Let $a \in \mathbb{N}$ such that $a + n^2$ can be expressed as the sum of two squares for all $n \in \mathbb{N}$. Prove that $a$ is the square of a natural number.

Jackie and Phil have two fair coins and a third coin that comes up heads with probability $ \frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $ \frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Find all sets $(a, b, c)$ of different positive integers $a$, $b$, $c$, for which: [b]*[/b] $2a - 1$ is a multiple of $b$; [b]*[/b] $2b - 1$ is a multiple of $c$; [b]*[/b] $2c - 1$ is a multiple of $a$.

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?

Compute the sum of all positive integers $n$ for which the expression \[\frac{n+7}{\sqrt{n-1}}\] is an integer.

Find three consecutive natural numbers, each of which is divisible by the square of the sum of its digits. Prove that there are no five such numbers in a row.

A positive integer $n$ is abundant if the sum of its proper divisors exceeds $n$. Show that every integer greater than $89 \times 315$ is the sum of two abundant numbers.

Suppose $a$ and $b$ are positive integers for which $8a^ab^b = 27a^bb^a$. Find $a^2 + b^2$.

Let \(n > 1\) be an integer and \(p\) be a prime. Prove that if \(n|p-1\) and \(p|n^3-1\), then \(4p-3\) is a perfect square.

Define a sequence of integers $a_0=m, a_1=n$ and $a_{k+1}=4a_k-5a_{k-1}$ for all $k \ge 1$. Suppose $p>5$ is a prime with $p \equiv 1 \pmod{4}$. Prove that it is possible to choose $m,n$ such that $p \nmid a_k$ for any $k \ge 0$.

Show that for no integers $a \geq 1, n \geq 1$ is the sum \[1+\frac{1}{1+a}+\frac{1}{1+2a}+\cdots+\frac{1}{1+na}\] an integer.

Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.

For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares. [b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$. [b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$. [b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$

The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 <m <6.5 \cdot 10^9$. Find $m$.

Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that \[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\] is an integer.

Find all four-digit perfect squares such that: $\bullet$ All your figures are less than $9$. $\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.

Given a prime number $p>5$. It is known that the length of the smallest period of the fraction $1/p$ is a multiple of three. This period (including possible leading zeros) was written on a strip of paper and cut into three equal-length parts $a$, $b$, $c$ (they may also have leading zeros). What could be the sum of the three periodic fractions: $0.(a)$, $0.(b)$, and $0.(c)$? [i]Proposed by A. Khrabrov[/i]

Determine the number of positive integers $a$, so that there exist nonnegative integers $x_0,x_1,\ldots,x_{2001}$ which satisfy the equation \[ \displaystyle a^{x_0} = \sum_{i=1}^{2001} a^{x_i} \]

How many pairs $(a, b)$ for integers $1 \le a, b \le 2015$ which satisfy that $a$ is divisible by $b + 1$ and $2016 - a$ is divisible by $b$.

Natural $p$ and $q$ are relatively prime. The $[0,1]$ is divided onto $(p+q)$ equal segments. Prove that every segment except two marginal contain exactly one from the $(p+q-2)$ numbers $$\{1/p, 2/p, ... , (p-1)/p, 1/q, 2/q, ... , (q-1)/q\}$$

Let $f(x) = 3x^2 + 1$. Prove that for any given positive integer $n$, the product $$f(1)\cdot f(2)\cdot\dots\cdot f(n)$$ has at most $n$ distinct prime divisors. [i]Proposed by Géza Kós[/i]

[b]p1.[/b] A rectangular pool has diagonal $17$ units and area $120$ units$^2$. Joey and Rachel start on opposite sides of the pool when Rachel starts chasing Joey. If Rachel runs $5$ units/sec faster than Joey, how long does it take for her to catch him? [b]p2. [/b] Alice plays a game with her standard deck of $52$ cards. She gives all of the cards number values where Aces are $1$’s, royal cards are $10$’s and all other cards are assigned their face value. Every turn she flips over the top card from her deck and creates a new pile. If the flipped card has value $v$, she places $12 - v$ cards on top of the flipped card. For example: if she flips the $3$ of diamonds then she places $9$ cards on top. Alice continues creating piles until she can no longer create a new pile. If the number of leftover cards is $4$ and there are $5$ piles, what is the sum of the flipped over cards? [b]p3.[/b] There are $5$ people standing at $(0, 0)$, $(3, 0)$, $(0, 3)$, $(-3, 0)$, and $(-3, 0)$ on a coordinate grid at a time $t = 0$ seconds. Each second, every person on the grid moves exactly $1$ unit up, down, left, or right. The person at the origin is infected with covid-$19$, and if someone who is not infected is at the same lattice point as a person who is infected, at any point in time, they will be infected from that point in time onwards. (Note that this means that if two people run into each other at a non-lattice point, such as $(0, 1.5)$, they will not infect each other.) What is the maximum possible number of infected people after $t = 7$ seconds? [b]p4.[/b] Kara gives Kaylie a ring with a circular diamond inscribed in a gold hexagon. The diameter of the diamond is $2$ mm. If diamonds cost $\$100/ mm ^2$ and gold costs $\$50 /mm ^2$ , what is the cost of the ring? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,a_3,...$ be an infinite sequence of positive integers. Suppose that a sequence $a_1,a_2,\ldots$ of positive integers satisfies $a_1=1$ and \[a_{n}=\sum_{n\neq d|n}a_d\] for every integer $n>1$. Prove that the exist infinitely many integers $k$ such that $a_k=k$.

Determine all positive integers $n$ such that for every positive devisor $ d $ of $n$, $d+1$ is devisor of $n+1$.

Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.