Olympiad problems

2006 Estonia Team Selection Test, 2

Tags: Circumcenter , geometry , projections , Concyclic , Cyclic , independent

The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.

2009 Balkan MO Shortlist, G3

Tags: diagonals , projections , geometry , midpoints , convex quadrilateral , Circumcenter , Isogonal conjugate

Let $ABCD$ be a convex quadrilateral, and $P$ be a point in its interior. The projections of $P$ on the sides of the quadrilateral lie on a circle with center $O$. Show that $O$ lies on the line through the midpoints of $AC$ and $BD$.

2016 Indonesia MO, 1

Tags: geometry , perpendicular , angle bisector , projections

Let $ABCD$ be a cyclic quadrilateral wih both diagonals perpendicular to each other and intersecting at point $O$. Let $E,F,G,H$ be the orthogonal projections of $O$ on sides $AB,BC,CD,DA$ respectively. a. Prove that $\angle EFG + \angle GHE = 180^o$ b. Prove that $OE$ bisects angle $\angle FEH$ .

2012 Czech-Polish-Slovak Junior Match, 3

Tags: circumcircle , Equilateral , fixed , area of a triangle , geometry , projections

Different points $A, B, C, D$ lie on a circle with a center at the point $O$ at such way that $\angle AOB$ $= \angle BOC =$ $\angle COD =$ $60^o$. Point $P$ lies on the shorter arc $BC$ of this circle. Points $K, L, M$ are projections of $P$ on lines $AO, BO, CO$ respectively . Show that (a) the triangle $KLM$ is equilateral, (b) the area of triangle $KLM$ does not depend on the choice of the position of point $P$ on the shorter arc $BC$

1985 Bundeswettbewerb Mathematik, 2

Tags: geometry , collinear , projections , altitude

Prove that in every triangle for each of its altitudes: If you project the foof of one altitude on the other two altitudes and on the other two sides of the triangle, those four projections lie on the same line.

2018 Yasinsky Geometry Olympiad, 6

Tags: geometry , Angle Chasing , equal angles , projections , altitude

$AH$ is the altitude of the acute triangle $ABC$, $K$ and $L$ are the feet of the perpendiculars, from point $H$ on sides $AB$ and $AC$ respectively. Prove that the angles $BKC$ and $BLC$ are equal.

1987 Mexico National Olympiad, 3

Tags: geometry , Locus , projections

Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on $\ell $, $B$ on $\ell ' $, and angle $\angle APB$ is right?

2012 Switzerland - Final Round, 10

Tags: geometry , Concyclic , projections , projection

Let $O$ be an inner point of an acute-angled triangle $ABC$. Let $A_1, B_1$ and $C_1$ be the projections of $O$ on the sides $BC, AC$ and $AB$ respectively . Let $P$ be the intersection of the perpendiculars on $B_1C_1$ and $A_1C_1$ from points$ A$ and $B$ respectilvey. Let $H$ be the projection of $P$ on $AB$. Show that points $A_1, B_1, C_1$ and $H$ lie on a circle.

Mathley 2014-15, 3

Tags: circumcircle , orthocenter , projections , game strategy

A point $P$ is interior to the triangle $ABC$ such that $AP \perp BC$. Let $E, F$ be the projections of $CA, AB$. Suppose that the tangents at $E, F$ of the circumcircle of triangle $AEF$ meets at a point on $BC$. Prove that $P$ is the orthocenter of triangle $ABC$. Do Thanh Son, High School of Natural Sciences, National University, Hanoi

2008 May Olympiad, 2

Tags: geometry , rectangle , projections

Let $ABCD$ be a rectangle and $P$ be a point on the side$ AD$ such that $\angle BPC = 90^o$. The perpendicular from $A$ on $BP$ cuts $BP$ at $M$ and the perpendicular from $D$ on $CP$ cuts $CP$ in $N$. Show that the center of the rectangle lies in the $MN$ segment.

1988 ITAMO, 5

Tags: 3D geometry , projections , parallelogram , geometry

Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?

Estonia Open Junior - geometry, 2009.2.4

Tags: ratio , geometry , projections

The triangle $ABC$ is $| BC | = a$ and $| AC | = b$. On the ray starting from vertex $C$ and passing the midpoint of side $AB$ , choose any point $D$ other than vertex $C$. Let $K$ and $L$ be the projections of $D$ on the lines $AC$ and $BC$, respectively, $K$ and $L$. Find the ratio $| DK | : | DL |$.

2011 Portugal MO, 5

Tags: geometry , incircle , projections , Angle Bisectors

Let $[ABC]$ be a triangle, $D$ be the orthogonal projection of $B$ on the bisector of $\angle ACB$ and $E$ the orthogonal projection of $C$ on the bisector of $\angle ABC$ . Prove that $DE$ intersects the sides $[AB]$ and $[AC]$ at the touchpoints of the circle inscribed in the triangle $[ABC]$.

2010 Balkan MO Shortlist, G4

Tags: Circumcenter , Euler Circle , geometry , midpoint , projections , collinear

Let $ABC$ be a given triangle and $\ell$ be a line that meets the lines $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$ respectively. Let $A'$ be the midpoint, of the segment connecting the projections of $A_1$ onto the lines $AB$ and $AC$. Construct, analogously the points $B'$ and $C'$. (a) Show that the points $A', B'$ and $C'$ are collinear on some line $\ell'$. (b) Show that if $\ell$ contains the circumcenter of the triangle $ABC$, then $\ell' $ contains the center of it's Euler circle.

Kharkiv City MO Seniors - geometry, 2019.11.5

Tags: geometry , projections , bisects segment , Kharkiv , altitudes

In the acute-angled triangle $ABC$, let $CD, AE$ be the altitudes. Points $F$ and $G$ are the projections of $A$ and $C$ on the line $DE$, respectively, $H$ and $K$ are the projections of $D$ and $E$ on the line $AC$, respectively. The lines $HF$ and $KG$ intersect at point $P$. Prove that line $BP$ bisects the segment $DE$.

VMEO IV 2015, 11.2

Tags: geometry , projections , Fixed point , fixed , isogonal

Let $ABC$ be a triangle with two isogonal points $ P$ and $Q$ . Let $D, E$ be the projection of $P$ on $AB$, $AC$. $G$ is the projection of $Q$ on $BC$. $U$ is the projection of $G$ on $DE$, $ L$ is the projection of $P$ on $AQ$, $K$ is the symmetric of $L$ wrt $UG$. Prove that $UK$ passes through a fixed point when $P$ and $Q$ vary.

2006 Estonia Team Selection Test, 2

Tags: Circumcenter , geometry , projections , Concyclic , Cyclic , independent

The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.10

Tags: geometry , 3D geometry , tetrahedron , projections

Given a tetrahedron $A_1A_2A_3A_4$ (not necessarily regulart). We shall call a point $N$ in space [i]Serve point[/i], if it's six projection points on the six edges of the tetrahedron lie on one plane. This plane we denote it by $a (N)$ and call the [i]Serve plane[/i] of the point $N$. By $B_{ij}$ denote, respectively, the midpoint of the edges $A_1A_j$, $1\le i <j \le 4$. For each point $M$, denote by $M_{ij}$ the points symmetric to $M$ with respect to $B_{ij},$ $1\le i <j \le 4$. Prove that if all points $M_{ij}$ are Serve points, then the point $M$ belongs to all Serve planes $a (M_{ij})$, $1\le i <j \le 4$.