Olympiad problems

Swiss NMO - geometry, 2012.3

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$ .

1998 Tournament Of Towns, 3

Tags: circles , geometry , tangential , tangent

Segment $AB$ intersects two equal circles, is parallel to the line joining their centres, and all the points of intersection of the segment and the circles lie between $A$ and $B$. From the point $A$ tangents to the circle nearest to $A$ are drawn, and from the point $B$ tangents to the circle nearest to $B$ are also drawn. It turns out that the quadrilateral formed by the four tangents extended contains both circles. Prove that a circle can be drawn so that it touches all four sides of the quadrilateral. (P Kozhevnikov)

2022 All-Russian Olympiad, 7

Tags: tangent , parallelogram , geometry

Point $E$ is marked on side $BC$ of parallelogram $ABCD$, and on the side $AD$ - point $F$ so that the circumscribed circle of $ABE$ is tangent to line segment $CF$. Prove that the circumcircle of triangle $CDF$ is tangent to line $AE$.

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 $\vartriangle BMI$.

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)

2019-IMOC, G4

Tags: tangent circle , geometry , circumcircle , tangent

$\vartriangle ABC$ is a scalene triangle with circumcircle $\Omega$. For a arbitrary $X$ in the plane, define $D_x,E_x, F_x$ to be the intersection of tangent line of $X$ (with respect to $BXC$) and $BC,CA,AB$, respectively. Let the intersection of $AX$ with $\Omega$ be $S_x$ and $T_x = D_xS_x \cap \Omega$. Show that $\Omega$ and circumcircle of $\vartriangle T_xE_xF_x$ are tangent to each other. [img]https://2.bp.blogspot.com/-rTMODHbs5Ac/XnYNQYjYzBI/AAAAAAAALeg/576nGDQ6NDA0-W5XqiNczNtI07cEZxPeQCK4BGAYYCw/s1600/imoc2019g4.png[/img]

2014 IMAC Arhimede, 2

Tags: tangential quadrilateral , geometry , concurrency , tangent

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.

2006 Sharygin Geometry Olympiad, 6

Tags: circles , construction , tangent , geometry

a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$. Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$. b) Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$. Draw through $A$ and $B$ a circle tangent to a circle of radius $r$.

2024 China Western Mathematical Olympiad, 3

Tags: geometry , tangent , 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$

2023 JBMO TST - Turkey, 3

Tags: geometry , tangent

Let $ABC$ is triangle and $D \in AB$,$E \in AC$ such that $DE//BC$. Let $(ABC)$ meets with $(BDE)$ and $(CDE)$ at the second time $K,L$ respectively. $BK$ and $CL$ intersect at $T$. Prove that $TA$ is tangent to the $(ABC)$

2018 Costa Rica - Final Round, 3

Tags: tangent , geometry

In the attached figure, point $C$ is the center of the circle, $AB$ is tangent to the circle, $P-C-P'$ and $AC\perp PP'$. If $AT = 2$ cm. and $AB = 4$ cm, calculate $BQ$ [img]https://cdn.artofproblemsolving.com/attachments/e/e/d47429b82fb87299c40f5224489313909cfd0f.png[/img] Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

V Soros Olympiad 1998 - 99 (Russia), 11.4

Tags: geometry , locus , tangent

A chord $AB$ is drawn in a circle. On its extensions beyond points $A$ and $B$, points $P$ and $Q$ respectively are taken such that $AP = BQ$. Through $P$ and $Q$ two tangents to the circle are drawn, intersecting at point $M$. Find the locus of points $M$ ($P$ and $Q$ move along a straight line and for any $P$ and $Q$ all possible pairs of tangents are taken, which determine four points from the desired locus of points) .

2018 Argentina National Olympiad, 6

Tags: geometry , parallelogram , circles , tangent

Let $ABCD$ be a parallelogram. An interior circle of the $ABCD$ is tangent to the lines $AB$ and $AD$ and intersects the diagonal $BD$ at $E$ and $F$. Prove that exists a circle that passes through $E$ and $F$ and is tangent to the lines $CB$ and $CD$.

2022 Korea Junior Math Olympiad, 6

Tags: circumcircle , concyclic , tangent , geometry

Let $ABC$ be a isosceles triangle with $\overline{AB}=\overline{AC}$. Let $D(\neq A, C)$ be a point on the side $AC$, and circle $\Omega$ is tangent to $BD$ at point $E$, and $AC$ at point $C$. Denote by $F(\neq E)$ the intersection of the line $AE$ and the circle $\Omega$, and $G(\neq a)$ the intersection of the line $AC$ and the circumcircle of the triangle $ABF$. Prove that points $D, E, F,$ and $G$ are concyclic.

