Olympiad problems

1986 Balkan MO, 4

Tags: perimeter , rectangle , conic , parallelogram , hyperbola , perpendicular bisector , geometry

Let $ABC$ a triangle and $P$ a point such that the triangles $PAB, PBC, PCA$ have the same area and the same perimeter. Prove that if: a) $P$ is in the interior of the triangle $ABC$ then $ABC$ is equilateral. b) $P$ is in the exterior of the triangle $ABC$ then $ABC$ is right angled triangle.

2015 AMC 12/AHSME, 24

Tags: pythagorean theorem , geometry , perpendicular bisector

Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$? $ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $

2013 China National Olympiad, 1

Tags: geometry , power of a point , perpendicular bisector , circumcircle

Two circles $K_1$ and $K_2$ of different radii intersect at two points $A$ and $B$, let $C$ and $D$ be two points on $K_1$ and $K_2$, respectively, such that $A$ is the midpoint of the segment $CD$. The extension of $DB$ meets $K_1$ at another point $E$, the extension of $CB$ meets $K_2$ at another point $F$. Let $l_1$ and $l_2$ be the perpendicular bisectors of $CD$ and $EF$, respectively. i) Show that $l_1$ and $l_2$ have a unique common point (denoted by $P$). ii) Prove that the lengths of $CA$, $AP$ and $PE$ are the side lengths of a right triangle.

1990 Baltic Way, 6

Tags: geometry , perpendicular bisector

Let $ABCD$ be a quadrilateral with $AD = BC$ and $\angle DAB + \angle ABC = 120^\circ$. An equilateral triangle $DPC$ is erected in the exterior of the quadrilateral. Prove that the triangle $APB$ is also equilateral.

2021 Turkey Team Selection Test, 5

Tags: menelaus theorem , geometry , perpendicular bisector

In a non isoceles triangle $ABC$, let the perpendicular bisector of $[BC]$ intersect $(ABC)$ at $M$ and $N$ respectively. Let the midpoints of $[AM]$ and $[AN]$ be $K$ and $L$ respectively. Let $(ABK)$ and $(ABL)$ intersect $AC$ again at $D$ and $E$ respectively, let $(ACK)$ and $(ACL)$ intersect $AB$ again at $F$ and $G$ respectively. Prove that the lines $DF$, $EG$ and $MN$ are concurrent.

2004 Germany Team Selection Test, 2

Tags: geometric transformation , geometry , parallelogram , circumcircle , perpendicular bisector , reflection

Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram.

2011 Iran MO (3rd Round), 2

Tags: cyclic quadrilateral , perpendicular bisector , trapezoid , geometry , circumcircle

In triangle $ABC$, $\omega$ is its circumcircle and $O$ is the center of this circle. Points $M$ and $N$ lie on sides $AB$ and $AC$ respectively. $\omega$ and the circumcircle of triangle $AMN$ intersect each other for the second time in $Q$. Let $P$ be the intersection point of $MN$ and $BC$. Prove that $PQ$ is tangent to $\omega$ iff $OM=ON$. [i]proposed by Mr.Etesami[/i]

2013 India IMO Training Camp, 2

Tags: circumcircle , cyclic quadrilateral , geometry , perpendicular bisector , trigonometry

Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$. Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively. Prove that $KL = MN$.

2019 Oral Moscow Geometry Olympiad, 4

Tags: tangent circle , geometry , perpendicular bisector , circumcircle

The perpendicular bisector of the bisector $BL$ of the triangle $ABC$ intersects the bisectors of its external angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the circle circumscribed around the triangle $PBQ$ is tangent to the circle circumscribed around the triangle $ABC$.

2001 Finnish National High School Mathematics Competition, 2

Tags: parabola , conic , perpendicular bisector , algebra , geometry

Equations of non-intersecting curves are $y = ax^2 + bx + c$ and $y = dx^2 + ex + f$ where $ad < 0.$ Prove that there is a line of the plane which does not meet either of the curves.

1995 Brazil National Olympiad, 1

Tags: geometry , rectangle , circumcircle , perpendicular bisector

$ABCD$ is a quadrilateral with a circumcircle centre $O$ and an inscribed circle centre $I$. The diagonals intersect at $S$. Show that if two of $O,I,S$ coincide, then it must be a square.

