Let $ ABCD $ be a convex quadrilateral. Let angle bisectors of $ \angle B $ and $ \angle C $ intersect at $ E $. Let $ AB $ intersect $ CD $ at $ F $. Prove that if $ AB+CD=BC $, then $A,D,E,F$ is cyclic.

If the altitude of an equilateral triangle is $ \sqrt {6}$, then the area is: $ \textbf{(A)}\ 2\sqrt {2} \qquad\textbf{(B)}\ 2\sqrt {3} \qquad\textbf{(C)}\ 3\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {2} \qquad\textbf{(E)}\ 12$

In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. If the radius if the incircle is $4$ units and if $BD, CE , AF$ are consecutive integers, find the sides of the triangle $ABC$.

Let $H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \}$ and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A\subseteq \{1,2,\dots, n\}$ satisfies $|A| \ge C\sqrt{n}$, then there exist $a,b\in A$ such that $a-b\in H$. (Here $\mathbb Z_{>0}$ is the set of positive integers, and $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)

Let $p = 2017$ be a prime. Given a positive integer $n$, let $T$ be the set of all $n\times n$ matrices with entries in $\mathbb{Z}/p\mathbb{Z}$. A function $f:T\rightarrow \mathbb{Z}/p\mathbb{Z}$ is called an $n$-[i]determinant[/i] if for every pair $1\le i, j\le n$ with $i\not= j$, \[f(A) = f(A'),\] where $A'$ is the matrix obtained by adding the $j$th row to the $i$th row. Let $a_n$ be the number of $n$-determinants. Over all $n\ge 1$, how many distinct remainders of $a_n$ are possible when divided by $\dfrac{(p^p - 1)(p^{p - 1} - 1)}{p - 1}$? [i]Proposed by Ashwin Sah[/i]

The locus of the midpoint of a line segment that is drawn from a given external point $ P$ to a given circle with center $ O$ and radius $ r$, is: $ \textbf{(A)}\ \text{a straight line perpendicular to }\overline{PO} \\ \textbf{(B)}\ \text{a straight line parallel to } \overline{PO} \\ \textbf{(C)}\ \text{a circle with center }P\text{ and radius }r \\ \textbf{(D)}\ \text{a circle with center at the midpoint of }\overline{PO}\text{ and radius }2r \\ \textbf{(E)}\ \text{a circle with center at the midpoint }\overline{PO}\text{ and radius }\frac{1}{2}r$

If $x, y$ and $2x + \frac{y}{2}$ are not zero, then \[ \left( 2x + \frac{y}{2} \right)\left[(2x)^{-1} + \left( \frac{y}{2} \right)^{-1} \right] \] equals $\textbf{(A) }1\qquad\textbf{(B) }xy^{-1}\qquad\textbf{(C) }x^{-1}y\qquad\textbf{(D) }(xy)^{-1}\qquad \textbf{(E) }\text{none of these}$

A diagonal of a convex hexagon is called [i]long[/i] if it decomposes the hexagon into two quadrangles. Each pair of [i]long[/i] diagonals decomposes the hexagon into two triangles and two quadrangles. Given is a hexagon with the property, that for each decomposition by two [i]long[/i] diagonals the resulting triangles are both isosceles with the side of the hexagon as base. Show that the hexagon has a circumcircle.

Let $ a$ and $ b$ be the roots of the equation $ x^2 \minus{} mx \plus{} 2 \equal{} 0$. Suppose that $ a \plus{} (1/b)$ and $ b \plus{} (1/a)$ are the roots of the equation $ x^2 \minus{} px \plus{} q \equal{} 0$. What is $ q$? $ \textbf{(A) } \frac 52 \qquad \textbf{(B) } \frac 72 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } \frac 92 \qquad \textbf{(E) } 8$

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn$?$ $\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$

Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$. Show that $|A|\cdot|B|\le|C|^2$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

Let $ ABCD $ be a convex quadrilateral, and $ P $ be a point that isn't found on any of the lines formed by the sides of the quadrilateral. Prove that the centers of mass of the triangles $ PAB, PBC, PCD $ and $ PDA, $ form a parallelogram, and calculate the legths of its sides in terms of its diagonals.

