Olympiad problems

Ukrainian From Tasks to Tasks - geometry, 2016.3

In fig. the bisectors of the angles $\angle DAC$, $ \angle EBD$, $\angle ACE$, $\angle BDA$ and $\angle CEB$ intersect at one point. Prove that the bisectors of the angles $\angle TPQ$, $\angle PQR$, $\angle QRS$, $\angle RST$ and $\angle STP$ also intersect at one point. [img]https://cdn.artofproblemsolving.com/attachments/6/e/870e4f20bc7fdcb37534f04541c45b1cd5034a.png[/img]

2022 Durer Math Competition Finals, 4

Tags: geometry , cyclic , concurrency

$ABCD$ is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let $O$ denote the centre of its circumcircle and $E$ the intersection of the diagonals. $J$ and $K$ denote the perpendicular projections of $E$ on the sides $AB$ and $BC$ . Let $F , G$ and $H$ be the midpoint line segments. Show that lines $GJ$ , $FB$ and $HK$ either pass through the same point or are parallel to each other.

2009 Switzerland - Final Round, 7

Tags: geometry , common tangent , circles , concurrency

Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.

2024 Israel National Olympiad (Gillis), P4

Tags: geometry , geometric transformation , triangle , circumcenter , reflection , concurrency

Acute triangle $ABC$ is inscribed in a circle with center $O$. The reflections of $O$ across the three altitudes of the triangle are called $U$, $V$, $W$: $U$ over the altitude from $A$, $V$ over the altitude from $B$, and $W$ over the altitude from $C$. Let $\ell_A$ be a line through $A$ parallel to $VW$, and define $\ell_B$, $\ell_C$ similarly. Prove that the three lines $\ell_A$, $\ell_B$, $\ell_C$ are concurrent.

KoMaL A Problems 2019/2020, A. 779

Tags: geometry , circles , incenter , tangent , concurrency , reflection

Two circles are given in the plane, $\Omega$ and inside it $\omega$. The center of $\omega$ is $I$. $P$ is a point moving on $\Omega$. The second intersection of the tangents from $P$ to $\omega$ and circle $\Omega$ are $Q$ and $R.$ The second intersection of circle $IQR$ and lines $PI$, $PQ$ and $PR$ are $J$, $S$ and $T,$ respectively. The reflection of point $J$ across line $ST$ is $K.$ Prove that lines $PK$ are concurrent.

1982 IMO, 2

Tags: geometry , triangle , midpoint , incircle , concurrency

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

2005 Sharygin Geometry Olympiad, 18

Tags: symmetry , geometry , line , circumcenter , concurrency

On the plane are three straight lines $\ell_1, \ell_2,\ell_3$, forming a triangle, and the point $O$ is marked, the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by $X_i$ the point symmetric to the point X with respect to the line $\ell_i, i = 1,2,3$. a) Prove that for an arbitrary point $M$ the straight lines connecting the midpoints of the segments $O_1O_2$ and $M_1M_2, O_2O_3$ and $M_2M_3, O_3O_1$ and $M_3M_1$ intersect at one point, b) where can this intersection point lie?

1929 Eotvos Mathematical Competition, 3

Tags: geometry , geometric inequalities , concurrency

Let $p, q$ and $r$ be three concurrent lines in the plane such that the angle between any two of them is $60^o$. Let $a$, $b$ and $c$ be real numbers such that $0 < a \le b \le c$. (a) Prove that the set of points whose distances from $p, q$ and $r$ are respectively less than $a, b$ and $c$ consists of the interior of a hexagon if and only if $a + b > c$. (b) Determine the length of the perimeter of this hexagon when $a + b > c$.

2021 IMO, 3

Tags: geometry , concurrency , circumcenter , moving points

Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.

2009 Federal Competition For Advanced Students, P1, 4

Tags: geometry , concurrency , midpoint , altitude

Let $D, E$, and $F$ be respectively the midpoints of the sides $BC, CA$, and $AB$ of $\vartriangle ABC$. Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$, and $H_aH_b$ respectively. Prove that $PD, QE$, and $RF$ are concurrent.

2016 Thailand TSTST, 2

Tags: geometry , tangent circle , circumcenter , concurrency

Let $\omega$ be a circle touching two parallel lines $\ell_1, \ell_2$, $\omega_1$ a circle touching $\ell_1$ at $A$ and $\omega$ externally at $C$, and $\omega_2$ a circle touching $\ell_2$ at $B$, $\omega$ externally at $D$, and $\omega_1$ externally at $E$. Prove that $AD, BC$ intersect at the circumcenter of $\vartriangle CDE$.

2002 Kazakhstan National Olympiad, 5

Tags: geometry , circumcircle , concurrency

On the plane is given the acute triangle $ ABC $. Let $ A_1 $ and $ B_1 $ be the feet of the altitudes of $ A $ and $ B $ drawn from those vertices, respectively. Tangents at points $ A_1 $ and $ B_1 $ drawn to the circumscribed circle of the triangle $ CA_1B_1 $ intersect at $ M $. Prove that the circles circumscribed around the triangles $ AMB_1 $, $ BMA_1 $ and $ CA_1B_1 $ have a common point.

1980 Dutch Mathematical Olympiad, 3

Tags: geometry , concurrency

Given is the non-right triangle $ABC$. $D,E$ and $F$ are the feet of the respective altitudes from $A,B$ and $C$. $P,Q$ and $R$ are the respective midpoints of the line segments $EF$, $FD$ and $DE$. $p \perp BC$ passes through $P$, $q \perp CA$ passes through $Q$ and $r \perp AB$ passes through $R$. Prove that the lines $p, q$ and $r$ pass through one point.

