Olympiad problems

2004 Junior Balkan Team Selection Tests - Romania, 2

Tags: trapezoid , incenter , rectangle , trigonometry , angle bisector , perpendicular bisector , geometry

Let $ABC$ be an isosceles triangle with $AB=AC$. Consider a variable point $P$ on the extension of the segment $BC$ beyound $B$ (in other words, $P$ lies on the line $BC$ such that the point $B$ lies inside the segment $PC$). Let $r_{1}$ be the radius of the incircle of the triangle $APB$, and let $r_{2}$ be the radius of the $P$-excircle of the triangle $APC$. Prove that the sum $r_{1}+r_{2}$ of these two radii remains constant when the point $P$ varies. [i]Remark.[/i] The $P$-excircle of the triangle $APC$ is defined as the circle which touches the side $AC$ and the [i]extensions[/i] of the sides $AP$ and $CP$.

2012 Uzbekistan National Olympiad, 5

Tags: incenter , geometric transformation , homothety , perpendicular bisector , geometry

Given points $A,B,C$ and $D$ lie a circle. $AC\cap BD=K$. $I_1, I_2,I_3$ and $I_4$ incenters of $ABK,BCK,CDK,DKA$. $M_1,M_2,M_3,M_4$ midpoints of arcs $AB,BC,CA,DA$ . Then prove that $M_1I_1,M_2I_2,M_3I_3,M_4I_4$ are concurrent.

2010 Sharygin Geometry Olympiad, 6

Tags: circumcircle , perpendicular bisector , geometry

Points $M$ and $N$ lie on the side $BC$ of the regular triangle $ABC$ ($M$ is between $B$ and $N$), and $\angle MAN=30^\circ.$ The circumcircles of triangles $AMC$ and $ANB$ meet at a point $K.$ Prove that the line $AK$ passes through the circumcenter of triangle $AMN.$

2009 Sharygin Geometry Olympiad, 10

Tags: circumcircle , trapezoid , perpendicular bisector , geometry

Let $ ABC$ be an acute triangle, $ CC_1$ its bisector, $ O$ its circumcenter. The perpendicular from $ C$ to $ AB$ meets line $ OC_1$ in a point lying on the circumcircle of $ AOB$. Determine angle $ C$.

2010 AMC 10, 19

Tags: geometry , trigonometry , quadratic , pythagorean theorem , trig identities , law of cosines , perpendicular bisector

A circle with center $ O$ has area $ 156\pi$. Triangle $ ABC$ is equilateral, $ \overline{BC}$ is a chord on the circle, $ OA \equal{} 4\sqrt3$, and point $ O$ is outside $ \triangle ABC$. What is the side length of $ \triangle ABC$? $ \textbf{(A)}\ 2\sqrt3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 4\sqrt3 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18$

2014 Contests, 3

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

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.

2001 Iran MO (2nd round), 2

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

In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?

2013 National Olympiad First Round, 13

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

Let $D$ and $E$ be points on side $[BC]$ of a triangle $ABC$ with circumcenter $O$ such that $D$ is between $B$ and $E$, $|AD|=|DB|=6$, and $|AE|=|EC|=8$. If $I$ is the incenter of triangle $ADE$ and $|AI|=5$, then what is $|IO|$? $ \textbf{(A)}\ \dfrac {29}{5} \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ \dfrac {23}{5} \qquad\textbf{(D)}\ \dfrac {21}{5} \qquad\textbf{(E)}\ \text{None of above} $

1989 India National Olympiad, 6

Tags: incenter , angle bisector , perpendicular bisector , geometry

Triangle $ ABC$ has incentre $ I$ and the incircle touches $ BC, CA$ at $ D, E$ respectively. Let $ BI$ meet $ DE$ at $ G$. Show that $ AG$ is perpendicular to $ BG$.

