Prove: [b](a)[/b] There are infinitely many positive intengers $n$, satisfying: $$\gcd(n,[\sqrt2n])=1.$$ [b](b)[/b] There are infinitely many positive intengers $n$, satisfying: $$\gcd(n,[\sqrt2n])>1.$$

The positive integers $a, b, c, d$ satisfy (i) $a + b + c + d = 2023$ (ii) $2023 \text{ } | \text{ } ab - cd$ (iii) $2023 \text{ } | \text{ } a^2 + b^2 + c^2 + d^2.$ Assuming that each of the numbers $a, b, c, d$ is divisible by $7$, prove that each of the numbers $a, b, c, d$ is divisible by $17$.

Prove that the product of five consecutive positive integers cannot be a perfect square.

Suppose that $ a,b$ are positive integers for which $ A\equal{}\frac{a\plus{}1}{b}\plus{}\frac{b}{a}$ is an integer.Prove that $ A\equal{}3$.

Pages of some book are numerated with numbers $1$ to $100$. From the book several double pages were ripped out and sum of enumerations of that pages is equal to $4949$. How many double pages were ripped out?

Find all natural numbers $n\ge1$ such that the implication $$(11\mid a^n+b^n)\implies(11\mid a\wedge11\mid b)$$holds for any two natural numbers $a$ and $b$.

Determine the gcd of all numbers of the form $p^8-1$, with p a prime above 5.

Let $ n$ be a positive integer. Show that the numbers \[ \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1}\] are congruent modulo $ 2^n$ to $ 1$, $ 3$, $ 5$, $ \ldots$, $ 2^n \minus{} 1$ in some order. [i]Proposed by Duskan Dukic, Serbia[/i]

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply? [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy? [b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert? [b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for every composite number $ n>4$, numbers $ kn$ divides $ (n\minus{}1)!$ for every integer $ k$ such that $ 1\le k\le \lfloor \sqrt{n\minus{}1} \rfloor$.

Given positive integers $m$, $a$, $b$, $(a,b)=1$. $A$ is a non-empty subset of the set of all positive integers, so that for every positive integer $n$ there is $an \in A$ and $bn \in A$. For all $A$ that satisfy the above condition, find the minimum of the value of $\left| A \cap \{ 1,2, \cdots,m \} \right|$

The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.

If natural numbers $ x$, $ y$, $ p$, $ n$, $ k$ with $ n > 1$ odd and $ p$ an odd prime satisfy $ x^n \plus{} y^n \equal{} p^k$, prove that $ n$ is a power of $ p$.

Find all ordered pairs $(a,b)$, of natural numbers, where $1<a,b\leq 100$, such that $\frac{1}{\log_{a}{10}}+\frac{1}{\log_{b}{10}}$ is a natural number.

Let $p$ be a prime number, determine all positive integers $(x, y, z)$ such that: $x^p + y^p = p^z$

Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.

Define the set $M_q=\{x \in \mathbb{Q} \mid x^3-2015x=q \}$ , where $q$ is an arbitrary rational number. [b]a)[/b] Show that there exists values for $q$ such that the set is null as well as values for which it has exactly one element. [b]b)[/b] Determine all the possible values for the cardinality of $M_q$

We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$. A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial. Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that $$P(n)|f(n)$$ for all positive integers $n$.

Let $({{A}_{n}})_{n=1}^{\infty }$ and $({{a}_{n}})_{n=1}^{\infty }$ be sequences of positive integers. Assume that for each positive integer $x$, there is a unique positive integer $N$ and a unique $N-tuple$ $({{x}_{1}},...,{{x}_{N}})$ such that $0\le {{x}_{k}}\le {{a}_{k}}$ for $k=1,2,...N$, ${{x}_{N}}\ne 0$, and $x=\sum\limits_{k=1}^{N}{{{A}_{k}}{{x}_{k}}}$. (a) Prove that ${{A}_{k}}=1$ for some $k$; (b) Prove that ${{A}_{k}}={{A}_{j}}\Leftrightarrow k=j$; (c) Prove that if ${{A}_{k}}\le {{A}_{j}}$, then $\left. {{A}_{k}} \right|{{A}_{j}}$.

Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) + \phi (b) + gcd (a, b)$. Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.

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.

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

[b]a)[/b] Show that if two non-negative integers $ p,q $ satisfy the property that both $ \sqrt{2p-q} $ and $ \sqrt{2p+q} $ are non-negative integers, then $ q $ is even. [b]b)[/b] Determine how many natural numbers $ m $ are there such that $ \sqrt{2m-4030} $ and $ \sqrt{2m+4030} $ are both natural.

Determine all pairs $(n,p)$ of nonnegative integers such that [list] [*] $p$ is a prime, [*] $n<2p$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.