Olympiad problems

1986 IMO Longlists, 51

Tags: inequalities , geometry , rectangle , trigonometry , cyclic quadrilateral , geometry proposed

Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.

2006 France Team Selection Test, 1

Tags: geometry , circumcircle , projective geometry , cyclic quadrilateral , angle bisector , geometry proposed

Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$. Prove that $(PS)$ and $(QR)$ are perpendicular.

2009 Moldova Team Selection Test, 3

Tags: geometry , circumcircle , search , cyclic quadrilateral , geometry proposed

[color=darkred]A circle $ \Omega_1$ is tangent outwardly to the circle $ \Omega_2$ of bigger radius. Line $ t_1$ is tangent at points $ A$ and $ D$ to the circles $ \Omega_1$ and $ \Omega_2$ respectively. Line $ t_2$, parallel to $ t_1$, is tangent to the circle $ \Omega_1$ and cuts $ \Omega_2$ at points $ E$ and $ F$. Point $ C$ belongs to the circle $ \Omega_2$ such that $ D$ and $ C$ are separated by the line $ EF$. Denote $ B$ the intersection of $ EF$ and $ CD$. Prove that circumcircle of $ ABC$ is tangent to the line $ AD$.[/color]

2010 Olympic Revenge, 3

Tags: geometry , geometric transformation , rotation , geometry proposed

Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions: $i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$. $ii)$ There are no two lines of $S$ which are parallel.

2013 Mexico National Olympiad, 6

Tags: inequalities , geometry , geometry proposed

Let $A_1A_2 ... A_8$ be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each $i = 1, ... , 8$, define $B_i$ as the intersection between segments $A_iA_{i+4}$ and $A_{i-1}A_{i+1}$, where $A_{j+8} = A_j$ and $B_{j+8} = B_j$ for all $j$. Show some number $i$, amongst 1, 2, 3, and 4 satisfies \[\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}\]

2008 Gheorghe Vranceanu, 2

Tags: geometry , circumcircle , perpendicular bisector , geometry proposed

Let $ D$ be an interior point of the side $ BC$ of a triangle $ ABC$, and let $ O_1$ and $ O_2$ be the circumcenters of triangles $ ABD$ and $ ADC$. The perpendicular bisector of the side $ AC$ meets the line $ AO_1$ at $ E$, and the perpendicular bisector of the side $ AB$ meets the line $ AO_2$ at $ F$. Prove that the bisectors of the angles $ \angle O_1EO_2$ and $ \angle O_1FO_2$ are orthogonal.

2009 Sharygin Geometry Olympiad, 15

Tags: geometry , trigonometry , geometry proposed

Given a circle and a point $ C$ not lying on this circle. Consider all triangles $ ABC$ such that points $ A$ and $ B$ lie on the given circle. Prove that the triangle of maximal area is isosceles.

2007 Indonesia TST, 1

Tags: geometry , circumcircle , cyclic quadrilateral , angle bisector , geometry proposed

Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.

2012 Kyoto University Entry Examination, 4

Tags: geometry proposed , geometry

Give the answer about the following propositions $(p),\ (q)$ whether they are true or not. If the answer is true, then give the proof and if the answer is false, then give the proof by giving the counter example. $(p)$ If we can form a triangle such that one of inner angles of the triangle is $60^\circ$ by choosing 3 points from the vertices of a regular $n$-polygon, then $n$ is a multiple of 3. $(q)$ In $\triangle{ABC},\ \triangle{A'B'C'}$, if $AB=A'B',\ BC=B'C',\ \angle{A}=\angle{A'}$, then these triangles are congruent. 30 points

2004 France Team Selection Test, 2

Tags: trigonometry , geometry , angle bisector , geometry proposed

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

1990 Iran MO (2nd round), 1

Tags: geometry , 3D geometry , sphere , geometry proposed

[b](a)[/b] Consider the set of all triangles $ABC$ which are inscribed in a circle with radius $R.$ When is $AB^2+BC^2+CA^2$ maximum? Find this maximum. [b](b)[/b] Consider the set of all tetragonals $ABCD$ which are inscribed in a sphere with radius $R.$ When is the sum of squares of the six edges of $ABCD$ maximum? Find this maximum, and in this case prove that all of the edges are equal.

