Given is a convex octagon $A_1A_2 \ldots A_8$. Given a triangulation $T$, one can take two triangles $\triangle A_iA_jA_k$ and $\triangle A_iA_kA_l$ and replace them with $\triangle A_iA_jA_l$ and $\triangle A_jA_lA_k$. Find the minimal number of operations $k$ we have to do so that for any pair of triangulations $T_1, T_2$, we can reach $T_2$ from $T_1$ using at most $k$ operations.

All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in. [img]https://cdn.artofproblemsolving.com/attachments/9/0/9a6af9cef6a4bb03fb4d3eef715f3fd77c74b3.png[/img]

Let $n{}$ be a positive integer. Show that there exists a polynomial $f{}$ of degree $n{}$ with integral coefficients such that \[f^2=(x^2-1)g^2+1,\] where $g{}$ is a polynomial with integral coefficients.

We have garland with $n$ lights. Some lights are on, some are off. In one move we can take some turned on light (only turned on) and turn off it and also change state of neigbour lights. We want to turn off all lights after some moves.. For what $n$ is it always possible?

Find all pairs of real numbers $(x, y)$ that satisfy the equation $(x + y)^2 = (x + 3) (y - 3)$.

Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality \[ \frac {1}{a^3 \plus{} b^3 \plus{} abc} \plus{} \frac {1}{b^3 \plus{} c^3 \plus{} abc} \plus{} \frac {1}{c^3 \plus{} a^3 \plus{} abc} \leq \frac {1}{abc} \] holds.

Let $\triangle ABC$ be an acute triangle with $AB =AC$ and let $D$ be a point on the side $BC$. The circle with centre $D$ passing through $C$ intersects $\odot(ABD)$ at points $P$ and $Q$, where $Q$ is the point closer to $B$. The line $BQ$ intersects $AD$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $PDXY$ is cyclic.

Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$.

Find the sum of all even numbers greater than 100000, that u can make only with the digits 0,2,4,6,8,9 without any digit repeating in any number.

Given that $P(x)$ is the least degree polynomial with rational coefficients such that \[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.

Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

We are given a polynomial $f(x)=x^6-11x^4+36x^2-36$. Prove that for an arbitrary prime number $p$, $f(x)\equiv 0\pmod{p}$ has a solution.

Into each box of a $ 2012 \times 2012 $ square grid, a real number greater than or equal to $ 0 $ and less than or equal to $ 1 $ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to $ 1 $, no matter how the grid is split into $2$ such rectangles. Determine the maximum possible value for the sum of all the $ 2012 \times 2012 $ numbers inserted into the boxes.

In a grid of $100\times 100$ squares, there is a mouse on the top-left square, and there is a piece of cheese in the bottom-right square. The mouse wants to move to the bottom-right square to eat the cheese. For each step, the mouse can move from one square to an adjacent square (two squares are considered adjacent if they share a common edge). Now, any divider can be placed on the common edge of two adjacent squares such that the mouse cannot directly move between these two adjacent squares. A placement of dividers is called "kind" if the mouse can still reach the cheese after the dividers are placed. Find the smallest positive integer $n$ such that, regardless of any "kind" placement of $2023$ dividers, the mouse can reach the cheese in at most $n$ steps.

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 $\underbrace{0...0}_\text{n-1}1$.

Let $\{U_{n,1},...,U_{n,n}\}_{n=1}^\infty$ be iid rv, uniformly distributed over [0,1] , and for $\alpha\geq 1$ consider the sets $\{[n^\alpha U_{n,1}],...,[n^\alpha U_{n,n}]\}$ , where [·] denotes the whole part. Prove that the elements of the sets $H_n\cap(\cup_{m=n+1}^\infty H_m)$ form an almost surely bounded sequence if and only if $\alpha>3$.

Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

There are $2022$ natural numbers written in a row. Product of any two adjacent numbers is a perfect cube. Prove that the product of the two extremes is also a perfect cube.

If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.

Given a $2\times 5$ rectangle is divided into unit squares as figure below. [img]https://cdn.artofproblemsolving.com/attachments/6/a/9432bbf40f6d89ee1cbb507e1a3f65326c6a13.png[/img] How many ways are there to write the letters $H, A, N, O, I$ into all of the unit squares, such that two neighbor squares (the squares with a common side) do not contain the same letters? (Each unit square is filled by only one letter and each letter may be used several times or not used as well.)

Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?

Positive real numbers, $a_1, .. ,a_6$ satisfy $a_1^2+..+a_6^2=2$. Think six squares that has side length of $a_i$ ($i=1,2,\ldots,6$). Show that the squares can be packed inside a square of length $2$, without overlapping.

Real numbers $x,y$ satisfy that $\begin{cases} (x-1)^3+1997(x-1)=-1\\ (y-1)^3+1997(y-1)=1 \end{cases}$, then $x+y=$________.

For which cardinalities $ \kappa$ do antimetric spaces of cardinality $ \kappa$ exist? $ (X,\varrho)$ is called an $ \textit{antimetric space}$ if $ X$ is a nonempty set, $ \varrho : X^2 \rightarrow [0,\infty)$ is a symmetric map, $ \varrho(x,y)\equal{}0$ holds iff $ x\equal{}y$, and for any three-element subset $ \{a_1,a_2,a_3 \}$ of $ X$ \[ \varrho(a_{1f},a_{2f})\plus{}\varrho(a_{2f},a_{3f}) < \varrho(a_{1f},a_{3f})\] holds for some permutation $ f$ of $ \{1,2,3 \}$. [i]V. Totik[/i]

Let $x,y,z$ be positive numbers with $x+y+z=1$. Show that $$yz+zx+xy\ge4\left(y^2z^2+z^2x^2+x^2y^2\right)+5xyz.$$When does equality hold?