We call a pair $(a,b)$ of positive integers, $a<391$, [i]pupusa[/i] if $$\textup{lcm}(a,b)>\textup{lcm}(a,391)$$ Find the minimum value of $b$ across all [i]pupusa[/i] pairs. Fun Fact: OMCC 2017 was held in El Salvador. [i]Pupusa[/i] is their national dish. It is a corn tortilla filled with cheese, meat, etc.

A natural number $n$, which is at most $500$, has the property that when one chooses at at random among the numbers $1, 2, 3,...,499, 500$, then the probability is $\frac{1}{100}$ for $m$ to add up to $n$. Determine the largest possible value of $n$.

Let $n\ge 1$ be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of $n$ spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either • places a stone in an empty space, or • removes a stone from a nonempty space $s,$ places a stone in the nearest empty space to the left of $s$ (if such a space exists), and places a stone in the nearest empty space to the right of $s$ (if such a space exists). Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?

Let $x,y,z \in \mathbb{R}$ such that $xyz = 1$. Prove that: $$\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).$$

Show that for all positive real numbers $x_1,x_2,\ldots,x_n$ with product 1, the following inequality holds \[ \frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1. \]

Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.

Prove that for every natural number $ n $, the number $ n^{2n} - n^{n+2} + n^n - 1 $ is divisible by $ (n - 1 )^3 $.

In a triangle $ ABC$, let $ M_{a}$, $ M_{b}$, $ M_{c}$ be the midpoints of the sides $ BC$, $ CA$, $ AB$, respectively, and $ T_{a}$, $ T_{b}$, $ T_{c}$ be the midpoints of the arcs $ BC$, $ CA$, $ AB$ of the circumcircle of $ ABC$, not containing the vertices $ A$, $ B$, $ C$, respectively. For $ i \in \left\{a, b, c\right\}$, let $ w_{i}$ be the circle with $ M_{i}T_{i}$ as diameter. Let $ p_{i}$ be the common external common tangent to the circles $ w_{j}$ and $ w_{k}$ (for all $ \left\{i, j, k\right\}= \left\{a, b, c\right\}$) such that $ w_{i}$ lies on the opposite side of $ p_{i}$ than $ w_{j}$ and $ w_{k}$ do. Prove that the lines $ p_{a}$, $ p_{b}$, $ p_{c}$ form a triangle similar to $ ABC$ and find the ratio of similitude. [i]Proposed by Tomas Jurik, Slovakia[/i]

In a tetrahedron, the six triangles determined by an edge of the tetrahedron and the midpoint of the opposite edge all have equal area. Prove that the tetrahedron is regular.

In the every cell of the board $5\times5$ there is one of the numbers: $-1$, $0$, $1$. It is true that in every $2 \times 2$ square there are three numbers summing up to $0$. Determine the maximal sum of all numbers in a board.

\[\int_1^\infty \frac{1}{x\sqrt{x^{2023}-1}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Find the least odd prime factor of $2019^8 + 1$.

Circle $\omega$ is inscribed in acute triangle $ABC$. Let $I$ denote the center of $\omega$, and let $D,E,F$ be the points of tangency of $\omega$ with $BC, CA, AB$ respectively. Let $M$ be the midpoint of $BC$, and $P$ be the intersection of the line through $I$ perpendicular to $AM$ and line $EF$. Suppose that $AP=9$, $EC=2EA$, and $BD=3$. Find the sum of all possible perimeters of $\triangle ABC$. [i]Proposed by Harry Kim[/i]

Let $ ABCD$ be a convex quadrilateral. Points $ P,Q$ and $ R$ are the feets of the perpendiculars from point $ D$ to lines $ BC, CA$ and $ AB$, respectively. Prove that $ PQ\equal{}QR$ if and only if the bisectors of the angles $ ABC$ and $ ADC$ meet on segment $ AC$.

Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have \[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}} \le \lambda\] Where $a_{n+i}=a_i,i=1,2,\ldots,k$

For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.) [i]Aprameya Tripathy[/i]

Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$. Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.

Given a convex $n$-gon with pairwise (mutually) non-parallel sides and a point inside it. Prove that there are not more than $n$ straight lines coming through that point and halving the area of the $n$-gon.

Let $H$ be the orthocentre and $G$ be the centroid of acute-angled triangle $ABC$ with $AB\ne AC$. The line $AG$ intersects the circumcircle of $ABC$ at $A$ and $P$. Let $P'$ be the reflection of $P$ in the line $BC$. Prove that $\angle CAB = 60$ if and only if $HG = GP'$

Let $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ be a function such that for every regular $ n $-gon $ A_1A_2 \ldots A_n $ we have $ f(A_1)+f(A_2)+\cdots +f(A_n)=0 $. Prove that $ f(x)=0 $ for all reals $ x $.

$B$ is a point on the line which is tangent to a circle at the point $A$. The line segment $AB$ is rotated about the centre of the circle through some angle to the line segment $A'B'$. Prove that the line $AA'$ passes through the midpoint of $BB'$.

Let \(ABC\) be a triangle. Let \(B'\) denote the reflection of \(b\) in the internal angle bisector \(l\) of \(\angle A\).Show that the circumcentre of the triangle \(CB'I\) lies on the line \(l\) where \(I\) is the incentre of \(ABC\).

For any positive integer $k$ consider the sequence $$a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},$$ where there are $n$ square-root signs on the right-hand side. (a) Show that the sequence converges, for every fixed integer $k\ge 1$. (b) Find $k$ such that the limit is an integer. Furthermore, prove that if $k$ is odd, then the limit is irrational.

In a three-dimensional Euclidean space, by $\overrightarrow{u_1}$ , $\overrightarrow{u_2}$ , $\overrightarrow{u_3}$ are denoted the three orthogonal unit vectors on the $x, y$, and $z$ axes, respectively. a) Prove that the point $P(t) = (1-t)\overrightarrow{u_1} +(2-3t)\overrightarrow{u_2} +(2t-1)\overrightarrow{u_3}$ , where $t$ takes all real values, describes a straight line (which we will denote by $L$). b) What describes the point $Q(t) = (1-t^2)\overrightarrow{u_1} +(2-3t^2)\overrightarrow{u_2} +(2t^2 -1)\overrightarrow{u_3}$ if $t$ takes all the real values? c) Find a vector parallel to $L$. d) For what values of $t$ is the point $P(t)$ on the plane $2x+ 3y + 2z +1 = 0$? e) Find the Cartesian equation of the plane parallel to the previous one and containing the point $Q(3)$. f) Find the Cartesian equation of the plane perpendicular to $L$ that contains the point $Q(2)$.

Define the function $f:\mathbb N_{>1}\to\mathbb N_{>1}$ such that $f(x)$ is the greatest prime factor of $x$. A sequence of positive integers $\{a_n\}$ satisfies $a_1=M>1$ and $$a_{n+1}=\begin{cases}a_n-f(a_n)&\text{if }a_n\text{ is composite.}\\a_n+k&\text{otherwise.}\end{cases}$$ Show that for any positive integers $M,k$, the sequence $\{a_n\}$ is bounded. (TAN768092100853)