2013 All-Russian Olympiad, 3

Tags: geometry , circumcircle , geometric transformation , homothety , Euler , symmetry , geometry proposed

The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively. The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and $ I_aC_1 $ at $ B_2 $. Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $. [i]L. Emelyanov, A. Polyansky[/i]

2006 Romania Team Selection Test, 1

Tags: geometry , circumcircle , geometric transformation , rotation , similar triangles , complex numbers , geometry proposed

Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral. [i]Valentin Vornicu[/i]

1997 Iran MO (2nd round), 2

Tags: geometry proposed , geometry

In triangle $ABC$, angles $B,C$ are acute. Point $D$ is on the side $BC$ such that $AD\perp{BC}$. Let the interior bisectors of $\angle B,\angle C$ meet $AD$ at $E,F$, respectively. If $BE=CF$, prove that $ABC$ is isosceles.

1986 IMO Longlists, 79

Tags: geometry proposed , geometry

Let $AA_1,BB_1, CC_1$ be the altitudes in an acute-angled triangle $ABC$, $K$ and $M$ are points on the line segments $A_1C_1$ and $B_1C_1$ respectively. Prove that if the angles $MAK$ and $CAA_1$ are equal, then the angle $C_1KM$ is bisected by $AK.$

2013 India IMO Training Camp, 3

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

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 Sharygin Geometry Olympiad, 8

Tags: geometry , incenter , trigonometry , rhombus , geometry proposed

A point $M$ lies on the side $BC$ of square $ABCD$. Let $X$, $Y$ , and $Z$ be the incenters of triangles $ABM$, $CMD$, and $AMD$ respectively. Let $H_x$, $H_y$, and $H_z$ be the orthocenters of triangles $AXB$, $CY D$, and $AZD$. Prove that $H_x$, $H_y$, and $H_z$ are collinear.

2007 Turkey Team Selection Test, 2

Tags: ratio , geometry , similar triangles , angle bisector , geometry proposed

Two different points $A$ and $B$ and a circle $\omega$ that passes through $A$ and $B$ are given. $P$ is a variable point on $\omega$ (different from $A$ and $B$). $M$ is a point such that $MP$ is the bisector of the angle $\angle{APB}$ ($M$ lies outside of $\omega$) and $MP=AP+BP$. Find the geometrical locus of $M$.

2005 Taiwan TST Round 1, 2

Tags: geometry , geometry proposed

Let $ABCD$ be a convex quadrilateral. Is it possible to find a point $P$ such that the segments drawn between $P$ and the midpoints of the sides of $ABCD$ divide the quadrilateral into four sections of equal area? If $P$ exists, is it unique?

1993 Korea - Final Round, 6

Tags: ratio , geometry , geometry proposed

Consider a triangle $ABC$ with $BC = a, CA = b, AB = c.$ Let $D$ be the midpoint of $BC$ and $E$ be the intersection of the bisector of $A$ with $BC$ . The circle through $A, D, E$ meets $AC, AB$ again at $F, G$ respectively. Let $H\not = B$ be a point on $AB$ with $BG = GH$ . Prove that triangles $EBH$ and $ABC$ are similar and ﬁnd the ratio of their areas.

2010 Pan African, 3

Tags: geometry , circumcircle , geometry proposed

In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.

2005 IMAR Test, 2

Tags: geometry , incenter , geometry proposed

Let $P$ be an arbitrary point on the side $BC$ of triangle $ABC$ and let $D$ be the tangency point between the incircle of the triangle $ABC$ and the side $BC$. If $Q$ and $R$ are respectively the incenters in the triangles $ABP$ and $ACP$, prove that $\angle QDR$ is a right angle. Prove that the triangle $QDR$ is isosceles if and only if $P$ is the foot of the altitude from $A$ in the triangle $ABC$.

2013 Tuymaada Olympiad, 2

Tags: geometry , vector , ratio , geometry proposed

$ABCDEF$ is a convex hexagon, such that in it $AC \parallel DF$, $BD \parallel AE$ and $CE \parallel BF$. Prove that \[AB^2+CD^2+EF^2=BC^2+DE^2+AF^2.\] [i]N. Sedrakyan[/i]

2013 China Team Selection Test, 2

Tags: geometry , circumcircle , geometric transformation , homothety , trigonometry , Euler , geometry proposed

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

1986 IMO Longlists, 5

Tags: function , geometry proposed , geometry

Let $ABC$ and $DEF$ be acute-angled triangles. Write $d = EF, e = FD, f = DE.$ Show that there exists a point $P$ in the interior of $ABC$ for which the value of the expression $X=d \cdot AP +e \cdot BP +f \cdot CP$ attains a minimum.