Olympiad problems

2023 Turkey MO (2nd round), 2

Let $ABC$ be a triangle and $P$ be an interior point. Let $\omega_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ internally and tangent to the circumcircle of $ABC$ at $A_1$ internally and let $\Gamma_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ externally and tangent to the circumcircle of $ABC$ at $A_2$ internally. Define $B_1$, $B_2$, $C_1$, $C_2$ analogously. Let $O$ be the circumcentre of $ABC$. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ and $OP$ are concurrent.

2025 Kyiv City MO Round 1, Problem 3

Tags: tangent , geometry

Point $ H $ is the orthocenter of the acute triangle $ ABC $, and $ AD $ is its altitude. Tangents are drawn from points $ B $ and $ C $ to the circle with center $ A $ and radius $ AD $, which do not coincide with the line $ BC $. These tangents intersect at point $ P $. Prove that the radius of the incircle of $ \triangle BCP $ is equal to $ HD $. [i]Proposed by Danylo Khilko[/i]

2005 Sharygin Geometry Olympiad, 10.5

Tags: circles , tangent , distance , geometry

Two circles of radius $1$ intersect at points $X, Y$, the distance between which is also equal to $1$. From point $C$ of one circle, tangents $CA, CB$ are drawn to the other. Line $CB$ will cross the first circle a second time at point $A'$. Find the distance $AA'$.

1990 IMO, 1

Tags: similar triangles , tangent , geometry , trigonometry

Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If \[ \frac {AM}{AB} \equal{} t, \] find $\frac {EG}{EF}$ in terms of $ t$.

2019 Azerbaijan BMO TST, 2

Tags: perpendicular , tangent , perpendicularity , orthocenter , perpendicular bisector , geometry , circumcircle

Let $ABC$ be a triangle inscribed in circle $\Gamma$ with center $O$. Let $H$ be the orthocenter of triangle $ABC$ and let $K$ be the midpoint of $OH$. Tangent of $\Gamma$ at $B$ intersects the perpendicular bisector of $AC$ at $L$. Tangent of $\Gamma$ at $C$ intersects the perpendicular bisector of $AB$ at $M$. Prove that $AK$ and $LM$ are perpendicular. by Michael Sarantis, Greece

2022 Mediterranean Mathematics Olympiad, 4

Tags: tangent , geometry

The triangle $ABC$ is inscribed in a circle $\gamma$ of center $O$, with $AB < AC$ . A point $D$ on the angle bisector of $\angle BAC$ and a point $E$ on segment $BC$ satisfy $OE$ is parallel to $AD$ and $DE \perp BC$. Point $K$ lies on the extension line of $EB$ such that $EA = EK$. A circle pass through points $A,K,D$ meets the extension line of $BC$ at point $P$, and meets the circle of center $O$ at point $Q\ne A$. Prove that the line $PQ$ is tangent to the circle $\gamma$.

1999 Spain Mathematical Olympiad, 1

Tags: geometry , parabola , tangent , conic , analytic geometry , area of a triangle

The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.

1989 All Soviet Union Mathematical Olympiad, 489

Tags: geometry , tangent , incircle , common tangent

The incircle of $ABC$ touches $AB$ at $M$. $N$ is any point on the segment $BC$. Show that the incircles of $AMN, BMN, ACN$ have a common tangent.

2001 Cuba MO, 4

Tags: tangent , ellipse , conic

The tangents at four different points of an arc of a circle less than $180^o$ intersect forming a convex quadrilateral $ABCD$. Prove that two of the vertices belong to an ellipse whose foci to the other two vertices.

2021 Saudi Arabia Training Tests, 19

Tags: geometry , circumcircle , tangent

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 $OD$

1956 Moscow Mathematical Olympiad, 343

Tags: tangent , tangential , parallel , concurrency

A quadrilateral is circumscribed around a circle. Prove that the straight lines connecting neighboring tangent points either meet on the extension of a diagonal of the quadrilateral or are parallel to it.

1940 Moscow Mathematical Olympiad, 057

Tags: circle construction , tangent , geometric construction , geometry , tangent circle

Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?

