The perimeter of a triangle is a natural number, its circumradius is equal to $\frac{65}{8}$, and the inradius is equal to $4$. Find the sides of the triangle.

Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.

Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$). Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$). Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.

In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$ Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$ Find the greatest integer that does not exceed $1000r.$

Acute triangle $\triangle ABC$ satises $AB < AC$. Let the circumcircle of this triangle be $O$, and the midpoint of $BC,CA,AB$ be $D,E,F$. Let $P$ be the intersection of the circle with $AB$ as its diameter and line $DF$, which is in the same side of $C$ with respect to $AB$. Let $Q$ be the intersection of the circle with $AC$ as its diameter and the line $DE$, which is in the same side of $B$ with respect to $AC$. Let $PQ \cap BC = R$, and let the line passing through $R$ and perpendicular to $BC$ meet $AO$ at $X$. Prove that $AX = XR$.

There is a triangle $ABC$. Its circumcircle and its circumcentre are given. Show how the orthocentre of $ABC$ may be constructed using only a straightedge (unmarked ruler). [The straightedge and paper may be assumed large enough for the construction to be completed]

$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$) Prove that $P$ and $I$ and $D$ are on a line.

Let $k$ be the incircle and let $l$ be the circumcircle of the triangle $ABC$. Prove that for each point $A'$ of the circle $l$, there exists a triangle $(A'B'C')$, inscribed in the circle $l$ and circumscribed about the circle $k.$

Let $AB$ be a diameter of the circunference $\omega$, let $C$ and $D$ be point in this circunference, such that $CD$ is perpedicular to $AB$. Let $E$ be the point of intersection of the segment $CD$ and the segment $AB$, and a point $P$ that is in the segment $CD, P$ is different of $E$. The lines $AP$ and $BP$ intersects $\omega$, in $F$ and $G$ respectively. If $O$ is the circumcenter of triangle $EFG$, show that the area of triangle $OCD$ is invariant, independent of the position of the point $P$.

Let $ k$ be the inscribed circle of non-isosceles triangle $ \triangle ABC$, which center is $ S$. Circle $ k$ touches sides $ BC,CA,AB$ in points $ P,Q,R$ respectively. Line $ QR$ intersects $ BC$ in point $ M$. Let a circle which contains points $ B$ and $ C$ touch $ k$ in point $ N$. Circumscribed circle of $ \triangle MNP$ intersects line $ AP$ in point $ L$, different from $ P$. Prove that points $ S,L$ and $ M$ are collinear.

Let $ABC$ be a triangle with incentre $I$. The angle bisectors $AI$, $BI$ and $CI$ meet $[BC]$, $[CA]$ and $[AB]$ at $D$, $E$ and $F$, respectively. The perpendicular bisector of $[AD]$ intersects the lines $BI$ and $CI$ at $M$ and $N$, respectively. Show that $A$, $I$, $M$ and $N$ lie on a circle.

In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

Points $Y,X$ lies on $AB,BC$ of $\triangle ABC$ and $X,Y,A,C$ are concyclic. $AX$ and $CY$ intersect in $O$. Points $M,N$ are midpoints of $AC$ and $XY$. Prove, that $BO$ is tangent to circumcircle of $\triangle MON$

An acute-angle non-isosceles triangle $ ABC $ is given. The point $ H $ is its orthocenter, the points $ O $ and $ I $ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $ OIH $ passes through the vertex $ A $. Prove that one of the angles of the triangle is $ 60^\circ $.

Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram. [i]Raja Oktovin, Pekanbaru[/i]

Given an isosceles triangle $ABC$ with base $AB$, cirumcenter $O$, incenter $S$ and $\angle C<60^\circ$. The circumcircle of $AOS$ intersects $AC$ at $D$. Prove that $SD\parallel BC$ and $AS\perp OD$.

Region $ABCDEFGHIJ$ consists of $13$ equal squares and is inscribed in rectangle $PQRS$ with $A$ on $\overline{PQ}$, $B$ on $\overline{QR}$, $E$ on $\overline{RS}$, and $H$ on $\overline{SP}$, as shown in the figure on the right. Given that $PQ=28$ and $QR=26$, determine, with proof, the area of region $ABCDEFGHIJ$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(12)); pair P=(0,0), Q=(0,28), R=(26,28), S=(26,0), B=(3,28); draw(P--Q--R--S--cycle); picture p = new picture; draw(p, (0,0)--(3,0)^^(0,-1)--(3,-1)^^(0,-2)--(5,-2)^^(0,-3)--(5,-3)^^(2,-4)--(3,-4)^^(2,-5)--(3,-5)); draw(p, (0,0)--(0,-3)^^(1,0)--(1,-3)^^(2,0)--(2,-5)^^(3,0)--(3,-5)^^(4,-2)--(4,-3)^^(5,-2)--(5,-3)); transform t = shift(B) * rotate(-aSin(1/26^.5)) * scale(26^.5); add(t*p); label("$P$",P,SW); label("$Q$",Q,NW); label("$R$",R,NE); label("$S$",S,SE); label("$A$",t*(0,-3),W); label("$B$",B,N); label("$C$",t*(3,0),plain.ENE); label("$D$",t*(3,-2),NE); label("$E$",t*(5,-2),plain.E); label("$F$",t*(5,-3),plain.SW); label("$G$",t*(3,-3),(0.81,-1.3)); label("$H$",t*(3,-5),plain.S); label("$I$",t*(2,-5),NW); label("$J$",t*(2,-3),SW);[/asy]

Let $AA_1 , CC_1$ be the altitudes of triangle $ABC, B_0$ the common point of the altitude from $B$ and the circumcircle of $ABC$; and $Q$ the common point of the circumcircles of $ABC$ and $A_1C_1B_0$, distinct from $B_0$. Prove that $BQ$ is the symmedian of $ABC$. [i]Proposed by D.Shvetsov[/i]

In triangle $ABC$, $AB=13$, $BC=15$, and $CA=14$. Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$

A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ in points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilateral $ABCD$ meet in point $X$, and the diagonals of $ABCD$ meet in point $Y$. Prove that points $X, Y, I$ and $M$ are collinear.

Let $\Gamma$ be the circumcircle of a triangle $ABC$ with $AB <BC$, and let $M$ be the midpoint from the side $AC$ . The median of side $AC$ cuts $\Gamma$ at points $X$ and $Y$ ($X$ in the arc $ABC$). The circumcircle of the triangle $BMY$ cuts the line $AB$ at $P$ ($P \ne B$) and the line $BC$ in $Q$ ($Q \ne B$). The circumcircles of the triangles $PBC$ and $QBA$ are cut in $R$ ($R \ne B$). Prove that points $X, B$ and $R$ are collinear.

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

The circumcentre $O$ of a given cyclic quadrilateral $ABCD$ lies inside the quadrilateral but not on the diagonal $AC$. The diagonals of the quadrilateral intersect at $I$. The circumcircle of the triangle $AOI$ meets the sides $AD$ and $AB$ at points $P$ and $Q$, respectively; the circumcircle of the triangle $COI$ meets the sides $CB$ and $CD$ at points $R$ and $S$, respectively. Prove that $PQRS$ is a parallelogram.

Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$. [b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$. [b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.

Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$. [i]Amol Aggarwal.[/i]