Find all integers $n$ with $1<n<1979$ having the following property: If $m$ is an integer coprime with $n$ and $1<m<n$, then $m$ is a prime number.

Given $n$ numbers $a_1,a_2,...,a_n$ ($n\geq 3$) where $a_i\in\{0,1\}$ for all $i=1,2.,,,.n$. Consider $n$ following $n$-tuples \[ \begin{aligned} S_1 & =(a_1,a_2,...,a_{n-1},a_n)\\ S_2 & =(a_2,a_3,...,a_n,a_1)\\ & \vdots\\ S_n & =(a_n,a_1,...,a_{n-2},a_{n-1}).\end{aligned}\] For each tuple $r=(b_1,b_2,...,b_n)$, let \[ \omega (r)=b_1\cdot 2^{n-1}+b_2\cdot 2^{n-2}+\cdots+b_n. \] Assume that the numbers $\omega (S_1),\omega (S_2),...,\omega (S_n)$ receive exactly $k$ different values. a) Prove that $k|n$ and $\frac{2^n-1}{2^k-1}|\omega (S_i)\quad\forall i=1,2,...,n.$ b) Let \[ \begin{aligned} M & =\max _{i=\overline{1,n}}\omega (S_i)\\ m & =\min _{i=\overline{1,n}}\omega (S_i). \end{aligned} \] Prove that \[ M-m\geq\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}. \]

Find all the natural numbers $x, y, z$ that satisfy simultaneously $$\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}$$

Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way: $(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$ $(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set. Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$

Find all prime numbers p such that $2^p+p^2 $ is also a prime number.

Let the lengths of three sides of a triangle be $l, m, n(l>m>n)$. If $\left\{\frac{3^l}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, find the minimum value of the perimeter of the triangle. Note: $\{x\}=x-[x]$ and $[x]$ denotes the integral part of number $x$.

Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter. $i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation. $ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.

Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$.

Solve in non-negative integers the equation $ 2^{a}3^{b} \plus{} 9 \equal{} c^{2}$

positive reals $a_1, a_2, . . . $ satisfying (i) $a_{n+1}=a_1^2\cdot a_2^2 \cdot . . . \cdot a_n^2-3$(all positive integers $n$) (ii) $\frac{1}{2}(a_1+\sqrt{a_2-1})$ is positive integer. prove that $\frac{1}{2}(a_1 \cdot a_2 \cdot . . . \cdot a_n + \sqrt{a_{n+1}-1})$ is positive integer

Denote $ S_n $ as being the sum of the squares of the first $ n\in\mathbb{N} $ terms of a given arithmetic sequence of natural numbers. [b]a)[/b] If $ p\ge 5 $ is a prime, then $ p\big| S_p. $ [b]b)[/b] $ S_5 $ is not a perfect square.

Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$. Find all local champions and determine their number. [i]Proposed by Zoran Sunic, USA[/i]

Let $F_n{}$ denote the $n{}$-th Fibonacci number. Prove that $3^{2023}$ divides \[3^2\cdot F_4+3^3\cdot F_6+3^4\cdot F_8+\dots+3^{2023}F_{4046}.\][i]Proposed by Dylan Toh[/i]

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]

Determine all pairs of positive integers $(m, n)$ for which there exists a bijective function \[f : \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_m \times \mathbb{Z}_n\]such that the vectors $f(\mathbf{v}) + \mathbf{v}$, as $\mathbf{v}$ runs through all of $\mathbb{Z}_m \times \mathbb{Z}_n$, are pairwise distinct. (For any integers $a$ and $b$, the vectors $[a, b], [a + m, b]$ and $[a, b + n]$ are treated as equal.) [i]Poland, Wojciech Nadara[/i]

Determine all primes $p$, for which there exist positive integers $m, n$, such that $p=m^2+n^2$ and $p|m^3+n^3+8mn$.

Let $n>100$ be a positive integer and originally the number $1$ is written on the blackboard. Petya and Vasya play the following game: every minute Petya represents the number of the board as a sum of two distinct positive fractions with coprime nominator and denominator and Vasya chooses which one to delete. Show that Petya can play in such a manner, that after $n$ moves, the denominator of the fraction left on the board is at most $2^n+50$, no matter how Vasya acts.

What is the smallest positive odd integer having the same number of positive divisors as $360$?

What is the largest whole number that is equal to the product of its digits?

Given $2N$ real numbers. It is known that if they are arbitrarily divided into two groups of $N$ numbers each then the products of the numbers of each group differ by $2$ at most. Is it necessarily true that if we arbitrarily place these numbers along a circle then there are two neighboring numbers that differ by $2$ at most, for a) $N=50$; (3 marks) b) $N=25$? (5 marks)

Let $a,b,m,n$ integers greater than 1. If $a^n-1$ and $b^m+1$ are both primes, give as much info as possible on $a,b,m,n$.

Show that $\phi(n)+\sigma(n) \ge 2n$ for all positive integers $n$.

Find all triples of positive integers $(a,b,c)$ such that the following equations are both true: I- $a^2+b^2=c^2$ II- $a^3+b^3+1=(c-1)^3$

Let s be a positive real. Consider a two-dimensional Cartesian coordinate system. A [i]lattice point[/i] is defined as a point whose coordinates in this system are both integers. At each lattice point of our coordinate system, there is a lamp. Initially, only the lamp in the origin of the Cartesian coordinate system is turned on; all other lamps are turned off. Each minute, we additionally turn on every lamp L for which there exists another lamp M such that - the lamp M is already turned on, and - the distance between the lamps L and M equals s. Prove that each lamp will be turned on after some time ... [b](a)[/b] ... if s = 13. [This was the problem for class 11.] [b](b)[/b] ... if s = 2005. [This was the problem for classes 12/13.] [b](c)[/b] ... if s is an integer of the form $s=p_1p_2...p_k$ if $p_1$, $p_2$, ..., $p_k$ are different primes which are all $\equiv 1\mod 4$. [This is my extension of the problem, generalizing both parts [b](a)[/b] and [b](b)[/b].] [b](d)[/b] ... if s is an integer whose prime factors are all $\equiv 1\mod 4$. [This is ZetaX's extension of the problem, and it is stronger than [b](c)[/b].] Darij

Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$