Olympiad problems

2011 IMAC Arhimede, 4

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

Inscribed circle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$ at the points $X$, $Y$ and $Z$, respectively. Let $AA_{1}$, $BB_{1}$ and $CC_{1}$ be the altitudes of the triangle $ABC$ and $M$, $N$ and $P$ be the incenters of triangles $AB_{1}C_{1}$, $BC_{1}A_{1}$ and $CA_{1}B_{1}$, respectively. a) Prove that $M$, $N$ and $P$ are orthocentres of triangles $AYZ$, $BZX$ and $CXY$, respectively. b) Prove that common external tangents of these incircles, different from triangle sides, are concurent at orthocentre of triangle $XYZ$.

2005 APMO, 3

Tags: algorithm , geometry proposed , geometry

Prove that there exists a triangle which can be cut into 2005 congruent triangles.

2017 Silk Road, 2

Tags: geometry proposed , geometry

The quadrilateral $ABCD$ is inscribed in the circle ω. The diagonals $AC$ and $BD$ intersect at the point $O$. On the segments $AO$ and $DO$, the points $E$ and $F$ are chosen, respectively. The straight line $EF$ intersects ω at the points $E_1$ and $F_1$. The circumscribed circles of the triangles $ADE$ and $BCF$ intersect the segment $EF$ at the points $E_2$ and $F_2$ respectively (assume that all the points $E, F, E_1, F_1, E_2$ and $F_2$ are different). Prove that $E_1E_2 = F_1F_2$. $(N. Sedrakyan)$

1982 IMO Longlists, 6

Tags: geometry proposed , geometry

On the three distinct lines $a, b$, and $c$ three points $A, B$, and $C$ are given, respectively. Construct three collinear points $X, Y,Z$ on lines $a, b, c$, respectively, such that $\frac{BY}{AX} = 2$ and $ \frac{CZ}{AX} = 3$.

2013 Canadian Mathematical Olympiad Qualification Repechage, 2

Tags: trigonometry , geometry , trig identities , Law of Sines , geometry proposed

In triangle $ABC$, $\angle A = 90^\circ$ and $\angle C = 70^\circ$. $F$ is point on $AB$ such that $\angle ACF = 30^\circ$, and $E$ is a point on $CA$ such that $\angle CF E = 20^\circ$. Prove that $BE$ bisects $\angle B$.

2004 Germany Team Selection Test, 2

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

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.

2013 Iran Team Selection Test, 17

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

In triangle $ABC$, $AD$ and $AH$ are the angle bisector and the altitude of vertex $A$, respectively. The perpendicular bisector of $AD$, intersects the semicircles with diameters $AB$ and $AC$ which are drawn outside triangle $ABC$ in $X$ and $Y$, respectively. Prove that the quadrilateral $XYDH$ is concyclic. [i]Proposed by Mahan Malihi[/i]

1985 IMO Longlists, 86

Tags: geometry , rectangle , geometry proposed , geometric inequality

Let $l$ denote the length of the smallest diagonal of all rectangles inscribed in a triangle $T$ . (By inscribed, we mean that all four vertices of the rectangle lie on the boundary of $T$ .) Determine the maximum value of $\frac{l^2}{S(T)}$ taken over all triangles ($S(T )$ denotes the area of triangle $T$ ).

2008 Irish Math Olympiad, 5

Tags: ratio , geometry , parallelogram , geometry proposed

A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$. Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.

1989 Turkey Team Selection Test, 3

Tags: geometry , perpendicular bisector , geometry proposed

Let $C_1$ and $C_2$ be given circles. Let $A_1$ on $C_1$ and $A_2$ on $C_2$ be fixed points. If chord $A_1P_1$ of $C_1$ is parallel to chord $A_2P_2$ of $C_2$, find the locus of the midpoint of $P_1P_2$.

2005 German National Olympiad, 5

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

[b](a)[/b] [Problem for class 11] Let r be the inradius and $r_a$, $r_b$, $r_c$ the exradii of a triangle ABC. Prove that $\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$. [b](b)[/b] [Problem for classes 12/13] Let r be the radius of the insphere and let $r_a$, $r_b$, $r_c$, $r_d$ the radii of the four exspheres of a tetrahedron ABCD. (An [i]exsphere[/i] of a tetrahedron is a sphere touching one sideface and the extensions of the three other sidefaces.) Prove that $\frac{2}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}+\frac{1}{r_d}$. I am really sorry for posting these, but else, Orl will probably post them. This time, we really did not have any challenging problem on the DeMO. But at least, the problems were simple enough that I solved all of them. ;) Darij

1974 IMO Longlists, 4

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

Let $K_a,K_b,K_c$ with centres $O_a,O_b,O_c$ be the excircles of a triangle $ABC$, touching the interiors of the sides $BC,CA,AB$ at points $T_a,T_b,T_c$ respectively. Prove that the lines $O_aT_a,O_bT_b,O_cT_c$ are concurrent in a point $P$ for which $PO_a=PO_b=PO_c=2R$ holds, where $R$ denotes the circumradius of $ABC$. Also prove that the circumcentre $O$ of $ABC$ is the midpoint of the segment $PI$, where $I$ is the incentre of $ABC$.

2005 Taiwan TST Round 3, 1

Tags: inequalities , geometry , circumcircle , geometry proposed

Let $P$ be a point in the interior of $\triangle ABC$. The lengths of the sides of $\triangle ABC$ is $a,b,c$, and the distance from $P$ to the sides of $\triangle ABC$ is $p,q,r$. Show that the circumradius $R$ of $\triangle ABC$ satisfies \[\displaystyle R\le \frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}.\] When does equality hold?

