Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

Prove that there exist an infinite number of triples $n-1 $,$n$,$n + 1$ such that (a) $n$ can be represented as the sum of two squares of natural numbers but neither of $n-1$ and $n+1$ can; (b) each of these three numbers can be represented as the sum of two squares. (V Senderov)

[b]p1.[/b] Find the largest integer which is a factor of all numbers of the form $n(n +1)(n + 2)$ where $n$ is any positive integer with unit digit $4$. Prove your claims. [b]p2.[/b] Each pair of the towns $A, B, C, D$ is joined by a single one way road. See example. Show that for any such arrangement, a salesman can plan a route starting at an appropriate town that: enables him to call on a customer in each of the towns. Note that it is not required that he return to his starting point. [img]https://cdn.artofproblemsolving.com/attachments/6/5/8c2cda79d2c1b1c859825f3df0163e65da761b.png[/img] [b]p3.[/b] $A$ and $B$ are two points on a circular race track . One runner starts at $A$ running counter clockwise, and, at the same time, a second runner starts from $B$ running clockwise. They meet first $100$ yds from A, measured along the track. They meet a second time at $B$ and the third time at $A$. Assuming constant speeds, now long is the track? [b]p4.[/b] $A$ and $B$ are points on the positive $x$ and positive $y$ axis, respectively, and $C$ is the point $(3,4)$. Prove that the perimeter of $\vartriangle ABC$ is greater than $10$. Suggestion: Reflect!! [b]p5.[/b] Let $A_1,A_2,...,A_8$ be a permutation of the integers $1,2,...,8$ so chosen that the eight sums $9 + A_1$, $10 + A_2$, $...$, $16 + A_8$ and the eight differences $9 -A_1$ , $10 - A_2$, $...$, $16 - A_8$ together comprise $16$ different numbers. Show that the same property holds for the eight numbers in reverse order. That is, show that the $16$ numbers $9 + A_8$, $10 + A_7$, $...$, $16 + A_1$ and $9 -A_8$ , $10 - A_7$, $...$, $16 - A_1$ are also pairwise different. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?

Solve in nonnegative integers the equation $5^t + 3^x4^y = z^2$.

Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.

One day while the Kubik family attends one of Michael's baseball games, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates Mitchell sees Tony and goes to join him. While playing around the creek, the two boys find an ordinary six-sided die buried in sediment. Mitchell washes it off in the water and challenges Tony to a contest. Each of the boys rolls the die exactly once. Mitchell's roll is $3$ higher than Tony's. "Let's play once more," says Tony. Let $a/b$ be the probability that the difference between the outcomes of the two dice is again exactly $3$ (regardless of which of the boys rolls higher), where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find the last five digits of $1^{100}+2^{100}+3^{100}+...+999999^{100}$

Given a prime number \( p \geq 3 \) and a positive integer \( m \), find the smallest positive integer \( n \) with the following property: for every positive integer \( a \), which is not divisible by \( p \), the sum of the natural divisors of \( a^n \) greater than 1 is divisible by \( p^m \).

Every positive integer is marked with a number from the set $\{ 0,1,2\}$, according to the following rule: $$\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0.$$ Let $S$ denote the sum of marks of the first $2019$ positive integers. Determine the maximum possible value of $S$. [i]Proposed by Ivan Novak[/i]

Determine all pairs $ (x,y)$ of positive integers satisfying the equation \[ x!\plus{}y!\equal{}x^{y}.\]

Let $p \geq 5$ be a prime and let \begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*} where $q_i$ are primes. Prove, \begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}

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

Let $n\ge 3$, $d$ be positive integers. For an integer $x$, denote $r(x)$ be the remainder of $x$ when divided by $n$ such that $0\le r(x)\le n-1$. Let $c$ be a positive integer with $1<c<n$ and $\gcd(c,n)=1$, and suppose $a_1, \cdots, a_d$ are positive integers with $a_1+\cdots+a_d\le n-1$. \\ (a) Prove that if $n<2d$, then $\displaystyle\sum_{i=1}^d r(ca_i)\ge n.$ \\ (b) For each $n$, find the smallest $d$ such that $\displaystyle\sum_{i=1}^d r(ca_i)\ge n$ always holds. [i]Proposed by Yeoh Zi Song & Anzo Teh Zhao Yang[/i]

Haydn picks two different integers between $1$ and $100$, inclusive, uniformly at random. The probability that their product is divisible by $4$ can be expressed in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$ holds. Find the minimum value of $k$.

A wizard thinks of a number from $1$ to $n$. You can ask the wizard any number of yes/no questions about the number. The wizard must answer all those questions, but not necessarily in the respective order. What is the least number of questions that must be asked in order to know what the number is for sure. (In terms of $n$.) Fresh translation.

Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$

Floyd the flea makes jumps on the positive integers. On the first day he can jump to any positive integer. From then on, every day he jumps to another number that is not more than twice his previous day's place. [list=a] [*]Show that Floyd can make infinitely many jumps in such a way that he never arrives at any number with the same sum of decimal digits as at a previous place.[/*] [*]Can the flea jump this way if we consider the sum of binary digits instead of decimal digits?[/*] [/list]

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

Let $T$ be a finite set of squarefree integers. (a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$. (b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists? [i]Allen Wang[/i]