Let $ABC$ be an acute triangle with altitude $AD$ ($D \in BC$). The line through $C$ parallel to $AB$ meets the perpendicular bisector of $AD$ at $G$. Show that $AC = BC$ if and only if $\angle AGC = 90^{\circ}$.

Three operations $f,g$ and $h$ are defined on subsets of the natural numbers $\mathbb{N}$ as follows: $f(n)=10n$, if $n$ is a positive integer; $g(n)=10n+4$, if $n$ is a positive integer; $h(n)=\frac{n}{2}$, if $n$ is an [i]even[/i] positive integer. Prove that, starting from $4$, every natural number can be constructed by performing a finite number of operations $f$, $g$ and $h$ in some order. $[$For example: $35=h(f(h(g(h(h(4)))))).]$

Let $n\ge 2$ be a positive integer, and suppose $a_1,a_2,\cdots ,a_n$ are positive real numbers whose sum is $1$ and whose squares add up to $S$. Prove that if $b_i=\tfrac{a^2_i}{S} \;(i=1,\cdots ,n)$, then for every $r>0$, we have $$\sum_{i=1}^n \frac{a_i}{{(1-a_i)}^r}\le \sum_{i=1}^n \frac{b_i}{{(1-b_i)}^r}.$$

Find the largest positive integer $n <30$ such that $\frac{1}{2}(n^8 + 3n^4 -4)$ is not divisible by the square of any prime number.

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the foot of the altitude from $A$ to $BC$. The inscribed circles of triangles $ABD$ and $ACD$ are tangent to $AD$ at $P$ and $Q$, respectively, and are tangent to $BC$ at $X$ and $Y$ , respectively. Let $PX$ and $QY$ meet at $Z$. Determine the area of triangle $XY Z$.

Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\min\{ab,bc,ca\}$. Find the largest possible value of $m$.

In equilateral triangle $ABC$ with side length $2$, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_1$ and $B_2$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $C_B$ at $C_1$ and $C_2$, respectively. Find the perimeter of the triangle formed by lines $A_1A_2$, $B_1B_2$, $C_1C_2$.

Maker and Breaker are building a wall. Maker has a supply of green cubical building blocks, and Breaker has a supply of red ones, all of the same size. On the ground, a row of $m$ squares has been marked in chalk as place-holders. Maker and Breaker now take turns in placing a block either directly on one of these squares, or on top of another block already in place, in such a way that the height of each column never exceeds $n$. Maker places the first block. Maker bets that he can form a green row, i.e. all $m$ blocks at a certain height are green. Breaker bets that he can prevent Maker from achieving this. Determine all pairs $(m,n)$ of positive integers for which Maker can make sure he wins the bet.

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Point $P$ lies in the interior of rectangle $ABCD$ such that $AP + CP = 27$, $BP - DP = 17$, and $\angle DAP \cong \angle DCP$. Compute the area of rectangle $ABCD$. [i]Proposed by Aaron Lin[/i]

Let $P$ and $Q$ be fixed points in the Euclidean plane. Consider another point $O_0$. Define $O_{i+1}$ as the center of the unique circle passing through $O_i$, $P$ and $Q$. (Assume that $O_i,P,Q$ are never collinear.) How many possible positions of $O_0$ satisfy that $O_{2021}=O_{0}$? [i]Proposed by Fei Peng[/i]

If $f$ is a monic cubic polynomial with $f(0)=-64$, and all roots of $f$ are non-negative real numbers, what is the largest possible value of $f(-1)$? (A polynomial is monic if it has a leading coefficient of $1$.)

Suppose that $a,b,c$ are positive integers such that $a^b$ divides $b^c$, and $a^c$ divides $c^b$. Prove that $a^2$ divides $bc$.