Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

The sequence $(x_n)$ is defined by formulas $$ x_1=c,\; x_{n+1} = cx_n + \sqrt{(c^2-1)(x_n^2-1)} \quad\text{ for }\quad n=1,2,\ldots$$ Prove that if $ c $ is a natural number, then all numbers $ x_n $ are natural.

Let $a_1,a_2,\cdots,a_6,a_7$ be seven positive integers. Let $S$ be the set of all numbers of the form $a_i^2+a_j^2$ where $1\leq i<j\leq 7$. Prove that there exist two elements of $S$ which have the same remainder on dividing by $36$.

In a sequence of natural numbers $ a_1,a_2,...,a_n$ every number $ a_k$ represents sum of the multiples of the $ k$ from sequence. Find all possible values for $ n$.

An integer $n\geq 3$ is [i]poli-pythagorean[/i] if there exist $n$ positive integers pairwise distinct such that we can order these numbers in the vertices of a regular $n$-gon such that the sum of the squares of consecutive vertices is also a perfect square. For instance, $3$ is poli-pythagorean, because if we write $44,117,240$ in the vertices of a triangle we notice: $$44^2+117^2=125^2, 117^2+240^2=267^2, 240^2+44^2=244^2$$ Determine all poli-pythagorean integers.

For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

In a quadratic table with $ 2007$ rows and $ 2007$ columns is an odd number written in each field. For $ 1\leq i\leq2007$ is $ Z_i$ the sum of the numbers in the $ i$-th row and for $ 1\leq j\leq2007$ is $ S_j$ the sum of the numbers in the $ j$-th column. $ A$ is the product of all $ Z_i$ and $ B$ the product of all $ S_j$. Show that $ A\plus{}B\neq0$

Prove that the following statement is true for two natural nos. $m,n$ if and only $v(m) = v(n)$ where $v(k)$ is the highest power of $2$ dividing $k$. $\exists$ a set $A$ of positive integers such that $(i)$ $x,y \in \mathbb{N}, |x-y| = m \implies x \in A $ or $y \in A$ $(ii)$ $x,y \in \mathbb{N}, |x-y| = n \implies x \not\in A $ or $y \not\in A$

The side lengths of a trapezoid are $\sqrt[4]{3}, \sqrt[4]{3}, \sqrt[4]{3}$, and $2 \cdot \sqrt[4]{3}$. Its area is the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$.

Determine if there exist pairwise distinct positive integers $a_1,a_2,\ldots,a_{101}$, $b_1$, $b_2$, \ldots, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,101\}$ the sum $\sum\limits_{i\in S}a_i$ divides $\left( 100!+\sum\limits_{i\in S}b_i \right)$.

Find all pairs of integers $(x,y)$ such that $$(x+1)(y+1)(x+y)(x^2+y^2)=16x^2y^2$$

Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab? $ \textbf{(A) }42\qquad\textbf{(B) }84\qquad\textbf{(C) }126\qquad\textbf{(D) }178\qquad\textbf{(E) }252$

Given a number $k\in \mathbb{N}$. $\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{1,2,\cdots,9\}$. For all $n\geq 0$ $$\left.\overline{a_{n}\cdots a_{1}a_{0}}+k \ \middle| \ \overline{b_{n}\cdots b_{1}b_{0}}+k \right. .$$ Prove that there is a number $1\leq t \leq 9$ and $N\in \mathbb{N}$ such that $b_n=ta_n$ for all $n\geq N$.\\ (Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$)

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

Suppose N is a $6$-digit number having base-$10$ representation $\underline{a}\text{ }\underline{b}\text{ }\underline{c}\text{ }\underline{d}\text{ }\underline{e}\text{ }\underline{f}$. If $N$ is $6/7$ of the number having base-$10$ representation $\underline{d}\text{ }\underline{e}\text{ }\underline{f}\text{ }\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, find $N$.

Find all nonzero subsets $A$ of the set $\left\{2,3,4,5,\cdots\right\}$ such that $\forall n\in A$, we have that $n^{2}+4$ and $\left\lfloor{\sqrt{n}\right\rfloor}+1$ are both in $A$.

Let $n$ be a natural number with more than $2021$ digits, none of which are $8$ or $9$. Suppose that $n$ has no common factors with $2021$. Prove that it is possible to increase one of the digits of $n$ by at most $2$ so that the resulting number is a multiple of 2021.

