The arithmetic mean of two different positive integers $x,y$ is a two digit integer. If one interchanges the digits, the geometric mean of these numbers is archieved. a) Find $x,y$. b) Show that a)'s solution is unique up to permutation if we work in base $g=10$, but that there is no solution in base $g=12$. c) Give more numbers $g$ such that a) can be solved; give more of them such that a) can't be solved, too.

Let $a,b,c$ be pairwise coprime integers such a that $\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=\tfrac{N}{a+b+c}$ for some positive integer $N.$ What is the sum of all possible values of $N.$

Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote $S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$, $S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$, Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$

Prove that the integer $145^{n} + 3114\cdot 138^{n}$ is divisible by $1981$ if $n=1981$, and that it is not divisible by $1981$ if $n=1980$.

Let $p{}$ be a prime number and $x_1,x_2,\ldots,x_p$ be integers for which $x_1^n+x_2^n+\cdots+x_p^n$ is divisible by $p{}$ for any positive integer $n{}$. Prove that $x_1-x_2$ is divisible by $p{}.$

How many ordered pairs of positive integers $(m, n)$ satisfy the system \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$. Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le 8$. [i]Proposed by Richard Henner[/i]

A function from the positive integers to the positive integers satisfies these properties 1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$. 2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$. Prove that $f(2)=2, f(3)=3, f(1999)=1999$.

(a) Find all triples of integers $(a,b,c)$ for which $\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$. (b) Determine all positive integers $n$ for which there exist positive integers $x_1,x_2,\ldots,x_n$ such that $1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}$.

$R$ is a ring such that $xy=yx$ for every $x,y\in R$ and if $ab=0$ then $a=0$ or $b=0$. if for every Ideal $I\subset R$ there exist $x_1,x_2,..,x_n$ in $R$ ($n$ is not constant) such that $I=(x_1,x_2,...,x_n)$, prove that every element in $R$ that is not $0$ and it's not a unit, is the product of finite irreducible elements.($\frac{100}{6}$ points)

Let $n$ be an integer greater than $2$. Prove that the $n$th power of the length of the hypotenuse of a right triangle is greater than the sum of the $n$th powers of the lengths of the legs.

Prove that: [b](a)[/b] If $(a_n)_{n\geq 1}$ is a strictly increasing sequence of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}$ is a constant as $n$ runs through all positive integers, then this constant is an integer greater than or equal to $4$; and [b](b)[/b] Given an integer $N\geq 4$, there exists a strictly increasing sequene $(a_n)_{n\geq 1}$ of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}=N$ for all indices $n$.

Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$

Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$

Determine the sum of the positive integers $n$ such that there exist primes $p,q,r$ satisfying $p^{n} + q^{2} = r^{2}$.

Let $n$ be a positive integer. Find the number of odd coefficients of the polynomial $(x^2-x+1)^n$.

Find all $n\in \mathbb Z^+$, so that \[ z_n = \underbrace{ 101\dots101}_{2n+1 \text{ digits} } \] is prime.

A positive integer $n$ is [i]friendly[/i] if the difference of each pair of neighbouring digits of $n$, written in base $10$, is exactly $1$. [i]For example, 6787 is friendly, but 211 and 901 are not.[/i] Find all odd natural numbers $m$ for which there exists a friendly integer divisible by $64m$.

Let $k \geq 2$ and $n_1, n_2, . . . , n_k \geq 1$ natural numbers having the property $n_2 | 2^{n_1} - 1, n_3 | 2^{n_2} -1 , \cdots, n_k | 2^{n_k-1}-1$, and $n_1 | 2^{n_k} - 1$. Show that $n_1 = n_2 = \cdots = n_k = 1.$

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$. [b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers. [b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$. [b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area? [img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$. Prove that $m> p$. (Karl Czakler)

Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?

Given a natural $n>1$ and its prime fatorization $n=p_1^{\alpha 1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}$, its [i]false derived[/i] is defined by $$f(n)=\alpha_1p_1^{\alpha_1-1}\alpha_2p_2^{\alpha_2-1}...\alpha_kp_k^{\alpha_k-1}.$$ Prove that there exist infinitely many naturals $n$ such that $f(n)=f(n-1)+1$.

Find all numbers between $100$ and $999$ which equal the sum of the cubes of their digits.

For any positive integer $n$, we define the integer $P(n)$ by : $P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$. Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.