Olympiad problems

2013 USA Team Selection Test, 3

Tags: geometry , circumcircle

Let $ABC$ be a scalene triangle with $\angle BCA = 90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK = BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL = AC$. The circumcircle of triangle $DKL$ intersects segment $AB$ at a second point $T$ (other than $D$). Prove that $\angle ACT = \angle BCT$.

1993 China Team Selection Test, 3

Tags: geometry , circumcircle , incenter , trigonometry , geometric transformation , reflection , ring theory

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

2002 Silk Road, 1

Tags: circumcircle , angle bisector , geometry

Let $ \triangle ABC$ be a triangle with incircle $ \omega(I,r)$and circumcircle $ \zeta(O,R)$.Let $ l_{a}$ be the angle bisector of $ \angle BAC$.Denote $ P\equal{}l_{a}\cap\zeta$.Let $ D$ be the point of tangency $ \omega$ with $ [BC]$.Denote $ Q\equal{}PD\cap\zeta$.Show that $ PI\equal{}QI$ if $ PD\equal{}r$.

2014 Taiwan TST Round 3, 3

Tags: geometry , circumcircle , incenter , geometric transformation , homothety , ratio , parallelogram

Let $M$ be any point on the circumcircle of triangle $ABC$. Suppose the tangents from $M$ to the incircle meet $BC$ at two points $X_1$ and $X_2$. Prove that the circumcircle of triangle $MX_1X_2$ intersects the circumcircle of $ABC$ again at the tangency point of the $A$-mixtilinear incircle.

1974 IMO, 2

Tags: trigonometry , circumcircle , geometry , triangle , trigonometric inequality

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]

2018 India Regional Mathematical Olympiad, 1

Tags: greatest common divisor , geometry , circumcircle

Let $ABC$ be a triangle with integer sides in which $AB<AC$. Let the tangent to the circumcircle of triangle $ABC$ at $A$ intersect the line $BC$ at $D$. Suppose $AD$ is also an integer. Prove that $\gcd(AB,AC)>1$.

Geometry Mathley 2011-12, 15.3

Tags: geometry , circumcircle , symmedian , perpendicular bisector

Triangle $ABC$ has circumcircle $(O,R)$, and orthocenter $H$. The symmedians through $A,B,C$ meet the perpendicular bisectors of $BC,CA,AB$ at $D,E, F$ respectively. Let $M,N, P$ be the perpendicular projections of H on the line $OD,OE,OF.$ Prove that $$\frac{OH^2}{R^2} =\frac{\overline{OM}}{\overline{OD}}+\frac{\overline{ON}}{\overline{OE}} +\frac{\overline{OP}}{\overline{OF}}$$ Đỗ Thanh Sơn

1985 IMO Longlists, 56

Tags: function , geometry , rhombus , circumcircle

Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$

2009 Belarus Team Selection Test, 2

Tags: circumcircle , homothety , trigonometry , quadrilateral , inversion , geometry

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

2000 Romania Team Selection Test, 2

Tags: geometry , circumcircle , perimeter , geometric inequality , triangle

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

2012 Grigore Moisil Intercounty, 3

Tags: circumcircle , geometric transformation , rotation , geometry

Let $ \Delta ABC$ be a triangle, with $ m(\angle A)=90^{\circ}$ and $ m(\angle B)=30^{\circ}.$ If $M$ is the middle of $[AB],$ $N$ is the middle of $[BC],$ and $P\in[BC],\ Q\in[MN],$ such that \[\frac{PB}{PC}=4\cdot\frac{QM}{QN}+3,\] prove that $ \Delta APQ$ is an equilateral triangle. [b]Author: MARIN BANCOȘ[/b] [b]Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 24.03.2012, 7th grade[/b]

2012 ELMO Shortlist, 2

Tags: circumcircle , power of a point , radical axis , cyclic quadrilateral , geometry

In triangle $ABC$, $P$ is a point on altitude $AD$. $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$, and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$. a) Prove $SX,TY, AD$ are concurrent at a point $Z$. b) Prove $Z$ is on $QR$ iff $Z=H$, where $H$ is the orthocenter of $ABC$. [i]Ray Li.[/i]

2012 Belarus Team Selection Test, 2

Tags: geometry , circumcircle , algebra , polynomial , quadratic , complex number