Let k be a positive integer. Find the number of non-negative integers n less than or equal to $10^k$ satisfying the following conditions: (i) n is divisible by 3; (ii) Each decimal digit of n is one of the digits 2,0,1 or 7.

All divisors of a positive integer $n$ are listed in the ascending order: $1=d_1<d_2< \ldots < d_k=n$. It turned out that the amount of pairs $(d_i,d_{i+1})$ of adjacent divisors such that $d_{i+1}$ is a multiple of $d_i$, is odd. Prove that $n=pm^2$, where $p$ is the smallest prime divisor of $n$, and $m$ is a positive integer.

Find all natural values of $n$ and $m$, such that $(n -1)2^{n - 1} + 5 = m^2 + 4m$.

[u]Round 1[/u] [b]1.1.[/b] A person asks for help every $3$ seconds. Over a time period of $5$ minutes, how many times will they ask for help? [b]1.2.[/b] In a big bag, there are $14$ red marbles, $15$ blue marbles, and$ 16$ white marbles. If Anuj takes a marble out of the bag each time without replacement, how many marbles does Anuj need to remove to be sure that he will have at least $3$ red marbles? [b]1.3.[/b] If Josh has $5$ distinct candies, how many ways can he pick $3$ of them to eat? [u]Round 2[/u] [b]2.1.[/b] Annie has a circular pizza. She makes $4$ straight cuts. What is the minimum number of slices of pizza that she can make? [b]2.2.[/b] What is the sum of the first $4$ prime numbers that can be written as the sum of two perfect squares? [b]2.3.[/b] Consider a regular octagon $ABCDEFGH$ inscribed in a circle of area $64\pi$. If the length of arc $ABC$ is $n\pi$, what is $n$? [u]Round 3[/u] [b]3.1.[/b] Let $ABCDEF$ be an equiangular hexagon with consecutive sides of length $6, 5, 3, 8$, and $3$. Find the length of the sixth side. [b]3.2.[/b] Jack writes all of the integers from $ 1$ to $ n$ on a blackboard except the even primes. He selects one of the numbers and erases all of its digits except the leftmost one. He adds up the new list of numbers and finds that the sum is $2020$. What was the number he chose? [b]3.3.[/b] Our original competition date was scheduled for April $11$, $2020$ which is a Saturday. The numbers $4116$ and $2020$ have the same remainder when divided by $x$. If $x$ is a prime number, find the sum of all possible $x$. [u]Round 4[/u] [b]4.1.[/b] The polynomials $5p^2 + 13pq + cq^2$ and $5p^2 + 13pq - cq^2$ where $c$ is a positive integer can both be factored into linear binomials with integer coefficients. Find $c$. [b]4.2.[/b] In a Cartesian coordinate plane, how many ways are there to get from $(0, 0)$ to $(2, 3)$ in $7$ moves, if each move consists of a moving one unit either up, down, left, or right? [b]4.3.[/b] Bob the Builder is building houses. On Monday he finds an empty field. Each day starting on Monday, he finishes building a house at noon. On the $n$th day, there is a $\frac{n}{8}$ chance that a storm will appear at $3:14$ PM and destroy all the houses on the field. At any given moment, Bob feels sad if and only if there is exactly $1$ house left on the field that is not destroyed. The probability that he will not be sad on Friday at $6$ PM can be expressed as $p/q$ in simplest form. Find $p + q$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784570p24468605]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime

Charlotte writes the integers $1,2,3,\ldots,2025$ on the board. Charlotte has two operations available: the GCD operation and the LCM operation. [list] [*]The GCD operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{gcd}(a, b)$. [*]The LCM operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{lcm}(a, b)$. [/list] An integer $N$ is called a [i]winning number[/i] if there exists a sequence of operations such that, at the end, the only integer left on the board is $N$. Find all winning integers among $\{1,2,3,\ldots,2025\}$ and, for each of them, determine the minimum number of GCD operations Charlotte must use. [b]Note:[/b] The number $\operatorname{gcd}(a, b)$ denotes the [i]greatest common divisor[/i] of $a$ and $b$, while the number $\operatorname{lcm}(a, b)$ denotes the [i]least common multiple[/i] of $a$ and $b$.

Determine all pairs of positive integers $(a,b)$ such that the following fraction is an integer: \[\frac{(a+b)^2}{4+4a(a-b)^2}.\]