$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+3)(n+4)$ is an integer. (Author: O. Podlipski) [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=427802](similar to Problem 5 of grade 9)[/url] Same problem for grades 10 and 11

Let $n$, $m$, and $k$ be positive integers satisfying $(n - 1)n(n + 1) = m^k$. Prove that $k = 1$.

An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$

Let $S \subset \{ 1, \dots, n \}$ be a nonempty set, where $n$ is a positive integer. We denote by $s$ the greatest common divisor of the elements of the set $S$. We assume that $s \not= 1$ and let $d$ be its smallest divisor greater than $1$. Let $T \subset \{ 1, \dots, n \}$ be a set such that $S \subset T$ and $|T| \ge 1 + \left[ \frac{n}{d} \right]$. Prove that the greatest common divisor of the elements in $T$ is $1$. ----------- [Second Version] Let $n(n \ge 1)$ be a positive integer and $U = \{ 1, \dots, n \}$. Let $S$ be a nonempty subset of $U$ and let $d (d \not= 1)$ be the smallest common divisor of all elements of the set $S$. Find the smallest positive integer $k$ such that for any subset $T$ of $U$, consisting of $k$ elements, with $S \subset T$, the greatest common divisor of all elements of $T$ is equal to $1$.

Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\dfrac{7}{2}$ times the original one.

A positive integer $k > 1$ is called nice if for any pair $(m, n)$ of positive integers satisfying the condition $kn + m | km + n$ we have $n | m$. 1. Prove that $5$ is a nice number. 2. Find all the nice numbers.

Recall that for an arbitrary prime $p$, we define a [b]primitive root[/b] modulo $p$ as an integer $r$ for which the least positive integer $v$ such that $r^{v}\equiv 1\pmod{p}$ is $p-1$.\\ Prove or disprove the following statement: [center] For every prime $p>2023$, there exists positive integers $1\leqslant a<b<c<p$\\ such that $a,b$ and $c$ are primitive roots modulo $p$ but $abc$ is not a primitive root modulo $p$. [/center]

A sequence of integers $1 = x_1 < x_2 < x_3 <...$ satisfies $x_{n+1} \le 2n$ for all $n$. Show that every positive integer $k$ can be written as $x_j -x_i$ for some $i, j$.

Find all positive integers $m$ and $n$ with no common divisor greater than 1 such that $m^3 + n^3$ divides $m^2 + 20mn + n^2$. [i](Professor Yongjin Song)[/i]

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

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.

3. Show that there do not exist positive integers $a$, $b$, $c$ and $d$, pairwise co-prime, such that $ab+cd$, $ac+bd$ and $ad+bc$ are odd divisors of the number $(a+b-c-d)(a-b+c-d)(a-b-c+d)$.

For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute $T =r_{100}+ r_{101} + r_{102}+...+r_{200}$

Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$ we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$

Determine the smallest integer $n \ge 3$ with the property that you can choose two of the numbers $1,2,\dots, n$ in such a way that their product is equal to the sum of the other $n - 2$ languages. What are the two numbers?

Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)

Find $10$ different positive integers such that each of them is a divisor of their sum (S Fomin, Leningrad)

Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.

Prove that if $n$ is a square-free positive integer, there are no coprime positive integers $x$ and $y$ such that $(x + y)^3$ divides $x^n+y^n$

An integer $a\ge1$ is called [i]Aegean[/i], if none of the numbers $a^{n+2}+3a^n+1$ with $n\ge1$ is prime. Prove that there are at least 500 Aegean integers in the set $\{1,2,\ldots,2018\}$. (Proposed by Gerhard Woeginger, Austria)

Find all primes $p$ such that $p^2 + 2543$ has at most $16$ divisors.

Given six irrational numbers, will it be possible to choose three such that the sum of any two of these three is irrational?

Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$.

Define the sequences $ a_n, b_n, c_n$ as follows. $ a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1$. If $ a_n$ is even then $ a_{n \plus{} 1} \equal{} \frac {a_n}{2}$, $ b_{n \plus{} 1} \equal{} 2b_n$, $ c_{n \plus{} 1} \equal{} c_n$. If $ a_n$ is odd, then $ a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n$, $ b_{n \plus{} 1} \equal{} b_n$, $ c_{n \plus{} 1} \equal{} b_n \plus{} c_n$. Find the number of positive integers $ k < 1995$ such that some $ a_n \equal{} 0$.