Let $p$ and $n$ be positive integers such that $p$ is prime and $1 + np$ is a perfect square. Prove that the number $n + 1$ can be expressed as the sum of $p$ perfect squares, where some of them can be equal.

An integer-sided triangle has angles $ p\theta$ and $ q\theta$, where $ p$ and $ q$ are relatively prime integers. Prove that $ \cos\theta$ is irrational.

A $2015$- digit natural number $A$ has the property that any $5$ of it's consecutive digits form a number divisible by $32$. Prove that $A$ is divisible by $2^{2015}$

Prove that the product of 4 consecutive natural numbers cannot be a perfect cube.

Determine whether it is possible to partition $\mathbb{Z}$ into triples $(a,b,c)$ such that, for every triple, $|a^3b + b^3c + c^3a|$ is perfect square.

Denote by $F_n$ the $n^{\text{th}}$ Fibonacci number $(F_1=F_2=1)$.Prove that if $a,b,c$ are positive integers such that $a| F_b,b|F_c,c|F_a$,then either $5$ divides each of $a,b,c$ or $12$ divides each of $a,b,c$.

Prove that there exists and infinity of multiples of $1997$ that have $1998$ as first four digits and last four digits.

In terms of the fixed non-negative integers $\alpha$ and $\beta$ determine the least upper bound of the ratio (or show that it is unbounded) \[ \frac{S(n)}{S(2^{\alpha}5^{\beta}n)} \] as $n$ varies through the positive integers, where $S(\cdot)$ denotes sum of digits in decimal representation.

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

a) Prove that there exists a sequence of digits $\{c_n\}_{n\geq 1}$ such that or each $n\geq 1$ no matter how we interlace $k_n$ digits, $1\leq k_n\leq 9$, between $c_n$ and $c_{n+1}$, the infinite sequence thus obtained does not represent the fractional part of a rational number. b) Prove that for $1\leq k_n\leq 10$ there is no such sequence $\{c_n\}_{n\geq 1}$. [i]Dan Schwartz[/i]

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

Find all functions $f:\mathbb N\to\mathbb N$ such that $$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.

Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that [list] [*]each row contains $n$ distinct consecutive integers in some order, [*]each column contains $m$ distinct consecutive integers in some order, and [*]each entry is less than or equal to $s$. [/list] [i]Proposed by Ankan Bhattacharya.[/i]

Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

There is number $N$ on the board. Every minute Ivan makes next operation: takes any number $a$ written on the board, erases it, then writes all divisors of $a$ except $a$( Can be same numbers on the board). After some time on the board there are $N^2$ numbers. For which $N$ is it possible?

Prove that, $$15x^2-7y^2=9$$ equation has any solutions in integers.

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

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

Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.

Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, find the largest positive integer $m$ for which such a partition exists.

The number $n$ is "good", if there is three divisors of $n$($d_1, d_2, d_3$), such that $d_1^2+d_2^2+d_3^2=n$ a) Prove that all good number is divisible by $3$ b) Determine if there are infinite good numbers.

Prove that the equation \[ x^z + y^z = z^z \] has no solutions in postive integers.

Show that the number \[p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2\] is a positive integer for any positive integer $n.$ Furthermore, show that the numbers $p_{2n-1}$ and $p_{2n}/5$ are perfect squares $($for any positive integer $n).$

Let $n$ be a natural number. Call $a$ the smallest natural number you need to subtract from $n$ to get a perfect square. Call $b$ the smallest natural number that you must add to $n$ to get a perfect square. Prove that $n - ab$ is a perfect square.

[b]p1.[/b] Solve the system of equations $$xy = 2x + 3y$$ $$yz = 2y + 3z$$ $$zx =2z+3x$$ [b]p2.[/b] For any integer $k$ greater than $1$ and any positive integer $n$ , prove that $n^k$ is the sum of $n$ consecutive odd integers. [b]p3.[/b] Determine all pairs of real numbers, $x_1$, $x_2$ with $|x_1|\le 1$ and $|x_2|\le 1$ which satisfy the inequality: $|x^2-1|\le |x-x_1||x-x_2|$ for all $x$ such that $|x| \ge 1$. [b]p4.[/b] Find the smallest positive integer having exactly $100$ different positive divisors. (The number $1$ counts as a divisor). [b]p5.[/b] $ABC$ is an equilateral triangle of side $3$ inches. $DB = AE = 1$ in. and $F$ is the point of intersection of segments $\overline{CD}$ and $\overline{BE}$ . Prove that $\overline{AF} \perp \overline{CD}$. [img]https://cdn.artofproblemsolving.com/attachments/f/a/568732d418f2b1aa8a4e8f53366df9fbc74bdb.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].