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

1987 IMO Longlists, 31

Tags: circumcircle , geometry

Construct a triangle $ABC$ given its side $a = BC$, its circumradius $R \ (2R \geq a)$, and the difference $\frac{1}{k} = \frac{1}{c}-\frac{1}{b}$, where $c = AB$ and $ b = AC.$

2004 India IMO Training Camp, 1

Tags: circumcircle , circumcenter , triangle , geometry , excenter

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

Durer Math Competition CD 1st Round - geometry, 2015.D4

Tags: geometry , circumcircle , altitude

The altitude of the acute triangle $ABC$ drawn from $A$ , intersects the side $BC$ at $A_1$ and the circumscribed circle at $A_2$ (different from $A$). Similarly, we get the points $B_1$, $B_2$, $C_1$, $C_2$. Prove that $$\frac{AA_2}{AA_1}+\frac{BB_2}{BB_1}+\frac{CC_2}{CC_1}= 4.$$

2005 District Olympiad, 4

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

In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.

2012 Polish MO Finals, 5

Tags: geometry , tangent , triangle , circumcircle

Point $O$ is a center of circumcircle of acute triangle $ABC$, bisector of angle $BAC$ cuts side $BC$ in point $D$. Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$. Lines $BM$ and $OA$ intersect in point $P$. Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$.

1967 IMO Shortlist, 2

Tags: geometry , 3d geometry , sphere , tetrahedron , circumcircle

Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.

2015 Saint Petersburg Mathematical Olympiad, 7

Tags: geometry , circumcircle , angle bisector

Let $BL$ be angle bisector of acute triangle $ABC$.Point $K$ choosen on $BL$ such that $\measuredangle AKC-\measuredangle ABC=90º$.point $S$ lies on the extention of $BL$ from $L$ such that $\measuredangle ASC=90º$.Point $T$ is diametrically opposite the point $K$ on the circumcircle of $\triangle AKC$.Prove that $ST$ passes through midpoint of arc $ABC$.(S. Berlov) [hide] :trampoline: my 100th post :trampoline: [/hide]

2010 Romania Team Selection Test, 3

Tags: circumcircle , power of a point , radical axis , geometry

Let $\gamma_1$ and $\gamma_2$ be two circles tangent at point $T$, and let $\ell_1$ and $\ell_2$ be two lines through $T$. The lines $\ell_1$ and $\ell_2$ meet again $\gamma_1$ at points $A$ and $B$, respectively, and $\gamma_2$ at points $A_1$ and $B_1$, respectively. Let further $X$ be a point in the complement of $\gamma_1 \cup \gamma_2 \cup \ell_1 \cup \ell_2$. The circles $ATX$ and $BTX$ meet again $\gamma_2$ at points $A_2$ and $B_2$, respectively. Prove that the lines $TX$, $A_1B_2$ and $A_2B_1$ are concurrent. [i]***[/i]

2016 Latvia National Olympiad, 2

Tags: geometry , circumcircle

The bisectors of the angles $\sphericalangle CAB$ and $\sphericalangle BCA$ intersect the circumcircle of $ABC$ in $P$ and $Q$ respectively. These bisectors intersect each other in point $I$. Prove that $PQ \perp BI$.

2008 China Team Selection Test, 1

Tags: circumcircle , trigonometry , ratio , cyclic quadrilateral , geometry , inequalities

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

2007 International Zhautykov Olympiad, 3

Tags: geometry , circumcircle , trapezoid , trigonometry , geometric transformation , homothety , ratio

Let $ABCDEF$ be a convex hexagon and it`s diagonals have one common point $M$. It is known that the circumcenters of triangles $MAB,MBC,MCD,MDE,MEF,MFA$ lie on a circle. Show that the quadrilaterals $ABDE,BCEF,CDFA$ have equal areas.

2021 Moldova Team Selection Test, 3

Tags: geometry , circumcircle

Acute triangle $ABC$ with $AB>BC$ is inscribed in circle $\Omega$. Points $D$ and $E$, that lie on $(BC)$ and $(AB)$ are the feet of altitudes from $A$ and $C$ in triangle $ABC$, and $M$ is the midpoint of the segment $DE$. Half-line $(AM$ intersects the circle $\Omega$ for the second time in $N$. Show that the circumcenter of triangle $MDN$ lies on the line $BC$.