, there are no $j,k,m,n\in A$ (not necessarily distinct from one another), with $j,k$ colored $\mathcal{R}$ and $m,n$ colored $\mathcal{S}$, so that $j$ is a divisor of $m$ and $n$ is a divisor of $k$, simultaneously. [/list] Prove that there exists a prime of $P$ so that all its multiples in $A$ are the same color.

The sum of $10$ positive integer numbers is equal to $300$. The product of their factorials is a perfect tenth power of some positive integer. Prove that all $10$ numbers are equal to each other. [i]Proposed by Pavlo Protsenko[/i]

A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.

[u]Round 1[/u] [b]p1.[/b] The temperature inside is $28^o$ F. After the temperature is increased by $5^o$ C, what will the new temperature in Fahrenheit be? [b]p2.[/b] Find the least positive integer value of $n$ such that $\sqrt{2021+n}$ is a perfect square. [b]p3.[/b] A heart consists of a square with two semicircles attached by their diameters as shown in the diagram. Given that one of the semicircles has a diameter of length $10$, then the area of the heart can be written as $a +b\pi$ where $a$ and $b$ are positive integers. Find $a +b$. [img]https://cdn.artofproblemsolving.com/attachments/7/b/d277d9ebad76f288504f0d5273e19df568bc44.png[/img] [u]Round 2[/u] [b]p4.[/b] An $L$-shaped tromino is a group of $3$ blocks (where blocks are squares) arranged in a $L$ shape, as pictured below to the left. How many ways are there to fill a $12$ by $2$ rectangle of blocks (pictured below to the right) with $L$-shaped trominos if the trominos can be rotated or reflected? [img]https://cdn.artofproblemsolving.com/attachments/d/c/cf37cdf9703ae0cd31c38af23b6874fddb3c12.png[/img] [b]p5.[/b] How many permutations of the word $PIKACHU$ are there such that no two vowels are next to each other? [b]p6.[/b] Find the number of primes $n$ such that there exists another prime $p$ such that both $n +p$ and $n-p$ are also prime numbers. [u]Round 3[/u] [b]p7.[/b] Maisy the Bear is at the origin of the Cartesian Plane. WhenMaisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Let $L(x, y)$ be the number of jumps it takes forMaisy to reach point (x, y). The sum of $L(x, y)$ over all lattice points $(x, y)$ with both coordinates between $0$ and $2020$, inclusive, is denoted as $S$. Find $\frac{S}{2020}$ . [b]p8.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past $B$ to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE = \sqrt3$. The area of $\vartriangle DEP$ can be written as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers such that $b$ is squarefree and $gcd (a,c) = 1$. Find $a +b +c$. [b]p9.[/b] Find the number of trailing zeroes at the end of $$\prod^{2021}_{i=1}(2021+i -1) = (2021)(2022)...(4041).$$ [u]Round 4[/u] [b]p10.[/b] Let $a, b$, and $c$ be side lengths of a rectangular prism with space diagonal $10$. Find the value of $$(a +b)^2 +(b +c)^2 +(c +a)^2 -(a +b +c)^2.$$ [b]p11.[/b] In a regular heptagon $ABCDEFG$, $\ell$ is a line through $E$ perpendicular to $DE$. There is a point $P$ on $\ell$ outside the heptagon such that $PA = BC$. Find the measure of $\angle EPA$. [b]p12.[/b] Dunan is being "$SUS$". The word "$SUS$" is a palindrome. Find the number of palindromes that can be written using some subset of the letters $\{S, U, S, S, Y, B, A, K, A\}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166494p28814284]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166500p28814367]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$p$ is a give odd prime, if $\sqrt{k^2-pk}$ is a positive integer, then the value of positive integer $k$ is________.

Are there any integers $a,b$ and $c$, not all of them $0$, such that $$a^2=2021b^2+2022c^2~~?$$ [i] (Cosmin Gavrilă)[/i]

Does there exist an infinite set $\displaystyle{M}$ consisting of positive integers such that for any $\displaystyle{a,b \in M}$ with $\displaystyle{a < b}$ the sum $\displaystyle{a + b}$ is square-free? [b]Note.[/b] A positive integer is called square-free if no perfect square greater than $\displaystyle{1}$ divides it. [i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]

Prove that $9$ divides $A_n=16^n+4^n-2$ for every nonnegative integer $n$.

Prove that $n + \big[ (\sqrt{2} + 1)^n\big] $ is odd for all positive integers $n$. $\big[ x \big]$ denotes the greatest integer function.

Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.

