Found problems: 353
2016 Stars of Mathematics, 3
Let $ ABC $ be a triangle, $ M_A $ be the midpoint of the side $ BC, $ and $ P_A $ be the orthogonal projection of $ A $ on $ BC. $ Similarly, define $ M_B,M_C,P_B,P_C. M_BM_C $ intersects $ P_BP_C $ at $ S_A, $ and the tangent of the circumcircle of $ ABC $ at $ A $ meets $ BC $ at $ T_A. $ Similarly, define $ S_B,S_C,T_B,T_C. $
Show that the perpendiculars through $ A,B,C, $ to $ S_AT_A,S_BT_B, $ respectively, $ S_CT_C, $ are concurent.
[i]Flavian Georgescu[/i]
2005 Argentina National Olympiad, 5
Let $AM$ and $AN$ be the lines tangent to a circle $\Gamma$ drawn from a point $A$ $(M$ and $N$ belong to the circle). A line through $A$ cuts $\Gamma$ at $B$ and $C$ with $B$ between $A$ and $C$, and $\frac{AB}{BC} =\frac23$. If $P$ is the intersection point of $AB$ and $MN$, calculate $\frac{AP}{CP}$.
2010 Belarus Team Selection Test, 6.2
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
1950 Moscow Mathematical Olympiad, 186
A spatial quadrilateral is circumscribed around a sphere. Prove that all the tangent points lie in one plane.
2012 Switzerland - Final Round, 3
The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $ \angle DPM = \angle BDC$ .
2024 Brazil EGMO TST, 4
Let $ABCD$ be a cyclic quadrilateral with all distinct sides that has an inscribed circle. The incircle of $ABCD$ has center $I$ and is tangent to $AB$, $BC$, $CD$, and $DA$ at points $W$, $X$, $Y$, and $Z$, respectively. Let $K$ be the intersection of the lines $WX$ and $YZ$. Prove that $KI$ is tangent to the circumcircle of triangle $AIC$.
2014 Sharygin Geometry Olympiad, 6
Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ are tangent to each other externally at point $O$. Points $X$ and $Y$ on $k_1$ and $k_2$ respectively are such that rays $O_1X$ and $O_2Y$ are parallel and codirectional. Prove that two tangents from $X$ to $k_2$ and two tangents from $Y$ to $k_1$ touch the same circle passing through $O$.
(V. Yasinsky)
1982 Austrian-Polish Competition, 2
Let $F$ be a closed convex region inside a circle $C$ with center $O$ and radius $1$. Furthermore, assume that from each point of $C$ one can draw two rays tangent to $F$ which form an angle of $60^o$. Prove that $F$ is the disc centered at $O$ with radius $1/2$.
2014 IMO Shortlist, G5
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
2014 Contests, 2
A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.
Kyiv City MO Seniors 2003+ geometry, 2016.11.4
The median $AM$ is drawn in the acute-angled triangle $ABC$ with different sides. Its extension intersects the circumscribed circle $w$ of this triangle at the point $P$. Let $A {{H} _ {1}}$ be the altitude $\Delta ABC$, $H$ be the point of intersection of its altitudes. The rays $MH$ and $P {{H} _ {1}}$ intersect the circle $w$ at the points $K$ and $T$, respectively. Prove that the circumscribed circle of $\Delta KT {{H} _ {1}}$ touches the segment $BC$.
(Hilko Danilo)
2001 Kazakhstan National Olympiad, 7
Two circles $ w_1 $ and $ w_2 $ intersect at two points $ P $ and $ Q $. The common tangent to $ w_1 $ and $ w_2 $, which is closer to the point $ P $ than to $ Q $, touches these circles at $ A $ and $ B $, respectively. The tangent to $ w_1 $ at the point $ P $ intersects $ w_2 $ at the point $ E $ (different from $ P $), and the tangent to $ w_2 $ at the point $ P $ intersects $ w_1 $ at $ F $ (different from $ P $). Let $ H $ and $ K $ be points on the rays $ AF $ and $ BE $, respectively, such that $ AH = AP $ and $ BK = BP $. Prove that the points $ A $, $ H $, $ Q $, $ K $ and $ B $ lie on the same circle.
2018 Dutch IMO TST, 4
In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.
2022 JBMO Shortlist, G6
Let $ABC$ be a right triangle with hypotenuse $BC$. The tangent to the circumcircle of triangle $ABC$ at $A$ intersects the line $BC$ at $T$. The points $D$ and $E$ are chosen so that $AD = BD, AE = CE,$ and $\angle CBD = \angle BCE < 90^{\circ}$. Prove that $D, E,$ and $T$ are collinear.