2006 All-Russian Olympiad Regional Round, 9.4

Tags: tangent , geometry

The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect the circumcircle of this triangle at points $A_0$ and $C_0$, respectively. A straight line passing through the center of the inscribed circle of a triangle $ABC$ is parallel to side $AC$ and intersects line $A_0C_0$ at point $P$. Prove that line $PB$ is tangent to the circumcircle of the triangle $ABC$.

2022 Singapore MO Open, Q1

Tags: symmedian , geometry , circumcircle , tangent

For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$. [i]Proposed by 61plus[/i]

2023 Yasinsky Geometry Olympiad, 5

Tags: angle bisector , tangent , geometry

The extension of the bisector of angle $A$ of triangle $ABC$ intersects with the circumscribed circle of this triangle at point $W$. A straight line is drawn through $W$, which is parallel to side $AB$ and intersects sides $BC$ and $AC$ , at points $N$ and $K$, respectively. Prove that the line $AW$ is tangent to the circumscribed circle of $\vartriangle CNW$. (Sergey Yakovlev)

2018 Dutch IMO TST, 2

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

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 $AB$.

2019 Balkan MO Shortlist, G3

Tags: tangent , circumcircle , geometry

Let $ABC$ be a scalene and acute triangle with circumcenter $O$. Let $\omega$ be the circle with center $A$, tangent to $BC$ at $D$. Suppose there are two points $F$ and $G$ on $\omega$ such that $FG \perp AO$, $\angle BFD = \angle DGC$ and the couples of points $(B,F)$ and $(C,G)$ are in different halfplanes with respect to the line $AD$. Show that the tangents to $\omega$ at $F$ and $G$ meet on the circumcircle of $ABC$.

Mexican Quarantine Mathematical Olympiad, #3

Tags: concyclic , tangent , geometry

Let $\Gamma_1$ and $\Gamma_2$ be circles intersecting at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at $C$ and $D$ respectively. Let $P$ be the intersection of the lines tangent to $\Gamma_1$ at $A$ and $C$, and let $Q$ be the intersection of the lines tangent to $\Gamma_2$ at $A$ and $D$. Let $X$ be the second intersection point of the circumcircles of $BCP$ and $BDQ$, and let $Y$ be the intersection of lines $AB$ and $PQ$. Prove that $C$, $D$, $X$ and $Y$ are concyclic. [i]Proposed by Ariel García[/i]

2008 Estonia Team Selection Test, 5

Tags: tangent , midpoint , circles , geometry

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 $BF$.

2021-IMOC, G8

Tags: tangent , geometry

Let $P$ be an arbitrary interior point of $\triangle ABC$, and $AP$, $BP$, $CP$ intersect $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Suppose that $M$ be the midpoint of $BC$, $\odot(AEF)$ and $\odot(ABC)$ intersect at $S$, $SD$ intersects $\odot(ABC)$ at $X$, and $XM$ intersects $\odot(ABC)$ at $Y$. Show that $AY$ is tangent to $\odot(AEF)$.

2017 IMO Shortlist, G2

Tags: geometry , circumcircle , tangent

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$. [i]Proposed by Charles Leytem, Luxembourg[/i]

2024 All-Russian Olympiad Regional Round, 11.7

Tags: derivative , quadratic , parabola , conic , algebra , tangent

Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: equal angles , tangent , geometry , circles

Two circles $C_1(O_1)$ and $C_2(O_2)$ with distinct radii meet at points $A$ and $B$. The tangent from $A$ to $C_1$ intersects the tangent from $B$ to $C_2$ at point $M$. Show that both circles are seen from $M$ under the same angle.

2004 German National Olympiad, 2

Tags: line , concurrency , circles , tangent , geometry

Let $k$ be a circle with center $M.$ There is another circle $k_1$ whose center $M_1$ lies on $k,$ and we denote the line through $M$ and $M_1$ by $g.$ Let $T$ be a point on $k_1$ and inside $k.$ The tangent $t$ to $k_1$ at $T$ intersects $k$ in two points $A$ and $B.$ Denote the tangents (diifferent from $t$) to $k_1$ passing through $A$ and $B$ by $a$ and $b$, respectively. Prove that the lines $a,b,$ and $g$ are either concurrent or parallel.

Novosibirsk Oral Geo Oly IX, 2020.6

Tags: equal angles , tangent , geometry

In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.