1999 CentroAmerican, 4

Tags: geometry , trapezoid , function , trigonometry , geometry proposed

In the trapezoid $ABCD$ with bases $AB$ and $CD$, let $M$ be the midpoint of side $DA$. If $BC=a$, $MC=b$ and $\angle MCB=150^\circ$, what is the area of trapezoid $ABCD$ as a function of $a$ and $b$?

2012 China Western Mathematical Olympiad, 4

Tags: inequalities , geometry , circumcircle , geometry proposed

$P$ is a point in the $\Delta ABC$, $\omega $ is the circumcircle of $\Delta ABC $. $BP \cap \omega = \left\{ {B,{B_1}} \right\}$,$CP \cap \omega = \left\{ {C,{C_1}} \right\}$, $PE \bot AC$，$PF \bot AB$. The radius of the inscribed circle and circumcircle of $\Delta ABC $ is $r,R$. Prove $\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}$.

2011 South East Mathematical Olympiad, 4

Tags: geometry , circumcircle , projective geometry , geometry proposed

Let $O$ be the circumcenter of triangle $ABC$ , a line passes through $O$ intersects sides $AB,AC$ at points $M,N$ , $E$ is the midpoint of $MC$ , $F$ is the midpoint of $NB$ , prove that : $\angle FOE= \angle BAC$

2005 Serbia Team Selection Test, 4

Tags: inequalities , trigonometry , geometry proposed , geometry

Let $T$ be the centroid of triangle $ABC$. Prove that \[ \frac 1{\sin \angle TAC} + \frac 1{\sin \angle TBC} \geq 4 \]

2019 IMEO, 1

Tags: geometry , geometry proposed , geometry solved , parallelogram , humpty point , humpty points , lengths

Let $ABC$ be a scalene triangle with circumcircle $\omega$. The tangent to $\omega$ at $A$ meets $BC$ at $D$. The $A$-median of triangle $ABC$ intersects $BC$ and $\omega$ at $M$ and $N$, respectively. Suppose that $K$ is a point such that $ADMK$ is a parallelogram. Prove that $KA = KN$. [i]Proposed by Alexandru Lopotenco (Moldova)[/i]

2008 Saint Petersburg Mathematical Olympiad, 3

Tags: geometry , angle bisector , geometry proposed

Pentagon $ABCDE$ has circle $S$ inscribed into it. Side $BC$ is tangent to $S$ at point $K$. If $AB=BC=CD$, prove that angle $EKB$ is a right angle.

2024 Abelkonkurransen Finale, 4a

Tags: geometry , circumcircle , geometry proposed , angles

The triangle $ABC$ with $AB < AC$ has an altitude $AD$. The points $E$ and $A$ lie on opposite sides of $BC$, with $E$ on the circumcircle of $ABC$. Furthermore, $AD = DE$ and $\angle ADO=\angle CDE$, where $O$ is the circumcentre of $ABC$. Determine $\angle BAC$.

2015 Mexico National Olympiad, 1

Tags: circumcircle , geometry proposed , geometry

Let $ABC$ be an acuted-angle triangle and let $H$ be it's orthocenter. Let $PQ$ be a segment through $H$ such that $P$ lies on $AB$ and $Q$ lies on $AC$ and such that $ \angle PHB= \angle CHQ$. Finally, in the circumcircle of $\triangle ABC$, consider $M$ such that $M$ is the mid point of the arc $BC$ that doesn't contain $A$. Prove that $MP=MQ$ Proposed by Eduardo Velasco/Marco Figueroa

1996 Romania Team Selection Test, 5

Tags: geometry , rectangle , geometry proposed

Let $A$ and $B$ be points on a circle $\mathcal{C}$ with center $O$ such that $\angle AOB = \dfrac {\pi}2$. Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at $A$ and $B$ respectively and are also externally tangent to one another. The circle $\mathcal{C}_3$ lies in the interior of $\angle AOB$ and it is tangent externally to $\mathcal{C}_1$, $\mathcal{C}_2$ at $P$ and $R$ and internally tangent to $\mathcal{C}$ at $S$. Evaluate the value of $\angle PSR$.

2012 CHKMO, 4

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

In $\triangle ABC$, $AB>AC$. In the circumcircle $(O)$ of $\triangle ABC$, $M$ is the midpoint of arc $BAC$. The incircle $(I)$ of $\triangle ABC$ touches $BC$ at $D$, the line through $D$ parallel to $AI$ intersects $(I)$ again at $P$. Prove that $AP$ and $IM$ intersect at a point on $(O)$.

2007 All-Russian Olympiad, 4

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

$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. [i]V. Astakhov[/i]

2007 Moldova Team Selection Test, 3

Tags: geometry , inradius , circumcircle , trigonometry , geometry proposed

Consider a triangle $ABC$, with corresponding sides $a,b,c$, inradius $r$ and circumradius $R$. If $r_{A}, r_{B}, r_{C}$ are the radii of the respective excircles of the triangle, show that \[a^{2}\left(\frac 2{r_{A}}-\frac{r}{r_{B}r_{C}}\right)+b^{2}\left(\frac 2{r_{B}}-\frac{r}{r_{A}r_{C}}\right)+c^{2}\left(\frac 2{r_{C}}-\frac{r}{r_{A}r_{B}}\right)=4(R+3r) \]