1954 Moscow Mathematical Olympiad, 279

Tags: geometry , concurrency , geometric inequality

Given four straight lines, $m_1, m_2, m_3, m_4$, intersecting at $O$ and numbered clockwise with $O$ as the center of the clock, we draw a line through an arbitrary point $A_1$ on $m_1$ parallel to $m_4$ until the line meets $m_2$ at $A_2$. We draw a line through $A_2$ parallel to $m_1$ until it meets $m_3$ at $A_3$. We also draw a line through $A_3$ parallel to $m_2$ until it meets $m_4$ at $A_4$. Now, we draw a line through$ A_4$ parallel to $m_3$ until it meets $m_1$ at $B$. Prove that a) $OB< \frac{OA_1}{2}$ . b) $OB \le \frac{OA_1}{4}$ . [img]https://cdn.artofproblemsolving.com/attachments/5/f/5ea08453605e02e7e1253fd7c74065a9ffbd8e.png[/img]

Ukrainian TYM Qualifying - geometry, XI.6

Tags: geometry , concurrency , equal segments , centroid

Prove that there exists a point $K$ in the plane of $\vartriangle ABC$ such that $$AK^2 - BC^2 = BK^2 - AC^2 = CK^2 - AB^2.$$ Let $Q, N, T$ be the points of intersection of the medians of the triangles $BKC, CKA, AKB$, respectively. Prove that the segments $AQ, BN$ and $CT$ are equal and have a common point.

Cono Sur Shortlist - geometry, 2018.G6

Tags: geometry , circumcircle , concurrency , tangent circle , orthocenter

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_A$ passes through the points $A$ and $H$ and is tangent to the circumcircle of the triangle $ABC$. Similarly, define the points $X_B$ and $X_C$. Let $O_A$, $O_B$ and $O_C$ be the reflections of $O$ with respect to sides $BC$, $CA$ and $AB$, respectively. Prove that the lines $O_AX_A$, $O_BX_B$ and $O_CX_C$ are concurrent.

2021 Sharygin Geometry Olympiad, 10-11.1

Tags: concurrency , geometry , right triangle

.Let $CH$ be an altitude of right-angled triangle $ABC$ ($\angle C = 90^o$), $HA_1$, $HB_1$ be the bisectors of angles $CHB$, $AHC$ respectively, and $E, F$ be the midpoints of $HB_1$ and $HA_1$ respectively. Prove that the lines $AE$ and $BF$ meet on the bisector of angle $ACB$.

1956 Moscow Mathematical Olympiad, 333

Tags: geometry , concurrency , altitude , circumcenter , circumcircle

Let $O$ be the center of the circle circumscribed around $\vartriangle ABC$, let $A_1, B_1, C_1$ be symmetric to $O$ through respective sides of $\vartriangle ABC$. Prove that all altitudes of $\vartriangle A_1B_1C_1$ pass through $O$, and all altitudes of $\vartriangle ABC$ pass through the center of the circle circumscribed around $\vartriangle A_1B_1C_1$.

1951 Kurschak Competition, 1

Tags: geometry , concurrency , square

$ABCD$ is a square. $E$ is a point on the side $BC$ such that $BE =1/3 BC$, and $F$ is a point on the ray $DC$ such that $CF =1/2 DC$. Prove that the lines $AE$ and $BF$ intersect on the circumcircle of the square. [img]https://cdn.artofproblemsolving.com/attachments/e/d/09a8235d0748ce4479e21a3bb09b0359de54b5.png[/img]

2022 Austrian MO Regional Competition, 3

Tags: geometry , incenter , concurrency

Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$. [i](Karl Czakler)[/i]

2008 Austria Beginners' Competition, 4

Tags: concurrency , angle , geometry

Let $ABC$ be an acute-angled triangle with the property that the bisector of $\angle BAC$, the altitude through $B$ and the perpendicular bisector of $AB$ intersect in one point. Determine the angle $\alpha = \angle BAC$.

2015 NZMOC Camp Selection Problems, 3

Tags: circumcircle , geometry , concurrency , arc

Let $ABC$ be an acute angled triangle. The arc between $A$ and $B$ of the circumcircle of $ABC$ is reflected through the line $AB$, and the arc between $A$ and $C$ of the circumcircle of $ABC$ is reflected over the line $AC$. Obviously these two reflected arcs intersect at the point $A$. Prove that they also intersect at another point inside the triangle $ABC$.

Champions Tournament Seniors - geometry, 2004.2

Tags: geometry , concurrency , circles

Two different circles $\omega_1$ ,$\omega_2$, with centers $O_1, O_2$ respectively intersect at the points $A, B$. The line $O_1B$ intersects $\omega_2$ at the point $F (F \ne B)$, and the line $O_2B$ intersects $\omega_1$ at the point $E (E\ne B)$. A line was drawn through the point $B$, parallel to the $EF$, which intersects $\omega_1$ at the point $M (M \ne B)$, and $\omega_2$ at the point $N (N\ne B)$. Prove that the lines $ME, AB$ and $NF$ intersect at one point.

2023 Indonesia TST, G

Tags: geometry , circumcircle , concurrency

Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$, $M$ is the midpoint of $AB$. A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$

2015 Sharygin Geometry Olympiad, 5

Tags: geometry , midpoint , concurrency

Let $BM$ be a median of nonisosceles right-angled triangle $ABC$ ($\angle B = 90^o$), and $Ha, Hc$ be the orthocenters of triangles $ABM, CBM$ respectively. Prove that lines $AH_c$ and $CH_a$ meet on the medial line of triangle $ABC$. (D. Svhetsov)