Olympiad problems

2014 IMAC Arhimede, 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.

2001 All-Russian Olympiad Regional Round, 11.3

Tags: geometry , tangent

Let $AD$ be the angle bisector of triangle $ABC$, and let the line $\ell$ touch circumcircles of triangles $ADB$ and $ADC$ at points $M$ and $N$ accordingly. Prove that the circle passing through the midpoints of the segments $BD$, $DC$ and $MN$ is tangent to the line $\ell$.

2008 Greece JBMO TST, 1

Tags: geometry , circumcenter , tangent , isosceles , circles

Given a point $A$ that lies on circle $c(o,R)$ (with center $O$ and radius $R$). Let $(e)$ be the tangent of the circle $c$ at point $A$ and a line $(d)$ that passes through point $O$ and intersects $(e)$ at point $M$ and the circle at points $B,C$ (let $B$ lie between $O$ and $A$). If $AM = R\sqrt3$ , prove that a) triangle $AMC$ is isosceles. b) circumcenter of triangle $AMC$ lies on circle $c$ .

2022 Costa Rica - Final Round, 6

Tags: tangent , geometry

Consider $ABC$ with $AC > AB$ and incenter $I$. The midpoints of $\overline{BC}$ and $\overline{AC}$ are $M$ and $N$, respectively. If $\overline{AI}$ is perpendicular to $\overline{IN}$, then prove that $\overline{AI}$ is tangent to the circumscribed circle of $\vartriangle BMI$.

2021 Saudi Arabia Training Tests, 19

Tags: tangent , circumcircle , geometry

Let $ABC$ be a triangle with $AB < AC$ inscribed in $(O)$. Tangent line at $A$ of $(O)$ cuts $BC$ at $D$. Take $H$ as the projection of $A$ on $OD$ and $E,F$ as projections of $H$ on $AB,AC$.Suppose that $EF$ cuts $(O)$ at $R,S$. Prove that $(HRS)$ is tangent to $OD$

2023 Yasinsky Geometry Olympiad, 6

Tags: tangent , geometry

Let $ABC$ be an isosceles triangle with $\angle BAC = 108^o$. The angle bisector of the $\angle ABC$ intersects the circumcircle of a triangle $ABC$ at the point $D$. Let $E$ be a point on segment $CB$ such that $AB =BE$. Prove that the perpendicular bisector of $CD$ is tangent to circumcircle of triangle $ABE$ . (Bohdan Zheliabovskyi)

2024 Sharygin Geometry Olympiad, 23

Tags: tangent , geometry , transformation

A point $P$ moves along a circle $\Omega$. Let $A$ and $B$ be two fixed points of $\Omega$, and $C$ be an arbitrary point inside $\Omega$. The common external tangents to the circumcircles of triangles $APC$ and $BCP$ meet at point $Q$. Prove that all points $Q$ lie on two fixed lines.

2011 China Northern MO, 6

Tags: geometry , tangent

As shown in figure, from a point $P$ exterior of circle $\odot O$, we draw tangent $PA$ and the secant $PBC$. Let $AD \perp PO$ Prove that $AC$ is tangent to the circumcircle of $\vartriangle ABD$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/32da6d4626bb3592cec19a4cf0202121ba64db.png[/img]

2001 All-Russian Olympiad Regional Round, 10.6

Tags: geometry , parallelogram , tangent

Given triangle $ABC$. Point $B_1$ is marked on line $AC$ so that $AB = AB_1$, while $B_1$ and $C$ are on the same side of $A$. Through points $C$, $B_1$ and the foot of the bisector of angle $A$ of triangle $ABC$, a circle $\omega$ is drawn, intersecting for second time the circle circumscribed around triangle $ABC$, at point $Q$. Prove that the tangent drawn to $\omega$ at point $Q$ is parallel to $AC$.

2008 Estonia Team Selection Test, 5

Tags: midpoint , tangent , geometry , circles

Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.

2014 Indonesia MO Shortlist, G6

Tags: geometry , circumcircle , tangent , tangent circle

Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

2003 Chile National Olympiad, 6

Tags: circumcircle , geometry , tangent

Consider a triangle $ ABC $. On the line $ AC $ take a point $ B_1 $ such that $ AB = AB_1 $ and in addition, $ B_1 $ and $ C $ are located on the same side of the line with respect to the point $ A $. The bisector of the angle $ A $ intersects the side $ BC $ at a point that we will denote as $ A_1 $. Let $ P $ and $ R $ be the circumscribed circles of the triangles $ ABC $ and $ A_1B_1C $ respectively. They intersect at points $ C $ and $ Q $. Prove that the tangent to the circle $ R $ at the point $ Q $ is parallel to the line $ AC $.

2021 Final Mathematical Cup, 2

Tags: geometry , tangent

The altitudes $BB_1$ and $CC_1$, are drawn in an acute triangle $ABC$. Let $X$ and $Y$ be the points, which are symmetrical to the points $B_1$ and $C_1$, with respect to the midpoints of the sides$ AB$ and $AC$ of the triangle $ABC$ respectively. Let's denote with $Z$ the point of intersection of the lines $BC$ and $XY$. Prove that the line $ZA$ is tangent to the circumscribed circle of the triangle $AXY$ .

