Olympiad problems

2022 Turkey Team Selection Test, 8

$ABC$ triangle with $|AB|<|BC|<|CA|$ has the incenter $I$. The orthocenters of triangles $IBC, IAC$ and $IAB$ are $H_A, H_A$ and $H_A$. $H_BH_C$ intersect $BC$ at $K_A$ and perpendicular line from $I$ to $H_BH_B$ intersect $BC$ at $L_A$. $K_B, L_B, K_C, L_C$ are defined similarly. Prove that $$|K_AL_A|=|K_BL_B|+|K_CL_C|$$

2005 Bulgaria Team Selection Test, 1

Tags: trigonometry , geometry proposed , geometry

Let $ABC$ be an acute triangle. Find the locus of the points $M$, in the interior of $\bigtriangleup ABC$, such that $AB-FG= \frac{MF.AG+MG.BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to the lines $BC$ and $AC$, respectively.

1995 Baltic Way, 19

Tags: geometry proposed , geometry

The following construction is used for training astronauts: A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a way that the touching point of the circles $C_2$ and $C_3$ remains at maximum distance from the touching point of the circles $C_1$ and $C_2$ at all times. How many revolutions (relative to the ground) does the astronaut perform together with the circle $C_3$ while the circle $C_2$ completes one full lap around the inside of circle $C_1$?

2010 Junior Balkan MO, 3

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

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

1997 Balkan MO, 1

Tags: geometry , trigonometry , inequalities , geometry proposed

Suppose that $O$ is a point inside a convex quadrilateral $ABCD$ such that \[ OA^2 + OB^2 + OC^2 + OD^2 = 2\mathcal A[ABCD] , \] where by $\mathcal A[ABCD]$ we have denoted the area of $ABCD$. Prove that $ABCD$ is a square and $O$ is its center. [i]Yugoslavia[/i]

2014 Contests, 4

Tags: geometry , circumcircle , trapezoid , geometry proposed

$ABC$ is an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on segment $BC$. Circumcircle of $\triangle BHC$ instersects with segments $AD$,$AE$ at $P$ and $Q$, respectively. Prove that if $BD^2+CD^2=2DP\cdot DA$ and $BE^2+CE^2=2EQ\cdot EA$, then $BP=CQ$.

1991 Iran MO (2nd round), 2

Tags: geometry , incenter , circumcircle , inequalities , geometry proposed

Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that \[\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3\]\[, IA'+IB'+IC' \geq IA+IB+IC\]

2006 Germany Team Selection Test, 1

Tags: geometry , geometric transformation , reflection , geometry proposed

Let $A$, $B$, $C$, $D$, $E$, $F$ be six points on a circle such that $AE\parallel BD$ and $BC\parallel DF$. Let $X$ be the reflection of the point $D$ in the line $CE$. Prove that the distance from the point $X$ to the line $EF$ equals to the distance from the point $B$ to the line $AC$.

2003 Bundeswettbewerb Mathematik, 3

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 $\measuredangle ADC$.

JBMO Geometry Collection, 2004

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

Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.

1987 Balkan MO, 4

Tags: trigonometry , geometry , similar triangles , geometry proposed

Two circles $K_{1}$ and $K_{2}$, centered at $O_{1}$ and $O_{2}$ with radii $1$ and $\sqrt{2}$ respectively, intersect at $A$ and $B$. Let $C$ be a point on $K_{2}$ such that the midpoint of $AC$ lies on $K_{1}$. Find the length of the segment $AC$ if $O_{1}O_{2}=2$

2011 Polish MO Finals, 2

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

In a tetrahedron $ABCD$, the four altitudes are concurrent at $H$. The line $DH$ intersects the plane $ABC$ at $P$ and the circumsphere of $ABCD$ at $Q\neq D$. Prove that $PQ=2HP$.

1996 Poland - Second Round, 2

Tags: trigonometry , geometry proposed , geometry

A circle with center O inscribed in a convex quadrilateral ABCD is tangent to the lines AB, BC, CD, DA at points K, L, M, N respectively. Assume that the lines KL and MN are not parallel and intersect at the point S. Prove that BD is perpendicular OS. I think it is very good and beautiful problem. I solved it without help. I'm wondering is it a well known theorem? Also I'm interested who is the creator of this problem? I'll be glad to see simple solution of this problem.

