Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT=CT=16$, $BC=22$, and $TX^2+TY^2+XY^2=1143$. Find $XY^2$.

A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length $ \textbf{(A)}\ 3\sqrt3 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 6\sqrt3 \qquad \textbf{(D)}\ 12\sqrt3 \qquad \textbf{(E)}\ \text{ none of these}$

The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.

Consider a triangle $ABC$ and a point $P$ in its interior. Lines $PA$, $PB$, $PC$ intersect $BC$, $CA$, $AB$ at $A', B', C'$ , respectively. Prove that $$\frac{BA'}{BC}+ \frac{CB'}{CA}+ \frac{AC'}{AB}= \frac32$$ if and only if at least two of the triangles $PAB$, $PBC$, $PCA$ have the same area.

Let $ P$, $ Q$, and $ R$ be the points on sides $ BC$, $ CA$, and $ AB$ of an acute triangle $ ABC$ such that triangle $ PQR$ is equilateral and has minimal area among all such equilateral triangles. Prove that the perpendiculars from $ A$ to line $ QR$, from $ B$ to line $ RP$, and from $ C$ to line $ PQ$ are concurrent.

Let $I$ be the incentre of a triangle $ABC$ and $\Gamma_a$ be the excircle opposite $A$ touching $BC$ at $D$. If $ID$ meets $\Gamma_a$ again at $S$, prove that $DS$ bisects $\angle BSC$.

Let $PT$ and $PB$ be two tangents to a circle, $T$ and $B$ on the circle. $AB$ is the diameter of the circle through $B$ and $TH$ is the perpendicular from $T$ to $AB$, $H$ on $AB$. Prove that $AP$ bisects $TH$.

In $\triangle ABC$, $AB>AC$. In the circumcircle $(O)$ of $\triangle ABC$, $M$ is the midpoint of arc $BAC$. The incircle $(I)$ of $\triangle ABC$ touches $BC$ at $D$, the line through $D$ parallel to $AI$ intersects $(I)$ again at $P$. Prove that $AP$ and $IM$ intersect at a point on $(O)$.

Let $ABC$ be a triangle inscribed in circle $(O)$, with its altitudes $BH_b, CH_c$ intersect at orthocenter $H$ ($H_b \in AC$, $H_c \in AB$). $H_bH_c$ meets $BC$ at $P$. Let $N$ be the midpoint of $AH, L$ be the orthogonal projection of $O$ on the symmedian with respect to angle $A$ of triangle $ABC$. Prove that $\angle NLP = 90^o$.

Let $A,B,C$ be non-collinear points. For each point $D$ of the ray $AC$, we denote by $E$ and $F$ the points of tangency of the incircle of $\triangle ABD$ with $AB$ and $AD$, respectively. Prove that, as point $D$ moves along the ray $AC$, the line $EF$ passes through a fixed point.

Let $ABC$ be an acute triangle and $G$ be the intersection of the meadians of triangle $ABC$. Let $D $be the foot of the altitude drawn from $A$ to $BC$. Draw a parallel line such that it is parallel to $BC$ and one of the points of it is $A$.Donate the point $S$ as the intersection of the parallel line and circumcircle $ABC$. Prove that $S,G,D$ are co-linear [asy] size(6cm); defaultpen(fontsize(10pt)); pair A = dir(50), S = dir(130), B = dir(200), C = dir(-20), G = (A+B+C)/3, D = foot(A, B, C); draw(A--B--C--cycle, black+linewidth(1)); draw(A--S^^A--D, magenta); draw(S--D, red+dashed); draw(circumcircle(A, B, C), heavymagenta); string[] names = {"$A$", "$B$", "$C$","$D$", "$G$","$S$"}; pair[] points = {A, B, C,D,G,S}; pair[] ll = {A, B, C,D, G,S}; int pt = names.length; for (int i=0; i<pt; ++i) dot(names[i], points[i], dir(ll[i])); [/asy]

Let $r$ be the inradius of an acute triangle. Prove that the sum of the distances from the orthocenter to the sides of the triangle does not exceed $3r$

The lengths of the four sides of an cyclic octagon are $4$ cm, the lengths of the other four sides are $6$ cm. Find the area of ​​the octagon.

Let $n\ge5$ be an integer. Find the biggest integer $k$ such that there always exists a $n$-gon with exactly $k$ interior right angles. (Find $k$ in terms of $n$).

In equilateral $\triangle ABC$, point $D$ lies on $\overline{BC}$ such that the radius of the circumcircle $\Gamma_1$ of $\triangle ACD$ is $7$ and the radius of the incircle $\Gamma_2$ of $\triangle{ABD}$ is $2$. Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at points $X$ and $Y$. Find $XY$.

In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.

The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$. Prove that the cosine of the angle between lines $(AC)$ and $(BD)$ is less than $|CD|/|AB|$.

Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality \[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\] holds.

An equilateral triangle with side 101 is placed on a plane so that one of its sides is horizontal and the triangle is above it. It is divided into smaller equilateral triangles with side 1 by segments parallel to its sides. All sides of these smaller triangles are colored red (including the entire border of the large triangle). An equilateral triangle on a plane is called a "mirror" triangle if its sides are parallel to the sides of the original triangle, but it lies below its horizontal side. What is the smallest number of contours of mirror triangles needed to cover all the red segments? (Mirror triangles may overlap and extend beyond the original triangle.) [i]Proposed by Dmitry Shiryayev[/i]

Let $ABCD$ be a convex, non-cyclic quadrilateral with $E$ the intersection of its diagonals. Given $\angle ABD+\angle DAC = \angle CBD+\angle DCA$, $AB = 10$, $BC = 15$, $AE = 7$, and $EC = 13$, find $BD$.

Let $\mathcal A = A_0A_1A_2A_3 \cdots A_{2013}A_{2014}$ be a [i]regular 2014-simplex[/i], meaning the $2015$ vertices of $\mathcal A$ lie in $2014$-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \le i < j \le 2014$. Let $O = (0,0,0,\dots,0)$, $A_0 = (1,0,0,\dots,0)$, and suppose $A_iO$ has length $1$ for $i=0,1,\dots,2014$. Set $P=(20,14,20,14,\dots,20,14)$. Find the remainder when \[PA_0^2 + PA_1^2 + \dots + PA_{2014}^2 \] is divided by $10^6$. [i]Proposed by Robin Park[/i]

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

Let $ABCD$ be a convex quadrilateral. Construct $S$ and $T$ on the side $AD$ and $AB$ respectively such that $AS=AT$. Construct $U$ and $V$ on the side $BC$ and $CD$ respectively such that $CU=CV$. Assume that $BT=BU$ and $ST, UV, BD$ are concurrent, prove that $AB+CD=BC+AD$.

A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?