2015 Greece Junior Math Olympiad, 4

Tags: geometry , isosceles , tangent

Let $ABC$ be an acute triangle with $AB\le AC$ and let $c(O,R)$ be it's circumscribed circle (with center $O$ and radius $R$). The perpendicular from vertex $A$ on the tangent of the circle passing through point $C$, intersect it at point $D$. a) If the triangle $ABC$ is isosceles with $AB=AC$, prove that $CD=BC/2$. b) If $CD=BC/2$, prove that the triangle $ABC$ is isosceles.

2018 Dutch IMO TST, 2

Tags: geometry , circumcircle , tangent , perpendicular , right triangle

Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.

2015 Singapore Senior Math Olympiad, 1

Tags: tangent , collinear , circumcircle , geometry

In an acute-angled triangle $ABC$, $M$ is a point on the side $BC$, the line $AM$ meets the circumcircle $\omega$ of $ABC$ at the point $Q$ distinct from $A$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to the diameter $AK$ of $\omega$ at the point $P$. Let $L$ be the point on $\omega$ distinct from $Q$ such that $PL$ is tangent to $\omega$ at $L$. Prove that $L,M$ and $K$ are collinear.

2010 Junior Balkan Team Selection Tests - Moldova, 3

Tags: geometry , circumcircle , collinear , tangent

The tangent to the circle circumscribed to the triangle $ABC$, taken through the vertex $A$, intersects the line $BC$ at the point $P$, and the tangents to the same circle, taken through $B$ and $C$, intersect the lines $AC$ and $AB$, respectively at the points $Q$ and $R$. Prove that the points $P, Q$ ¸ and $R$ are collinear.

2007 Greece JBMO TST, 3

Tags: geometry , rectangle , tangent , tangent circle

Let $ABCD$ be a rectangle with $AB=a >CD =b$. Given circles $(K_1,r_1) , (K_2,r_2)$ with $r_1<r_2$ tangent externally at point $K$ and also tangent to the sides of the rectangle, circle $(K_1,r_1)$ tangent to both $AD$ and $AB$, circle $(K_2,r_2)$ tangent to both $AB$ and $BC$. Let also the internal common tangent of those circles pass through point $D$. (i) Express sidelengths $a$ and $b$ in terms of $r_1$ and $r_2$. (ii) Calculate the ratios $\frac{r_1}{r_2}$ and $\frac{a}{b}$ . (iii) Find the length of $DK$ in terms of $r_1$ and $r_2$.

2014 IMO Shortlist, G5

Tags: geometry , circumcircle , quadrilateral , tangent , tangent circle

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 $TSH$.

2021 Iran Team Selection Test, 1

Tags: geometry , tangent

In acute scalene triangle $ABC$ the external angle bisector of $\angle BAC$ meet $BC$ at point $X$.Lines $l_b$ and $l_c$ which tangents of $B$ and $C$ with respect to $(ABC)$.The line pass through $X$ intersects $l_b$ and $l_c$ at points $Y$ and $Z$ respectively. Suppose $(AYB)\cap(AZC)=N$ and $l_b\cap l_c=D$. Show that $ND$ is angle bisector of $\angle YNZ$. Proposed by [i]Alireza Haghi[/i]

2024 China Western Mathematical Olympiad, 3

Tags: tangent , geometry , isogonal , phantom point , similarity

$AB,AC$ are tangent to $\Omega$ at $B$ and $C$, respectively. $D,E,F$ lie on segments $BC,CA,AB$ such that $AF<AE$ and $\angle FDB= \angle EDC$. The circumcircle of $\triangle FEC$ intersects $\Omega$ at $G$ and $C$. Show that $ \angle AEF= \angle BGD$

2017 Yasinsky Geometry Olympiad, 3

Tags: geometry , tangent , construction , arc

Given circle arc, whose center is an inaccessible point. $A$ is a point on this arc (see fig.). How to construct using compass and ruler without divisions, a tangent to given circle arc at point $A$ ? [img]https://1.bp.blogspot.com/-7oQBNJGLsVw/W6dYm4Xw7bI/AAAAAAAAJH8/sJ-rgAQZkW0kvlPOPwYiGjnOXGQZuDnRgCK4BGAYYCw/s1600/Yasinsky%2B2017%2BVIII-IX%2Bp3.png[/img]

Kyiv City MO Seniors 2003+ geometry, 2016.11.4

Tags: geometry , circumcircle , tangent , orthocenter

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)

Kyiv City MO Juniors 2003+ geometry, 2011.9.41

Tags: tangent , geometry , circumcircle , equal segments

The triangle $ABC$ is inscribed in a circle. At points $A$ and $B$ are tangents to this circle, which intersect at point $T$. A line drawn through the point $T$ parallel to the side $AC$ intersects the side $BC$ at the point $D$. Prove that $AD = CD$.

2008 Postal Coaching, 5

Tags: tangent , semicircle , parallel , circles , geometry

Let $\omega$ be the semicircle on diameter $AB$. A line parallel to $AB$ intersects $\omega$ at $C$ and $D$ so that $B$ and $C$ lie on opposite sides of $AD$. The line through $C$ parallel to $AD$ meets $\omega$ again in $E$. Lines $BE$ and $CD$ meet in $F$ and the line through $F$ parallel to $AD$ meets $AB$ in $P$. Prove that $PC$ is tangent to $\omega$.