The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$. Show that the line $CX$ bisects the angle $ACN$.

Let $I$ be the incentre of triangle $ABC$. A circle containing the points $B$ and $C$ meets the segments $BI$ and $CI$ at points $P$ and $Q$ respectively. It is known that $BP\cdot CQ=PI\cdot QI$. Prove that the circumcircle of the triangle $PQI$ is tangent to the circumcircle of $ABC$. [i]Proposed by S. Berlov[/i]

A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.

Let $H$ be the orthocenter of the acute-angled triangle $ABC$. Inside the segment $BC$ arbitrary point $D$ is selected. Let $P$ be such that $ADPH$ is a parallelogram. Prove that $\angle BCP< \angle BHP$.

The bisector of $\angle{BAD}$ in the parallellogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the centre of the circle passing through the points $C,\ K$ and $L$ lies on the circle passing through the points $B,\ C$ and $D$.

On the side $CD$ of parallelogram $ABCD$ a point $E$ is chosen. The perpendicular from $C$ to $BE$ and the perpendicular from $D$ to $AE$ intersect at $P$. Point $M$ is the midpoint of $PE$. Prove that the perpendicular from $M$ to $CD$ passes through the center of parallelogram $ABCD$. [i]Matsvei Zorka[/i]

Two parallel sides of a quadrilateral have integer lengths. Prove that this quadrilateral can be cut into congruent triangles. (A Shapovalov)

A given convex pentagon $ ABCDE$ has the property that the area of each of five triangles $ ABC, BCD, CDE, DEA$, and $ EAB$ is unity [i](equal to 1)[/i]. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.

A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom 52 = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$. [i]Proposed by Yang Liu[/i]

Given integer $n > 1$ and real number $t \geq 1$. $P$ is a parallelogram with four vertices $(0, 0), (0, t), (tF_{2n+1}, tF_{2n}), (tF_{2n+1}, tF_{2n} + t)$. Here, ${F_n}$ is the $n$-th term of Fibonacci sequence defined by $F_0 = 0, F_1 = 1$ and $F_{m+1} = F_m + F_{m-1}$. Let $L$ be the number of integral points (whose coordinates are integers) interior to $P$, and $M$ be the area of $P$, which is $t^2F_{2n+1}.$ [b][i]i)[/i][/b] Prove that for any integral point $(a, b)$, there exists a unique pair of integers $(j, k)$ such that$ j(F_{n+1}, F_n) + k(F_n, F_{n-1}) = (a, b)$, that is,$ jF_{n+1} + kF_n = a$ and $jF_n + kF_{n-1} = b.$ [i][b]ii)[/b][/i] Using [i][b]i)[/b][/i] or not, prove that $|\sqrt L-\sqrt M| \leq \sqrt 2.$

Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.

Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

In a triangle $ABC$ with $AB=AC$, $\omega$ is the circumcircle and $O$ its center. Let $D$ be a point on the extension of $BA$ beyond $A$. The circumcircle $\omega_{1}$ of triangle $OAD$ intersects the line $AC$ and the circle $\omega$ again at points $E$ and $G$, respectively. Point $H$ is such that $DAEH$ is a parallelogram. Line $EH$ meets circle $\omega_{1}$ again at point $J$. The line through $G$ perpendicular to $GB$ meets $\omega_{1}$ again at point $N$ and the line through $G$ perpendicular to $GJ$ meets $\omega$ again at point $L$. Prove that the points $L, N, H, G$ lie on a circle.

Let $ABCD$ be a parallelogram of center $O$. Prove that for any point $M\in (AB)$, there exist unique points $N\in (OC)$ and $P\in (OD)$ such that $O$ is the center of mass of $\triangle MNP$.

A [i]squiggle[/i] is composed of six equilateral triangles with side length $1$ as shown in the figure below. Determine all possible integers $n$ such that an equilateral triangle with side length $n$ can be fully covered with [i]squiggle[/i]s (rotations and reflections of [i]squiggle[/i]s are allowed, overlappings are not). [asy] import graph; size(100); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; draw((0,0)--(0.5,1),linewidth(2pt)); draw((0.5,1)--(1,0),linewidth(2pt)); draw((0,0)--(3,0),linewidth(2pt)); draw((1.5,1)--(2,0),linewidth(2pt)); draw((2,0)--(2.5,1),linewidth(2pt)); draw((0.5,1)--(2.5,1),linewidth(2pt)); draw((1,0)--(2,2),linewidth(2pt)); draw((2,2)--(3,0),linewidth(2pt)); dot((0,0),ds); dot((1,0),ds); dot((0.5,1),ds); dot((2,0),ds); dot((1.5,1),ds); dot((3,0),ds); dot((2.5,1),ds); dot((2,2),ds); clip((-4.28,-10.96)--(-4.28,6.28)--(16.2,6.28)--(16.2,-10.96)--cycle);[/asy]