In triangle $ABC$, the incircle touches $BC$ in $D$, $CA$ in $E$ and $AB$ in $F$. The bisector of $\angle BAC$ intersects $BC$ in $G$. The lines $BE$ and $CF$ intersect in $J$. The line through $J$ perpendicular to $EF$ intersects $BC$ in $K$. Prove that $\frac{GK}{DK}=\frac{AE}{CE}+\frac{AF}{BF}$

Triangles $OAB,OBC,OCD$ are isoceles triangles with $\angle OAB=\angle OBC=\angle OCD=\angle 90^o$. Find the area of the triangle $OAB$ if the area of the triangle $OCD$ is 12.

One of the midlines of a triangle is longer than one of its medians. Prove that the triangle has an obtuse angle.

Let $n>1$ be a natural number and $A_0A_1\ldots A_{2^n-2}$ be a regular polygon. Prove that \[\frac{1}{A_0A_1}=\frac{1}{A_0A_2}+\frac{1}{A_0A_4}+\frac{1}{A_0A_8}+\cdots+\frac{1}{A_0A_{2^{n-1}}}.\][i]Proposed by Le Hoang and Ngoc Thai (Vietnam)[/i]

Let $ABC$ be a triangle with $\angle B = 90^o$. Given that there exists a point $D$ on $AC$ such that $AD = DC$ and $BD = BC$, compute the value of the ratio $\frac{AB}{BC}$ .

A criminal is at point $X$, and three policemen at points $A, B$ and $C$ block him up, i.e. the point $X$ lies inside the triangle $ABC$. Each evening one of the policemen is replaced in the following way: a new policeman takes the position equidistant from three former policemen, after this one of the former policemen goes away so that three remaining policemen block up the criminal too. May the policemen after some time occupy again the points $A, B$ and $C$ (it is known that at any moment $X$ does not lie on a side of the triangle)? by V.Protasov

Find an ordered pair $(a,b)$ of real numbers for which $x^2+ax+b$ has a non-real root whose cube is $343$.

A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB,BC,CD,DA$ one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to $CD,DA,AB,BC,$ respectively. Prove that the centers of these four rectangles are vertices of a rectangle.

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 37$

Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.

6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.

In the figure below the triangles $BCD, CAE$ and $ABF$ are equilateral, and the triangle $ABC$ is right-angled with $\angle A = 90^o$. Prove that $|AD| = |EF|$. [img]https://1.bp.blogspot.com/-QMMhRdej1x8/XzP18QbsXOI/AAAAAAAAMUI/n53OsE8rwZcjB_zpKUXWXq6bg3o8GUfSwCLcBGAsYHQ/s0/2019%2Bmohr%2Bp5.png[/img]

Line segment $\overline{AE}$ of length $17$ bisects $\overline{DB}$ at a point $C$. If $\overline{AB} = 5$, $\overline{BC} = 6$ and $\angle BAC = 78^o$ degrees, calculate $\angle CDE$.

Let $ABC$ be an acute-angled triangle such that $AB+BC=4AC$. Let $D$ in $AC$ such that $BD$ is angle bisector of $\angle ABC$. In the segment $BD$, points $P$ and $Q$ are marked such that $BP=2DQ$. The perpendicular line to $BD$, passing by $Q$, cuts the segments $AB$ and $BC$ in $X$ and $Y$, respectively. Let $L$ be the parallel line to $AC$ passing by $P$. The point $B$ is in a different half-plane(with respect to the line $L$) of the points $X$ and $Y$. An ant starts a run in the point $X$, goes to a point in the line $AC$, after that goes to a point in the line $L$, returns to a point in the line $AC$ and finishes in the point $Y$. Prove that the least length of the ant's run is equal to $4XY$.

$ ABC$ is a triangle with $ \angle BAC \equal{} 10{}^\circ$, $ \angle ABC \equal{} 150{}^\circ$. Let $ X$ be a point on $ \left[AC\right]$ such that $ \left|AX\right| \equal{} \left|BC\right|$. Find $ \angle BXC$. $\textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 20^\circ \qquad\textbf{(C)}\ 25^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 35^\circ$

There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.

What is the smallest possible area of a rectangle that can completely contain the shape formed by joining six squares of side length $8$ cm as shown below? [asy] size(5cm,0); draw((0,2)--(0,3)); draw((1,1)--(1,3)); draw((2,0)--(2,3)); draw((3,0)--(3,2)); draw((4,0)--(4,1)); draw((2,0)--(4,0)); draw((1,1)--(4,1)); draw((0,2)--(3,2)); draw((0,3)--(2,3)); [/asy] $\text{(A) }384\text{ cm}^2\qquad\text{(B) }576\text{ cm}^2\qquad\text{(C) }672\text{ cm}^2\qquad\text{(D) }768\text{ cm}^2\qquad\text{(E) }832\text{ cm}^2$

In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$, find the area of the quadrilateral $AFED$.

Given a triangle $ABC$, a circle $\Omega$ is tangent to $AB,AC$ at $B,C,$ respectively. Point $D$ is the midpoint of $AC$, $O$ is the circumcenter of triangle $ABC$. A circle $\Gamma$ passing through $A,C$ intersects the minor arc $BC$ on $\Omega$ at $P$, and intersects $AB$ at $Q$. It is known that the midpoint $R$ of minor arc $PQ$ satisfies that $CR \perp AB$. Ray $PQ$ intersects line $AC$ at $L$, $M$ is the midpoint of $AL$, $N$ is the midpoint of $DR$, and $X$ is the projection of $M$ onto $ON$. Prove that the circumcircle of triangle $DNX$ passes through the center of $\Gamma$.

Let $n$ be a natural number. We define sequences $\langle a_i\rangle$ and $\langle b_i\rangle$ of integers as follows. We let $a_0=1$ and $b_0=n$. For $i>0$, we let $$\left( a_i,b_i\right)=\begin{cases} \left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\right) & \text{if } a_{i-1}<b_{i-1},\\ \left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\right) & \text{if } a_{i-1}>b_{i-1},\\ \left(a_{i-1},b_{i-1}\right) & \text{if } a_{i-1}=b_{i-1}.\end{cases}$$ Given that $a_k=b_k$ for some natural number $k$, prove that $n+3$ is a power of two.

Two circles $\omega_1$ and $\omega_2$, of equal radius intersect at different points $X_1$ and $X_2$. Consider a circle $\omega$ externally tangent to $\omega_1$ at $T_1$ and internally tangent to $\omega_2$ at point $T_2$. Prove that lines $X_1T_1$ and $X_2T_2$ intersect at a point lying on $\omega$.

The line touching the incircle of triangle $ABC$ and parallel to $BC$ meets the external bisector of angle $A$ at point $X$. Let $Y$ be the midpoint of arc $BAC$ of the circumcircle. Prove that the angle $XIY$ is right.

Consider rectangle $ABCD$ with $AB=30$ and $BC=60$. Construct circle $T$ whose diameter is $AD$. Construct circle $S$ whose diameter is $AB$. Let circles $S$ and $T$ intersect at $P$ such that $P\neq A$. Let $AP$ intersect $BC$ at $E$. Let $F$ be the point on $AB$ such that $EF$ is tangent to the circle with diameter $AD$. Find the area of triangle $AEF$.

The altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.