Olympiad problems

2005 Junior Balkan MO, 2

Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$. Prove that the lines $AP$ and $CS$ are parallel.

2009 Olympic Revenge, 3

Tags: geometry , incenter , geometry proposed

Let $ABC$ to be a triangle with incenter $I$. $\omega_{A}$, $\omega_{B}$ and $\omega_{C}$ are the incircles of the triangles $BIC$, $CIA$ and $AIB$, repectively. After all, $T$ is the tangent point between $\omega_{A}$ and $BC$. Prove that the other internal common tangent to $\omega_{B}$ and $\omega_{C}$ passes through the point $T$.

2014 Korea - Final Round, 2

Tags: geometry , circumcircle , perpendicular bisector , geometry proposed

Let $ABC$ be a isosceles triangle with $ AC = BC > AB$. Let $ E, F $ be the midpoints of segments $ AC, AB$, and let $l$ be the perpendicular bisector of $AC$. Let $ l $ meets $ AB$ at $K$, the line through $B$ parallel to $KC$ meets $AC$ at point $L$, and line $FL$ meets $ l$ at $W$. Let $ P $ be a point on segment $BF$. Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$. Line $FJ$ meets $l$ at $M$. Prove that $ AW = PW $ if and only if $B$ lies on the circumcircle of $EFM$.

1980 Canada National Olympiad, 3

Tags: geometry , perimeter , inradius , geometry proposed

Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.

1988 IberoAmerican, 1

Tags: geometry , trigonometry , arithmetic sequence , geometry proposed

The measure of the angles of a triangle are in arithmetic progression and the lengths of its altitudes are as well. Show that such a triangle is equilateral.

1977 IMO Longlists, 9

Tags: geometry , 3D geometry , tetrahedron , geometry proposed

Let $ABCD$ be a regular tetrahedron and $\mathbf{Z}$ an isometry mapping $A,B,C,D$ into $B,C,D,A$, respectively. Find the set $M$ of all points $X$ of the face $ABC$ whose distance from $\mathbf{Z}(X)$ is equal to a given number $t$. Find necessary and sufficient conditions for the set $M$ to be nonempty.

2002 Baltic Way, 14

Tags: geometry proposed , geometry

Let $L,M$ and $N$ be points on sides $AC,AB$ and $BC$ of triangle $ABC$, respectively, such that $BL$ is the bisector of angle $ABC$ and segments $AN,BL$ and $CM$ have a common point. Prove that if $\angle ALB=\angle MNB$ then $\angle LNM=90^{\circ}$.

2024 Austrian MO National Competition, 2

Tags: geometry , geometry proposed , semicircle , touching circles , cyclic quadrilateral

Let $h$ be a semicircle with diameter $AB$. The two circles $k_1$ and $k_2$, $k_1 \ne k_2$, touch the segment $AB$ at the points $C$ and $D$, respectively, and the semicircle $h$ fom the inside at the points $E$ and $F$, respectively. Prove that the four points $C$, $D$, $E$ and $F$ lie on a circle. [i](Walther Janous)[/i]

2020 Taiwan TST Round 1, 5

Tags: geometry proposed , geometry

Let $O$ be the center of the equilateral triangle $ABC$. Pick two points $P_1$ and $P_2$ other than $B$, $O$, $C$ on the circle $\odot(BOC)$ so that on this circle $B$, $P_1$, $P_2$, $O$, $C$ are placed in this order. Extensions of $BP_1$ and $CP_1$ intersects respectively with side $CA$ and $AB$ at points $R$ and $S$. Line $AP_1$ and $RS$ intersects at point $Q_1$. Analogously point $Q_2$ is defined. Let $\odot(OP_1Q_1)$ and $\odot(OP_2Q_2)$ meet again at point $U$ other than $O$. Prove that $2\,\angle Q_2UQ_1 + \angle Q_2OQ_1 = 360^\circ$. Remark. $\odot(XYZ)$ denotes the circumcircle of triangle $XYZ$.

2009 Middle European Mathematical Olympiad, 10

Tags: geometry , incenter , number theory , greatest common divisor , cyclic quadrilateral , geometry proposed

Suppose that $ ABCD$ is a cyclic quadrilateral and $ CD\equal{}DA$. Points $ E$ and $ F$ belong to the segments $ AB$ and $ BC$ respectively, and $ \angle ADC\equal{}2\angle EDF$. Segments $ DK$ and $ DM$ are height and median of triangle $ DEF$, respectively. $ L$ is the point symmetric to $ K$ with respect to $ M$. Prove that the lines $ DM$ and $ BL$ are parallel.

1976 Canada National Olympiad, 4

Tags: geometry , geometric transformation , homothety , geometry proposed

Let $ AB$ be a diameter of a circle, $ C$ be any fixed point between $ A$ and $ B$ on this diameter, and $ Q$ be a variable point on the circumference of the circle. Let $ P$ be the point on the line determined by $ Q$ and $ C$ for which $ \frac{AC}{CB}\equal{}\frac{QC}{CP}$. Describe, with proof, the locus of the point $ P$.

1996 Vietnam National Olympiad, 2

Tags: geometry , incenter , geometry proposed

The triangle ABC has BC=1 and $ \angle BAC \equal{} a$. Find the shortest distance between its incenter and its centroid. Denote this shortest distance by $ f(a)$. When a varies in the interval $ (\frac {\pi}{3},\pi)$, find the maximum value of $ f(a)$.

2008 Iran Team Selection Test, 10

Tags: geometry , circumcircle , incenter , geometry proposed

In the triangle $ ABC$, $ \angle B$ is greater than $ \angle C$. $ T$ is the midpoint of the arc $ BAC$ from the circumcircle of $ ABC$ and $ I$ is the incenter of $ ABC$. $ E$ is a point such that $ \angle AEI\equal{}90^\circ$ and $ AE\parallel BC$. $ TE$ intersects the circumcircle of $ ABC$ for the second time in $ P$. If $ \angle B\equal{}\angle IPB$, find the angle $ \angle A$.