2017 IMO Shortlist, G5

Tags: perpendicular bisector , geometry

Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.

1977 AMC 12/AHSME, 5

Tags: parabola , triangle inequality , geometry , conic , inequalities , perpendicular bisector

The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is $\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$ $\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$ $\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$ $\textbf{(D) }\text{an elllipse having positive area}\qquad$ $\textbf{(E) }\text{a parabola}$

2007 India IMO Training Camp, 1

Tags: euler , perpendicular bisector , geometry , trigonometry

Show that in a non-equilateral triangle, the following statements are equivalent: $(a)$ The angles of the triangle are in arithmetic progression. $(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.

2011 Tuymaada Olympiad, 3

Tags: geometry , perpendicular bisector

In a convex hexagon $AC'BA'CB'$, every two opposite sides are equal. Let $A_1$ denote the point of intersection of $BC$ with the perpendicular bisector of $AA'$. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.

1953 AMC 12/AHSME, 37

Tags: law of sines , trig identities , trigonometry , perpendicular bisector , circumcircle , geometry

The base of an isosceles triangle is $ 6$ inches and one of the equal sides is $ 12$ inches. The radius of the circle through the vertices of the triangle is: $ \textbf{(A)}\ \frac{7\sqrt{15}}{5} \qquad\textbf{(B)}\ 4\sqrt{3} \qquad\textbf{(C)}\ 3\sqrt{5} \qquad\textbf{(D)}\ 6\sqrt{3} \qquad\textbf{(E)}\ \text{none of these}$

Indonesia MO Shortlist - geometry, g11.8

Tags: perpendicular bisector , trigonometry , angle bisector , geometry , circumcircle

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$. (a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$. (b) Show that $ A_1KML$ is a cyclic quadrilateral.

2009 Kazakhstan National Olympiad, 2

Tags: perpendicular bisector , geometry

In triangle $ABC$ $AA_1; BB_1; CC_1$-altitudes. Let $I_1$ and $I_2$ be in-centers of triangles $AC_1B_1$ and $CA_1B_1$ respectively. Let in-circle of $ABC$ touch $AC$ in $B_2$. Prove, that quadrilateral $I_1I_2B_1B_2$ inscribed in a circle.

2014 Contests, 3

Tags: angle bisector , symmetry , geometry , perpendicular bisector , incenter

Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$. Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.

2013 India Regional Mathematical Olympiad, 3

Tags: perpendicular bisector , geometry

In an acute-angled triangle $ABC$ with $AB < AC$, the circle $\omega$ touches $AB$ at $B$ and passes through $C$ intersecting $AC$ again at $D$. Prove that the orthocentre of triangle $ABD$ lies on $\omega$ if and only if it lies on the perpendicular bisector of $BC$.

2018 Danube Mathematical Competition, 3

Tags: parallel , perpendicular bisector , angle bisector , circumcircle , geometry

Let $ABC$ be an acute non isosceles triangle. The angle bisector of angle $A$ meets again the circumcircle of the triangle $ABC$ in $D$. Let $O$ be the circumcenter of the triangle $ABC$. The angle bisectors of $\angle AOB$, and $\angle AOC$ meet the circle $\gamma$ of diameter $AD$ in $P$ and $Q$ respectively. The line $PQ$ meets the perpendicular bisector of $AD$ in $R$. Prove that $AR // BC$.

2014 PUMaC Geometry B, 4

Tags: circumcircle , perpendicular bisector , trigonometry , geometry

Let $O$ be the circumcenter of triangle $ABC$ with circumradius $15$. Let $G$ be the centroid of $ABC$ and let $M$ be the midpoint of $BC$. If $BC=18$ and $\angle MOA=150^\circ$, find the area of $OMG$.

2021 Romania National Olympiad, 1

Tags: perpendicular bisector , geometry , circles

Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre. [i]Ion Pătrașcu, Ion Cotoi[/i]

2013 India IMO Training Camp, 3

Tags: circumcircle , geometry , perpendicular bisector , angle bisector

In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.

2012 Federal Competition For Advanced Students, Part 1, 4

Tags: angle bisector , geometry , circumcircle , perpendicular bisector , incenter

Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$. Show that $\gamma$ is the second-largest angle in the triangle $ABC$.