Prove there exists a constant $k_0$ such that for any $k\ge k_0$, the equation \[a^{2n}+b^{4n}+2013=ka^nb^{2n}\] has no positive integer solutions $a,b,n$. [i]Proposed by István Pink.[/i]

$S$ is a subset of $\{1,2, . . . ,100\}$. What is the maximum number of elements in $S$ such that the product of any two of them is not a perfect square?

For any real number $p\geq1$ consider the set of all real numbers $x$ with \[p<x<\left(2+\sqrt{p+\frac{1}{4}}\right)^2.\] Prove that from any such set one can select four mutually distinct natural numbers $a, b, c, d$ with $ab=cd.$

For positive integer numbers $a$, $b$ and $c$ it is known that $a^2+b^2+c^2$ and $a^3+b^3+c^3$ are both divisible by $a+b+c$. In addition, $gcd(a+b+c, 6) = 1$. Prove that $a^5+b^5+c^5$ is divisible by $(a+b+c)^2$. [i] A. Antropov [/i]

Let $m,n,k$ be positive integers with $m\ge2$ and $k\ge\log_2(m-1)$. Prove that $$\prod_{s=1}^n\frac{ms-1}{ms}<\sqrt[2^{k+1}]{\frac1{2n+1}}.$$

Find all positive integers n with the following property: For every positive divisor $d$ of $n$, $d+1$ divides $n+1$.

Find all $n\in \mathbb{N}$, $n>1$ with the following property: All divisors of $n$ can be put in a rectangular table in such way that the sums of the numbers by rows are equal and the sums of the numbers by columns are also equal.

Let $X{}$ be a set of integers which can be partitioned into $N{}$ disjoint increasing arithmetic progressions (infinite in both directions), and cannot be partitioned into a smaller number of such progressions. Is such partition into $N{}$ progressions unique for every such $X{}$ if a) $N = 2{}$ and b) $N = 3$? [i]Viktor Kleptsyn[/i]

Find all natural numbers $n$ for which the following $5$ conditions hold: $(1)$ $n$ is not divisible by any perfect square bigger than $1$. $(2)$ $n$ has exactly one prime divisor of the form $4k+3$, $k\in \mathbb{N}_0$. $(3)$ Denote by $S(n)$ the sum of digits of $n$ and $d(n)$ as the number of positive divisors of $n$. Then we have that $S(n)+2=d(n)$. $(4)$ $n+3$ is a perfect square. $(5)$ $n$ does not have a prime divisor which has $4$ or more digits.

All the $5$-digit numbers from $11111$ to $99999$ are written on the cards. Those cards lies in a line in an arbitrary order. Prove that the resulting $444445$-digit number is not a power of two.

Show that there are infinitely many odd composite numbers in the sequence $1^1, 1^1 + 2^2, 1^1 + 2^2 + 3^3, 1^1 + 2^2 + 3^3 + 4^4, ...$ .

Find the smallest possible value of $x + y$ where $x, y \ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.

Can both $(x^2+y)$ and $(y^2+x)$ be exact squares for natural $x$ and $y$?

Determine all integers $m$ for which all solutions of the equation $3x^3-3x^2+m = 0$ are rational.

[b]6.1 [/b] Two people went from point A to point B. The first one walked along highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first of them arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway. [b]6.2.[/b] A pedestrian walks along the highway at a speed of 5 km/hour. Along this highway in both directions at the same speed Buses run, meeting every 5 minutes. At 12 o'clock the pedestrian noticed that the buses met near him and, Continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met near him again. It turned out that during this time the pedestrian encountered 4 buses more than overtook him. Find the speed of the bus [b]6.3. [/b] Prove that the difference $43^{43} - 17^{17}$ is divisible by $10$. [b]6.4. [/b] Two squares are cut out of the chessboard on the border of the board. When is it possible and when is it not possible to cover with the remaining squares of the board? shapes of the view without overlay? [b]6.5.[/b] The distance from city A to city B (by air) is 30 kilometers, from B to C - 80 kilometers, from C to D - 236 kilometers, from D to E - 86 kilometers, from E to A- 40 kilometers. Find the distance from E to C. [b]6.6.[/b] Is it possible to write down the numbers from $ 1$ to $1963$ in a series so that any two adjacent numbers and any two numbers located one after the other were mutually prime? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

Let $\phi$ be the Euler totient function. Let $\phi^k (n) = (\underbrace{\phi \circ ... \circ \phi}_{k})(n)$ be $\phi$ composed with itself $k$ times. Define $\theta (n) = min \{k \in N | \phi^k (n)=1 \}$ . For example, $\phi^1 (13) = \phi(13) = 12$ $\phi^2 (13) = \phi (\phi (13)) = 4$ $\phi^3 (13) = \phi(\phi(\phi(13))) = 2$ $\phi^4 (13) = \phi(\phi(\phi(\phi(13)))) = 1$ so $\theta (13) = 4$. Let $f(r) = \theta (13^r)$. Determine $f(2012)$.

Find all sets $A$ and $B$ that satisfy the following conditions: a) $A \cup B= \mathbb{Z}$; b) if $x \in A$ then $x-1 \in B$; c) if $x,y \in B$ then $x+y \in A$. [i]Laurentiu Panaitopol[/i]

For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.

For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

Find all positive integers $d$ that can be written in the form $$ d = \gcd(|x^2 - y| , |y^2 - z| , |z^2 - x|), $$ where $x, y, z$ are pairwise coprime positive integers such that $x^2 \neq y$, $y^2 \neq z$, and $z^2 \neq x$.

For a natural number $N$, consider all distinct perfect squares that can be obtained from $N$ by deleting one digit from its decimal representation. Prove that the number of such squares is bounded by some value that doesn't depend on $N$.

Are there infinitely many different natural numbers $a_1,a_2, a_3,...$ so that for every integer $k$ only finitely many of the numbers $a_1 + k$,$a_2 + k$,$a_3 + k$,$...$ are numbers prime?

Prove that there exist infinitely many positive integers which cannot be written in form $a^{d(a)}+b^{d(b)}$ for some positive integers $a$ and $b$ For positive integer $d(a)$ denotes number of positive divisors of $a$ [i]Proposed by Borna Vukorepa[/i]

For every positive integer $n$ we define $a_{n}$ as the last digit of the sum $1+2+\cdots+n$. Compute $a_{1}+a_{2}+\cdots+a_{1992}$.

It is known that for every integer $n > 1$ there is a prime number among the numbers $n+1,n+2,...,2n-1.$ Determine all positive integers $n$ with the following property: Every integer $m > 1$ less than $n$ and coprime to $n$ is prime.