On the side $DE$ and on the diagonal $BE$ of the regular pentagon $ABCDE$ we consider the squares $DEFG$ and $BEHI$. [list=a] [*] Prove that $A,I,$ and $G$ are collinear. [*] Prove that on this line lies also the centre $O$ of the square $BDJK$. [/list]

Let $ABC$ be a triangle and $I$ be its incenter. Let $D$ be the intersection of the exterior bisectors of $\angle BAC$ and $\angle BIC$, $E$ be the intersection of the exterior bisectors of $\angle ABC$ and $\angle AIC$, and $F$ be the intersection of the exterior bisectors of $\angle ACB$ and $\angle AIB$. Prove that $D$, $E$, $F$ are collinear

A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$, then it is also tangent to the edge $BD.$

Find an integral formula for the solution of the differential equation $$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$ for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$

Four orange lights are located at the points $(2,0)$, $(4,0)$, $(6,0)$ and $(8,0)$ in the $xy$-plane. Four yellow lights are located at the points $(1,0)$, $(3,0)$, $(5,0)$, $(7,0)$. Sparky chooses one or more of the lights to turn on. In how many ways can he do this such that the collection of illuminated lights is symmetric around some line parallel to the $y$-axis? [i]Proposed by Evan Chen[/i]

Triangle $ABC$ has an incenter $I$. Let points $X$, $Y$ be located on the line segments $AB$, $AC$ respectively, so that $BX \cdot AB = IB^2$ and $CY \cdot AC = IC^2$. Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$.

On a trip from the United States to Canada, Isabella took $ d$ U.S. dollars. At the border she exchanged them all, receiving $ 10$ Canadian dollars for every $ 7$ U.S. dollars. After spending $ 60$ Canadian dollars, she had $ d$ Canadian dollars left. What is the sum of the digits of $ d$? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

Find all positive integers $n$ such that $4^{n}+2007$ is a perfect square.

Alphonse and Beryl are playing a game. The game starts with two rectangles with integer side lengths. The players alternate turns, with Alphonse going first. On their turn, a player chooses one rectangle, and makes a cut parallel to a side, cutting the rectangle into two pieces, each of which has integer side lengths. The player then discards one of the three rectangles (either the one they did not cut, or one of the two pieces they cut) leaving two rectangles for the other player. A player loses if they cannot cut a rectangle. Determine who wins each of the following games: (a) The starting rectangles are $1 \times 2020$ and $2 \times 4040$. (b) The starting rectangles are $100 \times 100$ and $100 \times 500$.

Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.

Find the smallest natural number, which divides $2^{n}+15$ for some natural number $n$ and can be expressed in the form $3x^2-4xy+3y^2$ for some integers $x$ and $y$.

Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$.

Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$. Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$. Now let $L$ be the set of lines containing exactly one point of $S(T)$. Call a line $\ell_0$ passing through $O$ [i]bad[/i] if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$. (a) Prove that $L$ is nonempty. (b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$. [i]Proposed by David Yang[/i]

The circle $\omega_0$ touches the line at point A. Let $R$ be a given positive number. We consider various circles $\omega$ of radius $R$ that touch a line $\ell$ and have two different points in common with the circle $\omega_0$. Let $D$ be the touchpoint of the circle $\omega_0$ with the line $\ell$, and the points of intersection of the circles $\omega$ and $\omega_0$ are denoted by $B$ and $C$ (Assume that the distance from point $B$ to the line $\ell$ is greater than the distance from point $C$ to this line). Find the locus of the centers of the circumscribed circles of all such triangles $ABD$.

Joao had blue, white and red pearls and with them he made a necklace with $20$ pearls that has as many blue as white pearls. João noticed that, regardless of how he cut the necklace into two parts, both with an even number of pearls, one of the parts would always have more blue pearls than white ones. How many red pearls are in Joao's necklace?

$\triangle ABC$ is a triangle such that $\angle A = 60^{\circ}$. The incenter of $\triangle ABC$ is $I$. $AI$ intersects with $BC$ at $D$, $BI$ intersects with $CA$ at $E$, and $CI$ intersects with $AB$ at $F$, respectively. Also, the circumcircle of $\triangle DEF$ is $\omega$. The tangential line of $\omega$ at $E$ and $F$ intersects at $T$. Show that $\angle BTC \ge 60^{\circ}$

Represent the number $\sqrt[3]{1342\sqrt{167}+2005}$ in the form where it contains only addition, subtraction, multiplication, division and square roots.

Let $ n$ denote the smallest positive integer that is divisible by both $ 4$ and $ 9$, and whose base-$ 10$ representation consists of only $ 4$'s and $ 9$'s, with at least one of each. What are the last four digits of $ n$? $ \textbf{(A)}\ 4444\qquad \textbf{(B)}\ 4494\qquad \textbf{(C)}\ 4944\qquad \textbf{(D)}\ 9444\qquad \textbf{(E)}\ 9944$

Let $ABCD$ be a parallelogram and let $K$ and $L$ be points on the segments $BC$ and $CD$, respectively, such that $BK\cdot AD=DL\cdot AB$. Let the lines $DK$ and $BL$ intersect at $P$. Show that $\measuredangle DAP=\measuredangle BAC$.

A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$ $\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601$

An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? $\textbf{(A) }\frac{14}{75}\qquad\textbf{(B) }\frac{56}{225}\qquad\textbf{(C) }\frac{107}{400}\qquad\textbf{(D) }\frac{7}{25}\qquad\textbf{(E) }\frac{9}{25}$

Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity \[ \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b}\] must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j<k$, $a_j\neq a_k$. (So $1\in S$; $2\in S$ if and only if $a_1\neq a_2$; $3\in S$ if and only if $a_1\neq a_3$ and $a_2\neq a_3$; and so on.) In simplest form, let $\dfrac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.

Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.

The one hundred U.S. Senators are standing in a line in alphabetical order. Each senator either always tells the truth or always lies. The $i$th person in line says: "Of the $101-i$ people who are not ahead of me in line (including myself), more than half of them are truth-tellers.'' How many possibilities are there for the set of truth-tellers on the U.S. Senate? [i]Proposed by James Lin[/i]