Let $ABC$ be a triangle such that $|AB|=13 , |BC|=12$ and $|CA|=5$. Let the angle bisectors of $A$ and $B$ intersect at $I$ and meet the opposing sides at $D$ and $E$, respectively. The line passing through $I$ and the midpoint of $[DE]$ meets $[AB]$ at $F$. What is $|AF|$? $ \textbf{(A)}\ \dfrac{3}{2} \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \dfrac{5}{2} \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \dfrac{7}{2} $

Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest possible value of $c$? $\textbf{(A) }-2\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }4\qquad \textbf{(E) }6\qquad$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$. [i]George Xing.[/i]

There are two imposters and seven crewmates on Polus. How many ways are there for the nine people to split into three groups of three, such that each group has at least two crewmates? Assume that the two imposters and seven crewmates are all distinguishable from each other, but that the three groups are not distinguishable from each other.

With expressions containing the symbol $*$, the following transformations can be performed: 1) rewrite the expression in the form $x * (y * z) as ((1 * x) * y) * z$; 2) rewrite the expression in the form $x * 1$ as $x$. Conversions can only be performed with an integer expression, but not with its parts. For example, $(1 *1) * (1 *1)$ can be rewritten according to the first rule as $((1 * (1 * 1)) * 1) * 1$ (taking $x = 1 * 1$, $y = 1$ and $z = 1$), but not as $1 * (1 * 1)$ or $(1* 1) * 1$ (in the last two cases, the second rule would be applied separately to the left or right side $1 * 1$). Find all positive integers $n$ for which the expression $\underbrace{1 * (1 * (1 * (...* (1 * 1)...))}_{n units}$ it is possible to lead to a form in which there is not a single asterisk. Note. The expressions $(x * y) * $z and $x * (y * z)$ are considered different, also, in the general case, the expressions $x * y$ and $y * x$ are different.

Evaluate the sum \[ \left\lfloor \frac{2^0}{3} \right\rfloor + \left\lfloor \frac{2^1}{3} \right\rfloor + \left\lfloor \frac{2^2}{3} \right\rfloor + \cdots + \left\lfloor \frac{2^{1000}}{3} \right\rfloor. \]

In every unit square of a$ n \times n$ table ($n \ge 11$) a real number is written, such that the sum of the numbers in any $10 \times 10$ square is positive and the sum of the numbers in any $11\times 11$ square is negative. Determine all possible values for $n$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the following condition for any real numbers $x{}$ and $y$ $$f(x)+f(x+y) \leq f(xy)+f(y).$$

A positive integer $n$ is [i]brgorable[/i] if it is possible to arrange the numbers $1, 1, 2, 2, ..., n, n$ such that between any two $k$'s there are exactly $k$ numbers (for example, $n=2$ is not brgorable, but $n = 3$ is as demonstrated by $3, 1, 2, 1, 3, 2$). How many brgorable numbers are less than 2019?

Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.

$x_1,x_2,\cdots,x_n$ are positive real numbers. Prove that $$\frac{x_1^2}{x_2}+\frac{x_2^2}{x_3}+\cdots+\frac{x_n^2}{x_1}\geq x_1+x_2+\cdots x_n.$$

There are $N$ cities in a country, some of which are connected by two-way flights. No city is directly connected with every other city. For each pair $(A, B)$ of cities there is exactly one route using at most two flights between them. Prove that $N - 1$ is a square of an integer.

Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1\equal{}1$, $ a_2\equal{}\dfrac{4}{3}$, and \[ a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2}\] and \[ 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.\] [i]Fajar Yuliawan, Bandung[/i]

If $a$ is any number, $\lfloor a \rfloor$ is $a$ rounded down to the nearest integer. For example, $\lfloor \pi \rfloor =$ $3$. Show that the sequence $\lfloor \frac{2^{1}}{17} \rfloor$, $\lfloor \frac{2^{2}}{17} \rfloor$, $\lfloor \frac{2^{3}}{17} \rfloor$, $\dots$ contains infinitely many odd numbers.

If $ a\otimes b =\frac{a+b}{a-b} $ , then $ (6\otimes 4)\otimes 3 = $ = $ \text{(A)}\ 4\qquad\text{(B)}\ 13\qquad\text{(C)}\ 15\qquad\text{(D)}\ 30\qquad\text{(E)}\ 72 $

Do there exist two positive powers of $5$ such that the number obtained by writing one after the other is also a power of $5$?

Sammy and Tammy run laps around a circular track that has a radius of $1$ kilometer. They begin and end at the same point and at the same time. Sammy runs $3$ laps clockwise while Tammy runs $4$ laps counterclockwise. How many times during their run is the straight-line distance between Sammy and Tammy exactly $1$ kilometer? $\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21$

What is the largest $n$ such that a square cannot be partitioned into $n$ smaller, non-overlapping squares?

In an isosceles triangle $ABC, I$ is the center of the inscribed circle, $M_1$ is the midpoint of the side $BC, K_2, K_3$ are the points of contact of the inscribed circle of the triangle with segments $AC$ and $AB$, respectively. The point $P$ lies on the circumcircle of the triangle $BCI$, and the angle $M_1PI$ is right. Prove that the lines $BC, PI, K_2K_3$ intersect at one point. (Mikhail Plotnikov)

Let $k$ be a fixed circle in a given plane and a point $C$ out of the plane. Let $A$ be a random point from $k$ and $B$ be its diametrically opposite one in $k$. Find the geometric place of the center of the circumscribed circle of $ABC$.

Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circumcircle of triangle $AHC$ meets segments $AB$ and $BC$ at points $P$ and $Q$. Lines $PQ$ and $AC$ meet at point $R$. A point $K$ lies on the line $PH$ in such a way that $\angle KAC = 90^{\circ}$. Prove that $KR$ is perpendicular to one of the medians of triangle $ABC$.

An ancient noble family has $n$ members, each holding a different number of posts . As every year in December, they gather at a very specific place for a Council of War to be held, where also k, from the point of view of the high nobility, unimportant spammers speak up, which, due to their irrelevance, should and cannot be further differentiated. The Council is held as follows: those present speak one after the other, each one carefully put forward his request once. In addition, for reasons of respect, a nobleman never speaks right after a nobleman who holds more posts, while the common people disregarde such rules. Find the number of possible sequences of the Council of war.

Given a circumscribed trapezium $ ABCD$ with circumcircle $ \omega$ and 2 parallel sides $ AD,BC$ ($ BC<AD$). Tangent line of circle $ \omega$ at the point $ C$ meets with the line $ AD$ at point $ P$. $ PE$ is another tangent line of circle $ \omega$ and $ E\in\omega$. The line $ BP$ meets circle $ \omega$ at point $ K$. The line passing through the point $ C$ paralel to $ AB$ intersects with $ AE$ and $ AK$ at points $ N$ and $ M$ respectively. Prove that $ M$ is midpoint of segment $ CN$.

Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.

A circle is centered at point $O$ in the plane. Distinct pairs of points $A, B$ and $C, D$ are diametrically opposite on this circle. Point $P$ is chosen on line segment $AD$ such that line $BP$ hits the circle again at $M$ and line $AC$ at $X$ such that $M$ is the midpoint of $PX$. Now, the point $Y \neq X$ is taken for $BX = BY, CD \parallel XY$. IF $\angle PYB = 10^{\circ}$, find the measure of $\angle XCM$. Proposed by Albert Wang (awang11)