Olympiad problems

2009 Costa Rica - Final Round, 6

Let $ \Delta ABC$ with incircle $ \Gamma$, let $ D, E$ and $ F$ the tangency points of $ \Gamma$ with sides $ BC, AC$ and $ AB$, respectively and let $ P$ the intersection point of $ AD$ with $ \Gamma$. $ a)$ Prove that $ BC, EF$ and the straight line tangent to $ \Gamma$ for $ P$ concur at a point $ A'$. $ b)$ Define $ B'$ and $ C'$ in an anologous way than $ A'$. Prove that $ A'\minus{}B'\minus{}C'$

2008 Singapore Team Selection Test, 1

Tags: trigonometry , projective geometry , geometry

Let $(O)$ be a circle, and let $ABP$ be a line segment such that $A,B$ lie on $(O)$ and $P$ is a point outside $(O)$. Let $C$ be a point on $(O)$ such that $PC$ is tangent to $(O)$ and let $D$ be the point on $(O)$ such that $CD$ is a diameter of $(O)$ and intersects $AB$ inside $(O)$. Suppose that the lines $DB$ and $OP$ intersect at $E$. Prove that $AC$ is perpendicular to $CE$.

2014 Contests, 1

Tags: circumcircle , projective geometry , geometry

Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $BC$ intersects $AC$ at $X$ and $AB$ at $Y$. Also, $MX$ and $MY$ intersect $\Gamma$ again at $S$ and $T$, respectively. If $XY$ and $ST$ intersect at $P$, prove that $PA$ is tangent to $\Gamma$.

2008 Iran MO (3rd Round), 3

Tags: projective geometry , geometry

Let $ ABCD$ be a quadrilateral, and $ E$ be intersection points of $ AB,CD$ and $ AD,BC$ respectively. External bisectors of $ DAB$ and $ DCB$ intersect at $ P$, external bisectors of $ ABC$ and $ ADC$ intersect at $ Q$ and external bisectors of $ AED$ and $ AFB$ intersect at $ R$. Prove that $ P,Q,R$ are collinear.

2014 ELMO Shortlist, 12

Tags: circumcircle , ratio , symmetry , projective geometry , trigonometry , geometry

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

2014 Romania Team Selection Test, 1

Tags: geometry , circumcircle , projective geometry , trigonometry

Let $ABC$ a triangle and $O$ his circumcentre.The lines $OA$ and $BC$ intersect each other at $M$ ; the points $N$ and $P$ are defined in an analogous way.The tangent line in $A$ at the circumcircle of triangle $ABC$ intersect $NP$ in the point $X$ ; the points $Y$ and $Z$ are defined in an analogous way.Prove that the points $X$ , $Y$ and $Z$ are collinear.

Croatia MO (HMO) - geometry, 2011.7

Tags: circumcircle , perpendicular bisector , projective geometry , power of a point , radical axis , geometry

Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.

2022 Brazil Team Selection Test, 4

Tags: circumcircle , projective geometry , geometry

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

2006 Putnam, A1

Tags: analytic geometry , calculus , integration , trigonometry , geometry , projective geometry

Find the volume of the region of points $(x,y,z)$ such that \[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]

2012 ELMO Shortlist, 10

Tags: geometry , circumcircle , projective geometry , algebra , conic

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic. [i]David Yang.[/i]

2024 Brazil EGMO TST, 4

Tags: geometry , bicentric quadrilateral , tangent , projective geometry

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 $AIC$.

2005 China Team Selection Test, 1

Tags: geometry , projective geometry , concurrency

Point $P$ lies inside triangle $ABC$. Let the projections of $P$ onto sides $BC$,$CA$,$AB$ be $D$, $E$, $F$ respectively. Let the projections from $A$ to the lines $BP$ and $CP$ be $M$ and $N$ respectively. Prove that $ME$, $NF$ and $BC$ are concurrent.

2013 ELMO Shortlist, 10

Tags: circumcircle , ratio , symmetry , projective geometry , trigonometry , geometry

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

2005 IMO Shortlist, 6

Tags: geometry , homothety , reflection , trapezoid , projective geometry , polars

Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$. Let $\gamma$ be the incircle of triangle $ABC$. The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$. Let the lines passing through $K$ and $L$, parallel to $BC$, intersect the incircle $\gamma$ again in two points $X$ and $Y$. Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$. Prove that $BP = CQ$.

2002 Pan African, 5

Tags: geometry , circumcircle , projective geometry

Let $\triangle{ABC}$ be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P. Show that P lies on the altitude through the vertex C.

1991 Arnold's Trivium, 45

Tags: algebra , polynomial , projective geometry

Find the self-intersection index of the surface $x^4+y^4=1$ in the projective plane $\text{CP}^2$.

2012 Indonesia TST, 3

Tags: circumcircle , incenter , cyclic quadrilateral , projective geometry , geometry

Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.

2007 India IMO Training Camp, 1

Tags: ratio , geometry , projective geometry , homothety

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

2010 Germany Team Selection Test, 2

Tags: geometry , cyclic quadrilateral , projective geometry , tangent , power of a point

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]

2010 Belarus Team Selection Test, 6.2

Tags: geometry , cyclic quadrilateral , projective geometry , tangent , power of a point

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]

2012 USA Team Selection Test, 2

Tags: trigonometry , reflection , cyclic quadrilateral , angle bisector , projective geometry , geometry

In cyclic quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $P$. Let $E$ and $F$ be the respective feet of the perpendiculars from $P$ to lines $AB$ and $CD$. Segments $BF$ and $CE$ meet at $Q$. Prove that lines $PQ$ and $EF$ are perpendicular to each other.

2023 Brazil Team Selection Test, 4

Tags: hyperbola , geometry , projective geometry , conic

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

2008 Philippine MO, 3

Tags: projective geometry , circumcircle , geometry

Let $P$ be a point outside a circle $\Gamma$, and let the two tangent lines through $P$ touch $\Gamma$ at $A$ and $B$. Let $C$ be on the minor arc $AB$, and let ray $PC$ intersect $\Gamma$ again at $D$. Let $\ell$ be the line through $B$ and parallel to $PA$. $\ell$ intersects $AC$ and $AD$ at $E$ and $F$, respectively. Prove that $B$ is the midpoint of $EF$.

2012 Sharygin Geometry Olympiad, 19

Tags: geometry , 3d geometry , prism , projective geometry , incenter , geometric transformation , dilation

Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.

Cono Sur Shortlist - geometry, 2012.G4.2

Tags: rhombus , circumcircle , projective geometry , geometry

2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.