[b]p1.[/b] Frankie the frog likes to hop. On his first hop, he hops $1$ meter. On each successive hop, he hops twice as far as he did on the previous hop. For example, on his second hop, he hops $2$ meters, and on his third hop, he hops $4$ meters. How many meters, in total, has he travelled after $6$ hops? [b]p2.[/b] Anton flips $5$ fair coins. The probability that he gets an odd number of heads can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p3.[/b] April discovers that the quadratic polynomial $x^2 + 5x + 3$ has distinct roots $a$ and $b$. She also discovers that the quadratic polynomial $x^2 + 7x + 4$ has distinct roots $c$ and $d$. Compute $$ac + bc + bd + ad + a + b.$$ [b]p4.[/b] A rectangular picture frame that has a $2$ inch border can exactly fit a $10$ by $7$ inch photo. What is the total area of the frame's border around the photo, in square inches? [b]p5.[/b] Compute the median of the positive divisors of $9999$. [b]p6.[/b] Kaity only eats bread, pizza, and salad for her meals. However, she will refuse to have salad if she had pizza for the meal right before. Given that she eats $3$ meals a day (not necessarily distinct), in how many ways can we arrange her meals for the day? [b]p7.[/b] A triangle has side lengths $3$, $4$, and $x$, and another triangle has side lengths $3$, $4$, and $2x$. Assuming both triangles have positive area, compute the number of possible integer values for $x$. [b]p8.[/b] In the diagram below, the largest circle has radius $30$ and the other two white circles each have a radius of $15$. Compute the radius of the shaded circle. [img]https://cdn.artofproblemsolving.com/attachments/c/1/9eaf1064b2445edb15782278fc9c6efd1440b0.png[/img] [b]p9.[/b] What is the remainder when $2022$ is divided by $9$? [b]p10.[/b] For how many positive integers $x$ less than $2022$ is $x^3 - x^2 + x - 1$ prime? [b]p11.[/b] A sphere and cylinder have the same volume, and both have radius $10$. The height of the cylinder can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p12.[/b] Amanda, Brianna, Chad, and Derrick are playing a game where they pass around a red flag. Two players "interact" whenever one passes the flag to the other. How many different ways can the flag be passed among the players such that (1) each pair of players interacts exactly once, and (2) Amanda both starts and ends the game with the flag? [b]p13.[/b] Compute the value of $$\dfrac{12}{1 + \dfrac{12}{1+ \dfrac{12}{1+...}}}$$ [b]p14.[/b] Compute the sum of all positive integers $a$ such that $a^2 - 505$ is a perfect square. [b]p15.[/b] Alissa, Billy, Charles, Donovan, Eli, Faith, and Gerry each ask Sara a question. Sara must answer exactly $5$ of them, and must choose an order in which to answer the questions. Furthermore, Sara must answer Alissa and Billy's questions. In how many ways can Sara complete this task? [b]p16.[/b] The integers $-x$, $x^2 - 1$, and $x3$ form a non-decreasing arithmetic sequence (in that order). Compute the sum of all possible values of $x^3$. [b]p17.[/b] Moor and his $3$ other friends are trying to split burgers equally, but they will have $2$ left over. If they find another friend to split the burgers with, everyone can get an equal amount. What is the fewest number of burgers that Moor and his friends could have started with? [b]p18.[/b] Consider regular dodecagon $ABCDEFGHIJKL$ below. The ratio of the area of rectangle $AFGL$ to the area of the dodecagon can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/8/3/c38c10a9b2f445faae397d8a7bc4c8d3ed0290.png[/img] [b]p19.[/b] Compute the remainder when $3^{4^{5^6}}$ is divided by $4$. [b]p20.[/b] Fred is located at the middle of a $9$ by $11$ lattice (diagram below). At every second, he randomly moves to a neighboring point (left, right, up, or down), each with probability $1/4$. The probability that he is back at the middle after exactly $4$ seconds can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/c/f8e092e60f568ab7b28964d23b2ee02cdba7ad.png[/img] PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathbb{S}$ is the set of prime numbers that less or equal to 26. Is there any $a_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N}$ such that $$ gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for } 1\leq i \ne j \leq 6$$ and for every element $p$ of $\mathbb{S}$ there exists a pair of $ 1\leq k \ne l \leq 6$ such that $$s=gcd(a_k,a_l)?$$

Let the numbers $x_i \in \{-1, 1\}$ be given for $i = 1, 2,..., n$, satisfying $$x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0.$$ Prove that $n$ is divisible by $4$.

Determine the largest integer $n$ such that $2^n$ divides the decimal representation given by some permutation of the digits $2$, $0$, $1$, and $5$. (For example, $2^1$ divides $2150$. It may start with $0$.)