Found problems: 3585
Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q$, and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B$, respectively. The line parallel to line $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Suppose $PX = 10, PY = 14,$ and $PQ = 5$. Then the area of trapezoid $XABY$ is $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
In a convex hexagon $ABCDEF$ the diagonals $AD,BE,CF$ intersect at a point $M$. It is known that the triangles $ABM,BCM,CDM,DEM,EFM,FAM$ are acute. It is also known that the quadrilaterals $ABDE,BCEF,CDFA$ have the same area. Prove that the circumcenters of triangles $ABM,BCM,CDM,DEM,EFM,FAM$ are concyclic.
[i]Proposed by Nairy Sedrakyan[/i]
In $ABC$ triangle $I$ is incenter and incircle of $ABC$ tangents to $BC,AC,AB$ at $D,E,F$, respectively. If $AI$ intersects $DE$ and $DF$ at $P$ and $Q$, prove that the circumcenter of $DPQ$ triangle is the midpoint of $BC$.
Let $ AB$ be the diameter of semicircle $O$ ,
$C, D $ be points on the arc $AB$,
$P, Q$ be respectively the circumcenter of $\triangle OAC $ and $\triangle OBD $ .
Prove that:$CP\cdot CQ=DP \cdot DQ$.[asy]
import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black;
real h=sqrt(55/64);
pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B);
D(arc(O,1,0,180),darkgreen);
D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue);
D(O);
[/asy]
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon.
[i]Greece[/i]
Given a triangle $ABC$, a point $P$ is chosen on side $BC$. Points $M$ and $N$ lie on sides $AB$ and $AC$, respectively, such that $MP \parallel AC$ and $NP \parallel AB$. Point $P$ is reflected across $MN$ to point $Q$. Show that triangle $QMB$ is similar to triangle $CNQ$.
[i]Brian Hamrick.[/i]
Let $ABCD$ be a cyclic quadrilateral. Prove that $AC > BD$ if and only if $(AD-BC)(AB- CD) > 0$.
(V. Yasinsky)
In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively.
Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other.
[i]Proposed by Iman Maghsoudi[/i]
Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.
Let $ABC$ be an acute triangle and let $P,Q$ be points on $BC$ such that $\angle QAC =\angle ABC$ and $\angle PAB = \angle ACB$. We extend $AP$ to $M$ so that $ P$ is the midpoint of $AM$ and we extend $AQ$ to $N$ so that $Q$ is the midpoint of $AN$. If T is the intersection point of $BM$ and $CN$, show that quadrilateral $ABTC$ is cyclic.
Let right $\triangle ABC$ have $AC = 3$, $BC = 4$, and right angle at $C$. Let $D$ be the projection from $C$ to $\overline{AB}$. Let $\omega$ be a circle with center $D$ and radius $\overline{CD}$, and let $E$ be a variable point on the circumference of $\omega$. Let $F$ be the reflection of $E$ over point $D$, and let $O$ be the center of the circumcircle of $\triangle ABE$. Let $H$ be the intersection of the altitudes of $\triangle EFO$. As $E$ varies, the path of $H$ traces a region $\mathcal R$. The area of $\mathcal R$ can be written as $\tfrac{m\pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $\sqrt{m}+\sqrt{n}$.
[i]Proposed by ApraTrip[/i]
suppose that $O$ is the circumcenter of acute triangle $ABC$.
we have circle with center $O$ that is tangent too $BC$ that named $w$
suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$)
$T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$.
$S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$.
prove that $ST$ is tangent $ABC$.
In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$.
[i]Proposed by Iman Maghsoudi[/i]
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$.
Prove that line $AO$ passes through the midpoint of segment $BE$.
Let $ABCD$ be a cyclic quadrilateral and triangles $ACD, BCD$ are acute. Suppose that the lines $AB$ and $CD$ meet at $S$. Denote by $E$ the intersection of $AC, BD$. The circles $(ADE)$ and $(BC E)$ meet again at $F$.
a) Prove that $SF \perp EF.$
b) The point $G$ is taken out side of the quadrilateral $ABCD$ such that triangle $GAB$ and $FDC$ are similar. Prove that $GA+ FB = GB + FA$
In triangle $ABC$, $M,N$ and $P$ are midpoints of sides $BC,CA$ and $AB$. Point $K$ lies on segment $NP$ so that $AK$ bisects $\angle BKC$. Lines $MN,BK$ intersects at $E$ and lines $MP,CK$ intersects at $F$. Suppose that $H$ be the foot of perpendicular line from $A$ to $BC$ and $L$ the second intersection of circumcircle of triangles $AKH, HEF$. Prove that $MK,EF$ and $HL$ are concurrent.
[i]Proposed by Alireza Dadgarnia[/i]
Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$.
[i]Proposed by Ali Zamani, Hooman Fattahi[/i]
Let $ ABC$ be an acute angled triangle; let $ D,F$ be the midpoints of $ BC,AB$ respectively. Let the perpendicular from $ F$ to $ AC$ and the perpendicular from $ B$ ti $ BC$ meet in $ N$: Prove that $ ND$ is the circumradius of $ ABC$.
[15 points out of 100 for the 6 problems]
In an acute triangle $ABC$ the heights $AD, BE$ and $CF$ intersecting at $H{}$. Let $O{}$ be the circumcenter of the triangle $ABC$. The tangents to the circle $(ABC)$ drawn at $B{}$ and $C{}$ intersect at $T{}$. Let $K{}$ and $L{}$ be symmetric to $O{}$ with respect to $AB$ and $AC$ respectively. The circles $(DFK)$ and $(DEL)$ intersect at a point $P{}$ different from $D{}$. Prove that $P, D$ and $T{}$ lie on the same line.
[i]Proposed by Don Luu (Vietnam)[/i]
$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.
Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square?
(C.Pohoata, A.Zaslavsky)
A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.