Geometry Mathley 2011-12, 6.1

Tags: circumcircle , incenter , circumradius , excenter , distance , tangent , geometry

Show that the circumradius $R$ of a triangle $ABC$ equals the arithmetic mean of the oriented distances from its incenter $I$ and three excenters $I_a,I_b, I_c$ to any tangent $\tau$ to its circumcircle. In other words, if $\delta(P)$ denotes the distance from a point $P$ to $\tau$, then with appropriate choices of signs, we have $$\delta(I) \pm \delta_(I_a) \pm \delta_(I_b) \pm \delta_(I_c) = 4R$$ Luis González

2001 All-Russian Olympiad Regional Round, 11.3

Tags: tangent , geometry

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 $\ell$.

1985 Tournament Of Towns, (099) 3

Tags: tangent , concentric circles , area , circles , geometry

A teacher gives each student in the class the following task in their exercise book . "Take two concentric circles of radius $1$ and $10$ . To the smaller circle produce three tangents whose intersections $A, B$ and $C$ lie in the larger circle . Measure the area $S$ of triangle $ABC$, and areas $S_1 , S_2$ and $S_3$ , the three sector-like regions with vertices at $A, B$ and $C$ (as in the diagram). Find the value of $S_1 +S_2 +S_3 -S$." Prove that each student would obtain the same result . [img]https://1.bp.blogspot.com/-K3kHWWWgxgU/XWHRQ8WqqPI/AAAAAAAAKjE/0iO4-Yz6p9AcM2mklprX_M18xTyg9O81gCK4BGAYYCw/s200/TOT%2B1985%2BAutumn%2BJ3.png[/img] ( A . K . Tolpygo, Kiev)

Champions Tournament Seniors - geometry, 2002.2

Tags: tangent , geometry , secant , perpendicular

The point $P$ is outside the circle $\omega$ with center $O$. Lines $\ell_1$ and $\ell_2$ pass through a point $P$, $\ell_1$ touches the circle $\omega$ at the point $A$ and $\ell_2$ intersects $\omega$ at the points $B$ and $C$. Tangent to the circle $\omega$ at points $B$ and $C$ intersect at point $Q$. Let $K$ be the point of intersection of the lines $BC$ and $AQ$. Prove that $(OK) \perp (PQ)$.

1999 Argentina National Olympiad, 2

Tags: circles , geometry , tangent , segment

Let $C_1$ and $C_2$ be the outer circumferences of centers $O_1$ and $O_2$, respectively. The two tangents to the circumference $C_2$ are drawn by $O_1$, intersecting $C_1$ at $P$ and $P'$. The two tangents to the circumference $C_1$ are drawn by $O_2$, intersecting $C_2$ at $Q$ and $Q'$. Prove that the segment $PP'$ is equal to the segment $QQ'$.

Cono Sur Shortlist - geometry, 2021.G7

Tags: mixtilinear incircle , tangent , isosceles , geometry

Given an triangle $ABC$ isosceles at the vertex $A$, let $P$ and $Q$ be the touchpoints with $AB$ and $AC$, respectively with the circle $T$, which is tangent to both and is internally tangent to the circumcircle of $ABC$. Let $R$ and $S$ be the points of the circumscribed circle of $ABC$ such that $AP = AR = AS$ . Prove that $RS$ is tangent to $T$ .

2010 Regional Olympiad of Mexico Center Zone, 6

Tags: geometry , tangent

Let $ABC$ be an equilateral triangle and $D$ the midpoint of $BC$. Let $E$ and $F$ be points on $AC$ and $AB$ respectively such that $AF=CE$. $P=BE$ $\cap$ $CF$. Show that $\angle$$APF=$ $\angle$$BPD$

2000 Croatia National Olympiad, Problem 2

Tags: geometry , circles , tangent

Let $ABC$ be a triangle with $AB = AC$. With center in a point of the side $BC$, the circle $S$ is constructed that is tangent to the sides $AB$ and $AC$. Let $P$ and $Q$ be any points on the sides $AB$ and $AC$ respectively, such that $PQ$ is tangent to $S$. Show that $PB \cdot CQ = \left(\frac{BC}{2}\right)^2$

2006 Korea Junior Math Olympiad, 7

Tags: tangent , perpendicular , circles , geometry

A line through point $P$ outside of circle $O$ meets the said circle at $B,C$ ($PB < PC$). Let $PO$ meet circle $O$ at $Q,D$ (with $PQ < PD$). Let the line passing $Q$ and perpendicular to $BC$ meet circle $O$ at $A$. If $BD^2 = AD\cdot CP$, prove that $PA$ is a tangent to $O$.

2015 Peru IMO TST, 8

Tags: circles , inversion , tangent , geometry

Let $I$ be the incenter of the $ABC$ triangle. The circumference that passes through $I$ and has center in $A$ intersects the circumscribed circumference of the $ABC$ triangle at points $M$ and $N$. Prove that the line $MN$ is tangent to the inscribed circle of the $ABC$ triangle.

2017 BMT Spring, 13

Tags: geometric inequality , tangent , area of a triangle , geometry , area

Two points are located $10$ units apart, and a circle is drawn with radius $ r$ centered at one of the points. A tangent line to the circle is drawn from the other point. What value of $ r$ maximizes the area of the triangle formed by the two points and the point of tangency?