Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\] [i]Proposed by Holden Mui[/i]

Let $G$ be a finite group and $f: G \to G$ a map, such that $f (xy) = f (x) f (y)$ for all $x, y \in G$ and $f (x) = x^{-1}$ for more than $\frac34$ of all $x \in G$ is fulfilled. Show that $f (x) =x^{-1}$ even holds for all $x \in G$ holds.

The following figure shows a [i]walk[/i] of length 6: [asy] unitsize(20); for (int x = -5; x <= 5; ++x) for (int y = 0; y <= 5; ++y) dot((x, y)); label("$O$", (0, 0), S); draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3)); [/asy] This walk has three interesting properties: [list] [*] It starts at the origin, labelled $O$. [*] Each step is 1 unit north, east, or west. There are no south steps. [*] The walk never comes back to a point it has been to.[/list] Let's call a walk with these three properties a [i]northern walk[/i]. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?

In triangle $ABC$, $\angle BAC = 94^{\circ},\ \angle ACB = 39^{\circ}$. Prove that \[ BC^2 = AC^2 + AC\cdot AB\].

Let $ABCD$ be a square with side length $1$. If point $E$ is on $BC$, point $F$ is on $DC$, and triangle $AEF$ is equilateral, compute the side length of triangle $AEF$. (Note: if your answer has a square root inside a square root, you have not fully simplified your answer.)

The polynomial $x^8 +x^7$ is written on a blackboard. In a move, Peter can erase the polynomial $P(x)$ and write down $(x+1)P(x)$ or its derivative $P'(x).$ After a while, the linear polynomial $ax+b$ with $a\ne 0$ is written on the board. Prove that $a-b$ is divisible by $49.$

Suppose $\mathcal{E}_1 \neq \mathcal{E}_2$ are two intersecting ellipses with a common focus $X$; let the common external tangents of $\mathcal{E}_1$ and $\mathcal{E}_2$ intersect at a point $Y$. Further suppose that $X_1$ and $X_2$ are the other foci of $\mathcal{E}_1$ and $\mathcal{E}_2$, respectively, such that $X_1\in \mathcal{E}_2$ and $X_2\in \mathcal{E}_1$. If $X_1X_2=8, XX_2=7$, and $XX_1=9$, what is $XY^2$?

Find all pairs of integers $(m, n)$ which satisfy the equation \[(2n^2+5m-5n-mn)^2=m^3n\]

While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$. [hide="Clarifications"] [list] [*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect. [*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide] [i]Ray Li[/i]

Let $x, y, z$ be positive real numbers such that $\frac{1}{1 + x}+\frac{1}{1 + y}+\frac{1}{1 + z}= 2$. Prove that $8xyz \le 1$.

Let $ATHEM$ be a convex pentagon with $AT = 14$, $TH = MA = 20$, $HE = EM = 15$, and $\angle THE = \angle EMA = 90^{\circ}$. Find the area of $ATHEM$. [i]Proposed by Andrew Wu[/i]

Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak closes two adjacent digits by a black disc. Then Arutyun comes and says both closed digits (and their order). For which minimal $N$ they may show such a trick? [i]K. Knop, O. Leontieva[/i]

Describe all convex, inscriptible polygons which have the property that however we choose three distinct vertexes of of one of them, those vertexes form an isosceles triangle. [i]Gheorghe Iurea[/i]

Let $ \mathcal{F}$ be a family of hexagons $ H$ satisfying the following properties: i) $ H$ has parallel opposite sides. ii) Any 3 vertices of $ H$ can be covered with a strip of width 1. Determine the least $ \ell\in\mathbb{R}$ such that every hexagon belonging to $ \mathcal{F}$ can be covered with a strip of width $ \ell$. Note: A strip is the area bounded by two parallel lines separated by a distance $ \ell$. The lines belong to the strip, too.

Let $ f(x)$ be a polynomial and $ C$ be a real number. Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$.

$m,n$ are positive intergers and $x,y,z$ positive real numbers such that $0 \leq x,y,z \leq 1$. Let $m+n=p$. Prove that: $0 \leq x^p+y^p+z^p-x^m*y^n-y^m*z^n-z^m*x^n \leq 1$

Solve the system of equations: \[ x+y+z=a \] \[ x^2+y^2+z^2=b^2 \] \[ xy=z^2 \] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

In the non-isosceles $\vartriangle ABC$, the interior bisectors of vertices $B$ and $C$ are drawn, which cut the sides $AC$ and $AB$ at $E$ and $F$ respectively.The line $EF$ cuts the extension of side $BC$ at $T$. In the side$ BC$ a point D is located, so that $\frac{DB}{DC} = \frac{TB}{TC}$. Prove that $AT$ is the exterior bisector of angle $A$.

$a_n$ sequence is a arithmetic sequence with all terms be positive integers. (for $a_n$ non-constant sequence) Let $p_n$ is greatest prime divisor of $a_n$. Prove that $$(\frac{a_n}{p_n})$$ sequence is infinity. [hide]Note: If we find a $M>0$ constant such that $x_n \leq M$ for all $n \in {\mathbb N}$'s, $(x_n)$ sequence is non-infinite, but we can't find $M$, $(x_n)$ sequence is infinity [/hide]

Given an acute scalene triangle $ABC$ inscribed in circle $(O)$. Let $H$ be its orthocenter and $M$ be the midpoint of $BC$. Let $D$ lie on the opposite rays of $HA$ so that $BC=2DM$. Let $D'$ be the reflection of $D$ through line $BC$ and $X$ be the intersection of $AO$ and $MD$. a) Show that $AM$ bisects $D'X$. b) Similarly, we define the points $E,F$ like $D$ and $Y,Z$ like $X$. Let $S$ be the intersection of tangent lines from $B,C$ with respect to $(O)$. Let $G$ be the projection of the midpoint of $AS$ to the line $AO$. Show that there exists a point with the same power to all the circles $(BEY),(CFZ),(SGO)$ and $(O)$.

A $2022 \times 2022$ squareboard was divided into $L$ and $Z$ tetrominoes. Each tetromino consists of four squares, which can be rotated or flipped. Determine the least number of $Z$-tetrominoes necessary to cover the $2022 \times 2022$ squareboard.

Let $ABC$ be a triangle with $AB < AC$. Let $D$ be the intersection point of the internal bisector of angle $BAC$ and the circumcircle of $ABC$. Let $Z$ be the intersection point of the perpendicular bisector of $AC$ with the external bisector of angle $\angle{BAC}$. Prove that the midpoint of the segment $AB$ lies on the circumcircle of triangle $ADZ$. [i]Olimpiada de Matemáticas, Nicaragua[/i]

In Shvambrania there are $N$ towns, every two of which are connected by a road. These roads do not intersect. If necessary, some of them pass over or under others via bridges. An evil magician establishes one-way rules along the roads in such a way that if someone goes out of a certain town he is unable to come back. Prove that (a) It is possible to establish such rules. (b) There exists a town from which it is possible to reach any other town, and there exists a town from which it is not possible to go out. (c) There is one and only one route passing through all towns. (d) The magician can realise his intention in $N!$ ways. (LM Koganov, Moscow) PS. (a),(b),(c) for Juniors, (a),(b),(d) for Seniors

A positive integer $N$ is a [i]triple-double[/i] if there exists non-negative integers $a$, $b$, $c$ such that $2^a + 2^b + 2^c = N$. How many three-digit numbers are triple-doubles? [i]Proposed by Giacomo Rizzo[/i]