1998 Romania Team Selection Test, 2

Tags: geometry , rectangle , pigeonhole principle , geometry proposed

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

2009 Moldova Team Selection Test, 1

Tags: geometry , circumcircle , rotation , geometry proposed

[color=darkblue]Points $ X$, $ Y$ and $ Z$ are situated on the sides $ (BC)$, $ (CA)$ and $ (AB)$ of the triangles $ ABC$, such that triangles $ XYZ$ and $ ABC$ are similiar. Prove that circumcircle of $ AYZ$ passes through a fixed point.[/color]

1979 Canada National Olympiad, 2

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

It is known in Euclidean geometry that the sum of the angles of a triangle is constant. Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.

1996 Turkey MO (2nd round), 1

Tags: geometry proposed , geometry

A circle is tangent to sides $AD,\text{ }DC,\text{ }CB$ of a convex quadrilateral $ABCD$ at $\text{K},\text{ L},\text{ M}$ respectively. A line $l$, passing through $L$ and parallel to $AD$, meets $KM$ at $N$ and $KC$ at $P$. Prove that $PL=PN$.

2001 Turkey Team Selection Test, 2

Tags: geometry , geometric transformation , reflection , geometry proposed

A circle touches to diameter $AB$ of a unit circle with center $O$ at $T$ where $OT>1$. These circles intersect at two different points $C$ and $D$. The circle through $O$, $D$, and $C$ meet the line $AB$ at $P$ different from $O$. Show that \[|PA|\cdot |PB| = \dfrac {|PT|^2}{|OT|^2}.\]

1970 Vietnam National Olympiad, 5

Tags: geometry , 3D geometry , trigonometry , geometry proposed

A plane $p$ passes through a vertex of a cube so that the three edges at the vertex make equal angles with $p$. Find the cosine of this angle. Find the positions of the feet of the perpendiculars from the vertices of the cube onto $p$. There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane $p$.

2002 Romania National Olympiad, 4

Tags: geometry , 3D geometry , prism , geometry proposed

The right prism $[A_1A_2A_3\ldots A_nA_1'A_2'A_3'\ldots A_n'],n\in\mathbb{N},n\ge 3$, has a convex polygon as its base. It is known that $A_1A_2'\perp A_2A_3',A_2A_3'\perp A_3A_4',$$\ldots A_{n-1}A_n'\perp A_nA_1', A_nA_1'\perp A_1A_2'$. Show that: $a)$ $n=3$; $b)$ the prism is regular.

2008 German National Olympiad, 2

Tags: geometry , rectangle , geometry proposed

The triangle $ \triangle SFA$ has a right angle at $ F$. The points $ P$ and $ Q$ lie on the line $ SF$ such that the point $ P$ lies between $ S$ and $ F$ and the point $ F$ is the midpoint of the segment $ [PQ]$. The circle $ k_1$ is th incircle of the triangle $ \triangle SPA$. The circle $ k_2$ lies outside the triangle $ \triangle SQA$ and touches the segment $ [QA]$ and the lines $ SQ$ and $ SA$. Prove that the sum of the radii of the circles $ k_1$ and $ k_2$ equals the length of $ [FA]$.

2010 Turkey Team Selection Test, 2

Tags: geometry , circumcircle , analytic geometry , parallelogram , geometry proposed

For an interior point $D$ of a triangle $ABC,$ let $\Gamma_D$ denote the circle passing through the points $A, \: E, \: D, \: F$ if these points are concyclic where $BD \cap AC=\{E\}$ and $CD \cap AB=\{F\}.$ Show that all circles $\Gamma_D$ pass through a second common point different from $A$ as $D$ varies.

2014 Turkey Team Selection Test, 3

Tags: geometry , circumcircle , parallelogram , geometry proposed

Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds. Prove that \[ED=AC-AB \iff R=2r+r_a.\]

2011 ELMO Shortlist, 1

Tags: geometry , analytic geometry , geometry proposed

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle. [i]Evan O'Dorney.[/i]

2016 India National Olympiad, P1

Tags: geometry , geometry proposed

Let $ABC$ be a triangle in which $AB=AC$. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$.