Let $ABCD$ be a convex trapezoid such that $\angle ABC = \angle BCD = 90^o$, $AB = 3$, $BC = 6$, and $CD = 12$. Among all points $X$ inside the trapezoid satisfying $\angle XBC = \angle XDA$, compute the minimum possible value of $CX$.

Three distinct integers $a_1$, $a_2$, $a_3$ between $1$ and $21$, inclusive, are selected uniformly at random. The probability that the greatest common factor of $a_i-a_j$ and $21$ is $7$ for some positive integers $i $ and $j$, where $1 \le i \ne j \le3 $, can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\frac{a}{a+b^2}+\frac{b}{b+c^2}+\frac{c}{c+a^2} \le \frac{1}{4} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)$$

Let $ C_1,C_2 $ be two distinct concentric circles, and $ BA $ be a diameter of $ C_1. $ Choose the points $ M,N $ on $ C_1,C_2, $ respectively, but not on the line $ BA. $ [b]a)[/b] Show that there are unique points $ P,Q $ on $ MA,MB, $ respectively, so that $ N $ is the middle of $ PQ. $ [b]b)[/b] Prove that the value $ AP^2+BQ^2 $ does not depend on $ M,N. $ [i]Mihai Piticari, Mihail Bălună[/i]

[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps? [img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img] [b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side. [b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses . (a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$? (b) Is it possible to do it if $k=4$? [b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$. [b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$. [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that $$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$

Let $A,B,C$ be nonempty sets in $\mathbb R^n$. Suppose that $A$ is bounded, $C$ is closed and convex, and $A+B\subseteq A+C$. Prove that $B\subseteq C$. Recall that $E+F=\{e+f:e\in E,f\in F\}$ and $D\subseteq\mathbb R^n$ is convex iff $tx+(1-t)y\in D$ for all $x,y\in D$ and any $t\in[0,1]$.

Given a positive real number $t$ , determine the sets $A$ of real numbers containing $t$ , for which there exists a set $B$ of real numbers depending on $A$ , $|B| \geq 4$ , such that the elements of the set $AB =\{ ab \mid a\in A , b \in B \}$ form a finite arithmetic progression .

There are $2022$ black balls numbered $1$ to $2022$ and $2022$ white balls numbered $1$ to $2022$ as well. There are also $1011$ black boxes and white boxes each. In each box we put two balls that are the same color as the the box. Prove that no matter how the balls are distributed, we can always pick one ball from each box such that the $2022$ balls we chose have all the numbers from $1$ to $2022$.

Isabella the geologist discovers a diamond deep underground via an X-ray machine. The diamond has the shape of a convex cyclic pentagon $PABCD$ with $AD|| BC$. Soon after the discovery, her X-ray breaks, and she only recovers partial information about its dimensions. She knows that $AD = 70, BC = 55, PA : PD = 3 : 4$, and $PB : PC = 5 : 6$. Compute $PB$.

Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and \[f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.\] Prove that if $b + c < 1$, there is a real number $k$ such that \[f(n) \leq kn \qquad \text{ for all } n \qquad (1)\] while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$

a) Let $UVW$ , $U'V'W'$ be two triangles such that $ VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'.$ Prove that the angles $\angle VWU , \angle V'W'U'$ are equal or supplementary. b) $ABC$ is a triangle where $\angle A$ is [b]obtuse[/b]. take a point $P$ inside the triangle , and extend $AP,BP,CP$ to meet the sides $BC,CA,AB$ in $K,L,M$ respectively. Suppose that $PL = PM .$ 1) If $AP$ bisects $\angle A$ , then prove that $AB = AC$ . 2) Find the angles of the triangle $ABC$ if you know that $AK,BL,CM$ are angle bisectors of the triangle $ABC$ and that $2AK = BL$.

The median $AD$ of triangle $ABC$ intersects its inscribed circle (with center $O$) at the points $X$ and $Y$. Find the angle $XOY$ if $AC = AB + AD$. (A Fedotov)

How many distinct positive integers can be formed by choosing their digits from the string $04072019$?

A steel cube has edges of length $3$ cm, and a cone has a diameter of $8$ cm and a height of $24$ cm. The cube is placed in the cone so that one of its interior diagonals coincides with the axis of the cone. What is the distance, in cm, between the vertex of the cone and the closest vertex of the cube? [NEEDS DIAGRAM] $ \mathrm{(A) \ } \frac{12\sqrt6-3\sqrt3}{4} \qquad \mathrm{(B) \ } \frac{9\sqrt6-3\sqrt3}{2} \qquad \mathrm {(C) \ } 5\sqrt3 \qquad \mathrm{(D) \ } 6\sqrt6 - \sqrt3 \qquad \mathrm{(E) \ } 6\sqrt6$

Let $f,g:[0,1]\to(0,+\infty)$ be two continuous functions such that $f$ and $\frac gf$ are increasing. Prove that $$\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.$$