Proposed by [i]Nikola Velov, Macedonia[/i]
2009 Bosnia And Herzegovina - Regional Olympiad, 4
Let $C$ be a circle with center $O$ and radius $R$. From point $A$ of circle $C$ we construct a tangent $t$ on circle $C$. We construct line $d$ through point $O$ whch intersects tangent $t$ in point $M$ and circle $C$ in points $B$ and $D$ ($B$ lies between points $O$ and $M$). If $AM=R\sqrt{3}$, prove:
$a)$ Triangle $AMD$ is isosceles
$b)$ Circumcenter of $AMD$ lies on circle $C$
Cono Sur Shortlist - geometry, 2005.G6
Let $AM$ and $AN$ be the tangents to a circle $\Gamma$ drawn from a point $A$ ($M$ and $N$ lie on the circle). A line passing through $A$ cuts $\Gamma$ at $B$ and $C$, with B between $A$ and $C$ such that $AB: BC = 2: 3$. If $P$ is the intersection point of $AB$ and $MN$, calculate the ratio $AP: CP$ .
2024 Mozambican National MO Selection Test, P2
On a sheet divided into squares, each square measuring $2cm$, two circles are drawn such that both circles are inscribed in a square as in the figure below. Determine the minimum distance between the two circles.
1949 Putnam, B6
Let $C$ be a closed convex curve with a continuously turning tangent and let $O$ be a point inside $C.$ For each point $P$ on $C$ we define $T(P)$ as follows:
Draw the tangent to $C$ at $P$ and from $O$ drop the perpendicular to that tangent. Then $T(P)$ is the point at which $C$ intersects this perpendicular.
Starting now with a point $P_{0}$ on $C$, define points $P_n$ by $P_n =T(P_{n-1}).$ Prove that the points $P_{n}$ approach a limit and characterize all possible limit points. (You may assume that $T$ is continuous.)
1980 Austrian-Polish Competition, 9
Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$ and let $q_1,q_2$ be two rays from $C$. Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$. Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$. Prove that $M_1M_2 = N_1N_2$.
2010 German National Olympiad, 1
Given two circles $k$ and $l$ which intersect at two points. One of their common tangents touches $k$ at point $K$, while the other common tangent touches $l$ at $L.$ Let $A$ and $B$ be the intersections of the line $KL$ with the circles $k$ and $l$, respectively. Prove that $\overline{AK} = \overline{BL}.$
2019 Saudi Arabia IMO TST, 3
Let $ABC$ be an acute nonisosceles triangle with incenter $I$ and $(d)$ is an arbitrary line tangent to $(I)$ at $K$. The lines passes through $I$, perpendicular to $IA, IB, IC$ cut $(d)$ at $A_1, B_1,C_1$ respectively. Suppose that $(d)$ cuts $BC, CA, AB$ at $M,N, P$ respectively. The lines through $M,N,P$ and respectively parallel to the internal bisectors of $A, B, C$ in triangle $ABC$ meet each other to define a triange $XYZ$. Prove that three lines $AA_1, BB_1, CC_1$ are concurrent and $IK$ is tangent to the circle $(XY Z)$
2022 Taiwan TST Round 2, 5
Let $ABCDE$ be a pentagon inscribed in a circle $\Omega$. A line parallel to the segment $BC$ intersects $AB$ and $AC$ at points $S$ and $T$, respectively. Let $X$ be the intersection of the line $BE$ and $DS$, and $Y$ be the intersection of the line $CE$ and $DT$.
Prove that, if the line $AD$ is tangent to the circle $\odot(DXY)$, then the line $AE$ is tangent to the circle $\odot(EXY)$.
[i]Proposed by ltf0501.[/i]
2016 Saudi Arabia GMO TST, 1
Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK$ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Prove the following assertions:
a) If $A$ is the center of the circle $(IMN)$, then $BC$ is tangent to $(IMN)$.
b) If $I$ is the midpoint of $BC$, then $BC$ is equal to $4$ times of the distance between the centers of two circles $(ABK)$ and $(ACH)$.
2010 Peru MO (ONEM), 3
Consider $A, B$ and $C$ three collinear points of the plane such that $B$ is between $A$ and $C$. Let $S$ be the circle of diameter $AB$ and $L$ a line that passes through $C$, which does not intersect $S$ and is not perpendicular to line $AC$. The points $M$ and $N$ are, respectively, the feet of the altitudes drawn from $A$ and $B$ on the line $L$. From $C$ draw the two tangent lines to $S$, where $P$ is the closest tangency point to $L$. Prove that the quadrilateral $MPBC$ is cyclic if and only if the lines $MB$ and $AN$ are perpendicular.
2007 Oral Moscow Geometry Olympiad, 6
A circle and a point $P$ inside it are given. Two arbitrary perpendicular rays starting at point $P$ intersect the circle at points $A$ and $B$. Point $X$ is the projection of point $P$ onto line $AB, Y$ is the intersection point of tangents to the circle drawn through points $A$ and $B$. Prove that all lines $XY$ pass through the same point.
(A. Zaslavsky)