Let $ABC$ be a triangle and $D$ be a point on segment $AC$. The circumscribed circle of the triangle $BDC$ cuts $AB$ again at $E$ and the circumference circle of the triangle $ABD$ cuts $BC$ again at $F$. Prove that $AE = CF$ if and only if $BD$ is the interior bisector of $\angle ABC$.

A rhombus is formed by two radii and two chords of a circle whose radius is $ 16$ feet. The area of the rhombus in square feet is: $ \textbf{(A)}\ 128 \qquad\textbf{(B)}\ 128\sqrt {3} \qquad\textbf{(C)}\ 256 \qquad\textbf{(D)}\ 512 \qquad\textbf{(E)}\ 512\sqrt {3}$

Let $n \geq 3$ be an integer. Consider the set $A=\{1,2,3,\ldots,n\}$, in each move, we replace the numbers $i, j$ by the numbers $i+j$ and $|i-j|$. After doing such moves all of the numbers are equal to $k$. Find all possible values for $k$.

In a triangle $ABC$, it is drawn a circumference with center in the incenter $I$ and that meet twice each of the sides of the triangle: the segment $BC$ on $D$ and $P$ (where $D$ is nearer two $B$); the segment $CA$ on $E$ and $Q$ (where $E$ is nearer to $C$); and the segment $AB$ on $F$ and $R$ ( where $F$ is nearer to $A$). Let $S$ be the point of intersection of the diagonals of the quadrilateral $EQFR$. Let $T$ be the point of intersection of the diagonals of the quadrilateral $FRDP$. Let $U$ be the point of intersection of the diagonals of the quadrilateral $DPEQ$. Show that the circumcircle to the triangle $\triangle{FRT}$, $\triangle{DPU}$ and $\triangle{EQS}$ have a unique point in common.

The incenter of a triangle is the midpoint of the line segment of length $4$ joining the centroid and the orthocenter of the triangle. Determine the maximum possible area of the triangle.

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

Let consider the following function set $$F=\{f\ |\ f:\{1,\ 2,\ \cdots,\ n\}\to \{1,\ 2,\ \cdots,\ n\} \}$$ [list=1] [*] Find $|F|$ [*] For $n=2k$ prove that $|F|< e{(4k)}^{k}$ [*] Find $n$, if $|F|=540$ and $n=2k$ [/list]

Let $ABCD$ be a convex quadrilateral, and $M$, $N$, and $P$ be the midpoints of diagonals $AC$ and $BD$, and side $AD$, respectively. Also, suppose that $\angle{ABC} + \angle{DCB} = 90$ and that $AB = 6$, $CD = 8$. Calculate the perimeter of triangle $MNP$.

Prove that for any integers $ n> m> 0 $ the number $ 2 ^n-1 $ has a prime divisor not dividing $ 2 ^m-1 $.

Mark the Martian and Bark the Bartian live on planet Blok, in the year $2019$. Mark and Bark decide to play a game on a $10 \times 10$ grid of cells. First, Mark randomly generates a subset $S$ of $\{1, 2, \dots, 2019\}$ with $|S|=100$. Then, Bark writes each of the $100$ integers in a different cell of the $10 \times 10$ grid. Afterwards, Bark constructs a solid out of this grid in the following way: for each grid cell, if the number written on it is $n$, then she stacks $n$ $1 \times 1 \times 1$ blocks on top of one other in that cell. Let $B$ be the largest possible surface area of the resulting solid, including the bottom of the solid, over all possible ways Bark could have inserted the $100$ integers into the grid of cells. Find the expected value of $B$ over all possible sets $S$ Mark could have generated. [i]Proposed by Yang Liu[/i]

The vertices of a $ 3 \minus{} 4 \minus{} 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles? [asy]unitsize(5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair B=(0,0), C=(5,0); pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0]; draw(A--B--C--cycle); draw(Circle(C,3)); draw(Circle(A,1)); draw(Circle(B,2)); label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("3",midpoint(B--A),NW); label("4",midpoint(A--C),NE); label("5",midpoint(B--C),S);[/asy]$ \textbf{(A) } 12\pi\qquad \textbf{(B) } \frac {25\pi}{2}\qquad \textbf{(C) } 13\pi\qquad \textbf{(D) } \frac {27\pi}{2}\qquad \textbf{(E) } 14\pi$

In a triangle $ABC$, point $P$ is the incenter and $A'$, $B'$, $C'$ its orthogonal projections on $BC$, $CA$, $AB$, respectively. Show that $\angle B'A'C'$ is acute.

Prove that there exists no innite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$: $p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.

Show that, if $\sqrt{2}$ is written in decimal notation, there is at least one nonzero digit at the interval of 1,000,000-th and 3,000,000-th digits.

Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD\equal{}CE\equal{}BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Can a figure have a greater than $1$ and finite number of centers of symmetry?

In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle DFP$.

Let $S$ a finite subset of $\mathbb{N}$. For every positive integer $i$, let $A_{i}$ the number of partitions of $i$ with all parts in $ \mathbb{N}-S$. Prove that there exists $M\in \mathbb{N}$ such that $A_{i+1}>A_{i}$ for all $i>M$. ($ \mathbb{N}$ is the set of positive integers)

Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that: $$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$

Determine the number of functions $f : Z^+ \to Z^+$ so that for all positive integers $x$ we have $f(f(x)) = f(x + 1)$, and $\max (f(2), . . . , f(14)) \le f(1) - 2 = 12$.

Triangle $\vartriangle ABC$ is isosceles with $AC = AB$, $BC = 1$, and $\angle BAC = 36^o$. Let $\omega$ be a circle with center B and radius $r_{\omega}= \frac{P_{ABC}}{4}$, where $P_{ABC}$ denotes the perimeter of $\vartriangle ABC$. Let $\omega$ intersect line $AB$ at $P$ and line $BC$ at $Q$. Let $I_B$ be the center of the excircle with of $\vartriangle ABC$ with respect to point $B$, and let $BI_B$ intersect $P Q$ at $S$. We draw a tangent line from $S$ to $\odot I_B$ that intersects $\odot I_B$ at point $T$. Compute the length of ST.

Calculate the numerical value of $1 \times 1 + 2 \times 2 - 2$. $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .\] [i]Proposed by Azerbaijan[/i] [hide=Second Suggested Version]Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .\][/hide]

For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?