2007 All-Russian Olympiad, 6

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

Let $ABC$ be an acute triangle. The points $M$ and $N$ are midpoints of $AB$ and $BC$ respectively, and $BH$ is an altitude of $ABC$. The circumcircles of $AHN$ and $CHM$ meet in $P$ where $P\ne H$. Prove that $PH$ passes through the midpoint of $MN$. [i]V. Filimonov[/i]

1959 IMO Shortlist, 5

Tags: geometry , circumcircle , perpendicular bisector , angle bisector

An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$. a) Prove that $N$ and $N'$ coincide; b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$; c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.

2014 PUMaC Geometry A, 4

Tags: geometric transformation , reflection , trigonometry , cyclic quadrilateral , perpendicular bisector , trig identities , geometry

Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.

2010 Stanford Mathematics Tournament, 17

Tags: geometry , perpendicular bisector

An equilateral triangle is inscribed inside of a circle of radius $R$. Find the side length of the triangle

2004 Postal Coaching, 10

Tags: geometric transformation , reflection , perpendicular bisector , angle bisector , geometry

A convex quadrilateral $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Show that at least two circles have the same size.

2014 Saudi Arabia IMO TST, 4

Tags: geometry , circumcircle , incenter , geometric transformation , reflection , perpendicular bisector , angle bisector

Points $A_1,~ B_1,~ C_1$ lie on the sides $BC,~ AC$ and $AB$ of a triangle $ABC$, respectively, such that $AB_1 -AC_1 = CA_1 -CB_1 = BC_1 -BA_1$. Let $I_A,~ I_B,~ I_C$ be the incenters of triangles $AB_1C_1,~ A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcenter of triangle $I_AI_BI_C$, is the incenter of triangle $ABC$.

2011 Turkey MO (2nd round), 2

Tags: circumcircle , trigonometry , geometric transformation , reflection , perpendicular bisector , cyclic quadrilateral , geometry

Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.

1996 Brazil National Olympiad, 4

Tags: geometry , circumcircle , vector , ratio , trigonometry , perpendicular bisector , geometric transformation

$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.

2014 India Regional Mathematical Olympiad, 1

Tags: circumcircle , incenter , perpendicular bisector , geometry

In an acute-angled triangle $ABC, \angle ABC$ is the largest angle. The perpendicular bisectors of $BC$ and $BA$ intersect AC at $X$ and $Y$ respectively. Prove that circumcentre of triangle $ABC$ is incentre of triangle $BXY$ .

2011 Tokio University Entry Examination, 4

Tags: analytic geometry , graphing lines , slope , perpendicular bisector , geometry

Take a point $P\left(\frac 12,\ \frac 14\right)$ on the coordinate plane. Let two points $Q(\alpha ,\ \alpha ^ 2),\ R(\beta ,\ \beta ^2)$ move in such a way that 3 points $P,\ Q,\ R$ form an isosceles triangle with the base $QR$, find the locus of the barycenter $G(X,\ Y)$ of $\triangle{PQR}$. [i]2011 Tokyo University entrance exam[/i]

1995 Brazil National Olympiad, 1

Tags: geometry , circumcircle , rectangle , 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.

2014 Contests, 1

Tags: geometry , circumcircle , perpendicular bisector

In the figure of [url]http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view[/url] $\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$. The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$, and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$. If $DE \parallel O_1A$, prove that $DC \perp CO_2$.

1991 Hungary-Israel Binational, 2

Tags: inradius , perimeter , perpendicular bisector , geometry

The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.

2013 Czech-Polish-Slovak Junior Match, 5

Tags: geometry , perpendicular bisector , midpoint , circumcenter

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Point $P$ lies on the segment $AB$, and points $S_1$ and $S_2$ are the centers of the circumcircles of $APC$ and $BPC$, respectively. Show that the midpoint of segment $S_1S_2$ lies on the perpendicular bisector of segment $CM$.

2025 Poland - First Round, 7

Tags: geometry , perpendicular bisector

Circles $o_1, o_2$ with equal radii intersect at points $A, B$. Points $C, D, E, F$ lie in this order on one line, with $C, E$ lying on $o_1$ and $D, F$ on $o_2$. Perpendicular bisectors of $CD$ and $EF$ intersect $AB$ at $X, Y$ respectively. Prove that $AX=BY$.

2011 China Team Selection Test, 1

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

Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.