In a triangle $ABC ~(\overline{AB} < \overline{AC})$, points $D (\neq A, B)$ and $E (\neq A, C)$ lies on side $AB$ and $AC$ respectively. Point $P$ satisfies $\overline{PB}=\overline{PD}, \overline{PC}=\overline{PE}$. $X (\neq A, C)$ is on the arc $AC$ of the circumcircle of triangle $ABC$ not including $B$. Let $Y (\neq A)$ be the intersection of circumcircle of triangle $ADE$ and line $XA$. Prove that $\overline{PX} = \overline{PY}$.

Let there be $320$ points arranged on a circle, labeled $1$, $2$, $3$, $\dots$, $8$, $1$, $2$, $3$, $\dots$, $8$, $\dots$ in order. Line segments may only be drawn to connect points labelled with the same number. What the largest number of non-intersecting line segments one can draw? (Two segments sharing the same endpoint are considered to be intersecting).

$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.

The set $S$ consists of five integers. If pairs of distinct elements of $S$ are added, the following ten sums are obtained: 1967,1972,1973,1974,1975,1980,1983,1984,1989,1991. What are the elements of $S$?

We are given an angle $0^o < \phi \le 180^o$ and a circular disc. An ant begins its journey from an interior point of the disc, travelling in a straight line in a certain direction. When it reaches the edge of the disc, it does the following: it turns clockwise by the angle $\phi $, and if its new direction does not point towards the interior of the disc, it turns by the angle $\phi $ again, and repeats this until it faces the interior. Then it continues its journey in this new direction and turns as before every time when it reaches the edge. For what values of $\phi $ is it true that for any starting point and initial direction the ant eventually returns to its starting position?

Cut the $10$ cm $\times 20$ cm rectangle into two pieces with one straight cut so that they can fit inside the $19.5$ cm diameter circle without intersecting.

Let $ABCD$ be a convex quadrilateral, such that $ A$, $ B$, $C$, and $D$ lie on a circle, with $\angle DAB < \angle ABC$. Let $I$ be the intersection of the bisector of $\angle ABC$ with the bisector of $\angle BAD$. Let $\ell$ be the parallel line to $CD$ passing through point $I$. Suppose $\ell$ cuts segments $DA$ and $BC$ at $ L$ and $J$, respectively. Prove that $AL + JB = LJ$.

Given circle $\Omega_1$ and $\Omega_2$ interesting at $P$ and $Q$. $X$ and $Y$ on line $PQ$ such that $X, P, Q, Y$ in that order. Point $A$ and $B$ on $\Omega_1$ and $\Omega_2$ respectively such that the intersections of $\Omega_1$ with $AX$ and $AY$, intersections of $\Omega_2$ with $BX$ and $BY$ are all in one line. $l$. Prove that $AB, l$ and perpendicular bisector of $PQ$ are concurrent.

Alice and Bob are playing the following game. Each turn Alice suggests an integer and Bob writes down either that number or the sum of that number with all previously written numbers. Is it always possible for Alice to ensure that at some moment among the written numbers there are [list=a] [*]at least a hundred copies of number 5? [*]at least a hundred copies of number 10? [/list] [i]Andrey Arzhantsev[/i]

John takes seven positive integers $a_1,a_2,...,a_7$ and writes the numbers $a_i a_j$, $a_i+a_j$ and $|a_i -a_j |$ for all $i \ne j$ on the blackboard. Find the greatest possible number of distinct odd integers on the blackboard.

In quadrilateral $ABCD$, $BC=8$, $CD=12$, $AD=10$, and $m\angle A= m\angle B = 60^\circ$. Given that $AB=p + \sqrt{q}$, where $p$ and $q$ are positive integers, find $p+q$.

Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.

The product of 22 integers is 1. Show that their sum can not be 0.

For all $x,y,z$ positive real numbers, find the all $c$ positive real numbers that providing $$\frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2$$

Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied: $ (1)$ $ n$ is not a perfect square; $ (2)$ $ a^{3}$ divides $ n^{2}$.

A number $x$ is "Tlahuica" if there exist prime numbers $p_1,\ p_2,\ \dots,\ p_k$ such that \[x=\frac{1}{p_1}+\frac{1}{p_2}+\dots+\frac{1}{p_k}.\] Find the largest Tlahuica number $x$ such that $0<x<1$ and there exists a positive integer $m\leq 2022$ such that $mx$ is an integer.

A triangle has side lengths $a, b, c$. Prove that $$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$

Suppose that you are given the foot of the altitude from vertex $A$ of a scalene triangle $ABC$, the midpoint of the arc with endpoints $B$ and $C$, not containing $A$ of the circumscribed circle of $ABC$, and also a third point $P$. Construct the triangle from these three points if $P$ is the a) orthocenter b) centroid c) incenter of the triangle.

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

A sum of $335$ pairwise distinct positive integers equals $100000$. a) What is the least number of uneven integers in that sum? b) What is the greatest number of uneven integers in that sum?

Find all positive integers $n$ for which there exist real numbers $x_1, x_2,. . . , x_n$ satisfying all of the following conditions: (i) $-1 <x_i <1,$ for all $1\leq i \leq n.$ (ii) $ x_1 + x_2 + ... + x_n = 0.$ (iii) $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$

A number is called [i]6-composite[/i] if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is [i]composite[/i] if it has a factor not equal to 1 or itself. In particular, 1 is not composite.) [i]Ray Li.[/i]

Let $a,b,c$ be real numbers $a + b +c = 0$. Show that [list=a] [*] $\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^3 + b^3 + c^3}{3} = \frac{a^5 + b^5 + c^5}{5}$. [*] $\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^5 + b^5 + c^5}{5} = \frac{a^7 + b^7 + c^7}{7}$. [/list] [I]Folklore[/I]

Starting with a sequence of $n 1's$, you can insert plus signs to get various sums. For example, when $n = 10$, you can get the sum $1 + 1 + 1 + 11 + 11 + 111 = 136$, and the sum $1 + 1 + 11 + 111 + 111 = 235$. Find the number of values of $n$ so that the sum of $1111$ is possible.