2009 Iran Team Selection Test, 10

Tags: geometry , circumcircle , incenter , ratio , trigonometry , cyclic quadrilateral , geometry proposed

Let $ ABC$ be a triangle and $ AB\ne AC$ . $ D$ is a point on $ BC$ such that $ BA \equal{} BD$ and $ B$ is between $ C$ and $ D$ . Let $ I_{c}$ be center of the circle which touches $ AB$ and the extensions of $ AC$ and $ BC$ . $ CI_{c}$ intersect the circumcircle of $ ABC$ again at $ T$ . If $ \angle TDI_{c} \equal{} \frac {\angle B \plus{} \angle C}{4}$ then find $ \angle A$

2011 All-Russian Olympiad, 2

Tags: geometry , parallelogram , circumcircle , geometric transformation , reflection , symmetry , geometry proposed

On side $BC$ of parallelogram $ABCD$ ($A$ is acute) lies point $T$ so that triangle $ATD$ is an acute triangle. Let $O_1$, $O_2$, and $O_3$ be the circumcenters of triangles $ABT$, $DAT$, and $CDT$ respectively. Prove that the orthocenter of triangle $O_1O_2O_3$ lies on line $AD$.

2005 Canada National Olympiad, 4

Tags: geometry , circumcircle , perimeter , inequalities , inradius , geometry proposed

Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$.

1992 Turkey Team Selection Test, 1

Tags: geometry , cyclic quadrilateral , geometry proposed

The feet of perpendiculars from the intersection point of the diagonals of cyclic quadrilateral $ABCD$ to the sides $AB,BC,CD,DA$ are $P,Q,R,S$, respectively. Prove $PQ+RS=QR+SP$.

2000 239 Open Mathematical Olympiad, 6

Tags: geometry , circumcircle , geometry proposed

Let ABCD be a convex quadrilateral, and let M and N be the midpoints of its sides AD and BC, respectively. Assume that the points A, B, M, N are concyclic, and the circumcircle of triangle BMC touches the line AB. Show that the circumcircle of triangle AND touches the line AB, too. Darij

2002 Tournament Of Towns, 1

Tags: geometry proposed , geometry

In a convex $2002\text{-gon}$ several diagonals are drawn so that they do not intersect inside the polygon. As a result the polygon splits into $2000$ triangles. Isit possible that exactly $1000$ triangles have diagonals for all their three sides?

2009 District Round (Round II), 3

Tags: trigonometry , geometry proposed , geometry

$A,B,C$ are the three angles in a triangle such that $2\sin B\sin (A+B)-\cos A=1$, $2\sin C\sin (B+C)-\cos B=0$ find the three angles.

1999 All-Russian Olympiad, 3

Tags: geometry , incenter , geometric transformation , reflection , geometry proposed

The incircle of $\triangle ABC$ touch $AB$,$BC$,$CA$ at $K$,$L$,$M$. The common external tangents to the incircles of $\triangle AMK$,$\triangle BKL$,$\triangle CLM$, distinct from the sides of $\triangle ABC$, are drawn. Show that these three lines are concurrent.

2009 Regional Competition For Advanced Students, 3

Tags: AoPSwiki , geometry proposed , geometry

Let $ D$, $ E$, $ F$ be the feet of the altitudes wrt sides $ BC$, $ CA$, $ AB$ of acute-angled triangle $ \triangle ABC$, respectively. In triangle $ \triangle CFB$, let $ P$ be the foot of the altitude wrt side $ BC$. Define $ Q$ and $ R$ wrt triangles $ \triangle ADC$ and $ \triangle BEA$ analogously. Prove that lines $ AP$, $ BQ$, $ CR$ don't intersect in one common point.

2005 Kazakhstan National Olympiad, 2

Tags: geometry proposed , geometry

The line parallel to side $AC$ of a right triangle $ABC$ $(\angle C=90^\circ)$ intersects sides $AB$ and $BC$ at $M$ and $N$, respectively, so that the $CN / BN = AC / BC = 2$. Let $O$ be the intersection point of the segments $AN$ and $CM$ and $K$ be a point on the segment $ON$ such that $MO + OK = KN$. The perpendicular line to $AN$ at point $K$ and the bisector of triangle $ABC$ of $\angle B$ meet at point $T$. Find the angle $\angle MTB$.

2007 Iran MO (2nd Round), 2

Tags: geometry , geometry proposed

Tow circles $C,D$ are exterior tangent to each other at point $P$. Point $A$ is in the circle $C$. We draw $2$ tangents $AM,AN$ from $A$ to the circle $D$ ($M,N$ are the tangency points.). The second meet points of $AM,AN$ with $C$ are $E,F$, respectively. Prove that $\frac{PE}{PF}=\frac{ME}{NF}$.

2002 Iran MO (3rd Round), 6

Tags: conics , parabola , ellipse , geometry , parallelogram , geometry proposed

$M$ is midpoint of $BC$.$P$ is an arbitary point on $BC$. $C_{1}$ is tangent to big circle.Suppose radius of $C_{1}$ is $r_{1}$ Radius of $C_{4}$ is equal to radius of $C_{1}$ and $C_{4}$ is tangent to $BC$ at P. $C_{2}$ and $C_{3}$ are tangent to big circle and line $BC$ and circle $C_{4}$. [img]http://aycu01.webshots.com/image/4120/2005120338156776027_rs.jpg[/img] Prove : \[r_{1}+r_{2}+r_{3}=R\] ($R$ radius of big circle)