Find the minimum value of $xy+x+y+\frac{1}{xy}+\frac{1}{x}+\frac{1}{y}$ for $x, y>0$ real.

Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

Construct on the real projective plane a continuous curve, consisting of simple points, which is not a straight line and is intersected in a single point by every tangent and every secant of a given conic. [i]F. Karteszi[/i]

Find all pairs of integers $(a,b)$ such that $(b^2+7(a-b))^2=a^{3}b$.

A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$ The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.

In the cyclic quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $P$. Let $O$ be the center of the circumcircle $ABCD$, and $E$ a point of the extension of $OC$ beyond $C$. A parallel line to $CD$ is drawn through $E$ that cuts the extension of $OD$ beyonf $D$ at $F$. Let $Q$ be a point interior to $ABCD$, such that $\angle AFQ = \angle BEQ$ and $\angle FAQ = \angle EBQ$. Prove that $PQ \perp CD$.

The positive integer $a$ is relatively prime with $10$. Prove that for any positive integer $n$, there exists a power of $a$ whose last $n$ digits are $\underbrace{0...0}_\text{n-1}1$.

Farmer John is grazing his cows at the origin. There is a river that runs east to west $50$ feet north of the origin. The barn is $100$ feet to the south and $80$ feet to the east of the origin. Farmer John leads his cows to the river to take a swim, then the cows leave the river from the same place they entered and Farmer John leads them to the barn. He does this using the shortest path possible, and the total distance he travels is $d$ feet. Find the value of $d$.

Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?

Find all natural numbers such that each is equal to the sum of the factorials of its digits. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Ana and Beto are playing a game. Ana writes a whole number on the board. Beto then has the right to erase the number and add $2$ to it, or erase the number and subtract $3$, as many times as he wants. Beto wins if he can get $2021$ after a finite number of stages; otherwise, Ana wins. Which player has a winning strategy?

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

Let $A,B\in M_2(\mathbb Z)$ s.t. $AB=\begin{pmatrix}1&2005\\0&1\end{pmatrix}$. Prove that there is a matrix $C\in M_2(\mathbb Z)$ s.t. $BA=C^{2005}$. [i]Dinu Serbanescu[/i]

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for any two positive integers $a$ and $b$ we have $$ f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1)$$ Where $f^n(m)$ is defined in the standard iterative manner.

Let $a$, $b$ and $c$ be given positive integers which are two by two coprime. A positive integer $n$ is called [i]sozopolian[/i], if it [u]can’t[/u] be written as $n=bcx+cay+abz$ where $x$, $y$, $z$ are also positive integers. Find the number of [i]sozopolian[/i] numbers as a function of $a$, $b$ and $c$.

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

Consider the $4 \times 4$ “multiplication table” below. The numbers in the first column multiplied by the numbers in the first row give the remaining numbers in the table. For example, the $3$ in the first column times the $4$ in the first row give the $12 (= 3 \cdot 4)$ in the cell that is in the 3rd row and 4th column. [asy] size(3cm); for (int x=0; x<=4; ++x) draw((x, 0) -- (x, 4), linewidth(.5pt)); for (int y=0; y<=4; ++y) draw((0, y) -- (4, y), linewidth(.5pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); void foo(int x, int y, string n) { label(n, (x+0.5, y+0.5)); } foo(0, 3, "1"); foo(1, 3, "2"); foo(2, 3, "3"); foo(3, 3, "4"); foo(0, 2, "2"); foo(1, 2, "4"); foo(2, 2, "6"); foo(3, 2, "8"); foo(0, 1, "3"); foo(1, 1, "6"); foo(2, 1, "9"); foo(3, 1, "12"); foo(0, 0, "4"); foo(1, 0, "8"); foo(2, 0, "12"); foo(3, 0, "16"); [/asy] We create a path from the upper-left square to the lower-right square by always moving one cell either to the right or down. For example, here is one such possible path, with all the numbers along the path circled: [asy] import graph; size(3cm); for (int x=0; x<=4; ++x) draw((x, 0) -- (x, 4), linewidth(.5pt)); for (int y=0; y<=4; ++y) draw((0, y) -- (4, y), linewidth(.5pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); void foo(int x, int y, string n) { label(n, (x+0.5, y+0.5)); } draw(Circle((0.5,3.5),0.5)); draw(Circle((1.5,3.5),0.5)); draw(Circle((2.5,3.5),0.5)); draw(Circle((2.5,2.5),0.5)); draw(Circle((3.5,2.5),0.5)); draw(Circle((3.5,1.5),0.5)); draw(Circle((3.5,0.5),0.5)); foo(0, 3, "1"); foo(1, 3, "2"); foo(2, 3, "3"); foo(3, 3, "4"); foo(0, 2, "2"); foo(1, 2, "4"); foo(2, 2, "6"); foo(3, 2, "8"); foo(0, 1, "3"); foo(1, 1, "6"); foo(2, 1, "9"); foo(3, 1, "12"); foo(0, 0, "4"); foo(1, 0, "8"); foo(2, 0, "12"); foo(3, 0, "16"); [/asy] If we add up the circled numbers in the example above (including the start and end squares), we get $48$. Considering all such possible paths: (a) What is the smallest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct. (b) What is the largest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct.

Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.

Seventeen cities are served by four airlines. It is noted that there is direct service (without stops) between any two cities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.

Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $

We call a tiling of an $m \times n$ rectangle with corners (see figure below) "regular" if there is no sub-rectangle which is tiled with corners. Prove that if for some $m$ and $n$ there exists a "regular" tiling of the $m \times n$ rectangular then there exists a "regular" tiling also for the $2m \times 2n $ rectangle.

The robot crawls the meter in a straight line, puts a flag on and turns by an angle $a <180^o$ clockwise. After that, everything is repeated. Prove that all flags are on the same circle.

We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions: [b]a)[/b] every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares. [b]b)[/b] the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division. [color=#008000]Moderator says: see [url]https://artofproblemsolving.com/community/c6h58591[/url][/color]

Prove that there are do not exist natural numbers $k, m$ such that numbers $k^2+2m$, $m^2+2k$ to be squares of integers.