Olympiad problems

1993 ITAMO, 4

Let $P$ be a point in the plane of a triangle $ABC$, different from its circumcenter. Prove that the triangle whose vertices are the projections of $P$ on the perpendicular bisectors of the sides of $ABC$, is similar to $ABC$.

2008 Oral Moscow Geometry Olympiad, 4

Tags: geometry , Concyclic , projections , cyclic quadrilateral

A circle can be circumscribed around the quadrilateral $ABCD$. Point $P$ is the foot of the perpendicular drawn from point $A$ on line $BC$, and respectively $Q$ from $A$ on $DC$, $R$ from $D$ on $AB$ and $T$ from $D$ on $BC$ . Prove that points $P,Q,R$ and $T$ lie on the same circle. (A. Myakishev)

2006 Sharygin Geometry Olympiad, 24

Tags: geometry , Locus , 3D geometry , projections , Plane , circle , sphere

a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$. b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane

2006 MOP Homework, 2

Tags: geometry , equal angles , orthocenter , projections , perpendicular , right angle

Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.

2013 Sharygin Geometry Olympiad, 3

Tags: geometry , convex polygon , projections

Each vertex of a convex polygon is projected to all nonadjacent sidelines. Can it happen that each of these projections lies outside the corresponding side?

1974 Vietnam National Olympiad, 3

Tags: ratio , projections , geometry , equal angles , constant

Let $ABC$ be a triangle with $A = 90^o, AH$ the altitude, $P,Q$ the feet of the perpendiculars from $H$ to $AB,AC$ respectively. Let $M$ be a variable point on the line $PQ$. The line through $M$ perpendicular to $MH$ meets the lines $AB,AC$ at $R, S$ respectively. i) Prove that circumcircle of $ARS$ always passes the fixed point $H$. ii) Let $M_1$ be another position of $M$ with corresponding points $R_1, S_1$. Prove that the ratio $RR_1/SS_1$ is constant. iii) The point $K$ is symmetric to $H$ with respect to $M$. The line through $K$ perpendicular to the line $PQ$ meets the line $RS$ at $D$. Prove that$ \angle BHR = \angle DHR, \angle DHS = \angle CHS$.

2006 Sharygin Geometry Olympiad, 8.6

Tags: geometry , circumcircle , Circumcenter , interior , projections

A triangle $ABC$ and a point $P$ inside it are given. $A', B', C'$ are the projections of $P$ onto the straight lines ot the sides $BC,CA,AB$. Prove that the center of the circle circumscribed around the triangle $A'B'C'$ lies inside the triangle $ABC$.

2010 Bundeswettbewerb Mathematik, 3

Tags: Concyclic , collinear , projections , altitude , perpendicular , geometry

Given an acute-angled triangle $ABC$. Let $CB$ be the altitude and $E$ a random point on the line $CD$. Finally, let $P, Q, R$ and $S$ are the projections of $D$ on the straight lines $AC, AE, BE$ and $BC$. Prove that the points $P, Q, R$ and $S$ lie either on a circle or on one straight line.

Brazil L2 Finals (OBM) - geometry, 2013.3

Tags: geometry , circumcircle , projections , perpendicular , midpoints

Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.

2006 Sharygin Geometry Olympiad, 10.2

Tags: symmetry , Concyclic , projections , geometry

The projections of the point $X$ onto the sides of the $ABCD$ quadrangle lie on the same circle. $Y$ is a point symmetric to $X$ with respect to the center of this circle. Prove that the projections of the point $B$ onto the lines $AX,XC, CY, YA$ also lie on the same circle.

2015 Thailand TSTST, 2

Tags: geometry , equal segments , projections

In any $\vartriangle ABC, \ell$ is any line through $C$ and points $P, Q$. If $BP, AQ$ are perpendicular to the line $\ell$ and $M$ is the midpoint of the line segment $AB$, then prove that $MP = MQ$

May Olympiad L2 - geometry, 2008.2

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.

2010 Sharygin Geometry Olympiad, 4

Tags: geometry , parallelogram , concentric circles , concentric , projections , inscribed

Projections of two points to the sidelines of a quadrilateral lie on two concentric circles (projections of each point form a cyclic quadrilateral and the radii of circles are different). Prove that this quadrilateral is a parallelogram.

2023 Sharygin Geometry Olympiad, 21

Tags: geometry , humpty points , radical axis , projections , othorcenter , Sharygin Geometry Olympiad , Sharygin 2023

Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.

VMEO III 2006, 10.1

Tags: geometry , angles , ratio , projections

Given a triangle $ABC$ ($AB \ne AC$). Let $ P$ be a point in the plane containing triangle $ABC$ satisfying the following property: If the projections of $ P$ onto $AB$,$AC$ are $C_1$,$B_1$ respectively, then $\frac{PB}{PC}=\frac{PC_1}{PB_1}=\frac{AB}{AC}$ or $\frac{PB}{PC}=\frac{PB_1}{PC_1}=\frac{AB}{AC}$. Prove that $\angle PBC + \angle PCB = \angle BAC$.

2010 Saudi Arabia BMO TST, 4

Tags: geometry , projections

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Denote by $P, Q, R, S$ the orthogonal projections of $O$ onto $AB$ , $BC$ ,$CD$ , $DA$, respectively. Prove that $$PA \cdot AB + RC \cdot CD =\frac12 (AD^2 + BC^2)$$ if and only if $$QB \cdot BC + SD \cdot DA = \frac12(AB ^2 + CD^2)$$

2021-IMOC qualification, G1

Tags: geometry , Circumcenter , incenter , projections , circumcircle

Let $O$ be the circumcenter and $I$ be the incenter of $\vartriangle$, $P$ is the reflection from $I$ through $O$, the foot of perpendicular from $P$ to $BC,CA,AB$ is $X,Y,Z$, respectively. Prove that $AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2$.

1997 Romania National Olympiad, 3

Tags: geometry , projections

A point $A_0$ and two lines $d_1$ and $d_2$ are given in the space. For each nonnegative integer $n$ we denote by $B_n$ the projection of $A_n$ on $d_2,$ and by $A_{n+1}$ the projection of $B_n$ on $d_1.$ Prove that there exist two segments $[A'A''] \subset d_1$ and $[B'B''] \subset d_2$ of length $0.001$ and a nonnegative integer $N$ such that $A_n \in [A'A'']$ and $B_n \in [B'B'']$ for any $n \ge N.$

2011 Sharygin Geometry Olympiad, 23

Tags: geometry , Nine Point Circle , projections , midpoints , Circumcenter , collinear

Given are triangle $ABC$ and line $\ell$ intersecting $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. Point $A'$ is the midpoint of the segment between the projections of $A_1$ to $AB$ and $AC$. Points $B'$ and $C'$ are defined similarly. (a) Prove that $A', B'$ and $C'$ lie on some line $\ell'$. (b) Suppose $\ell$ passes through the circumcenter of $\triangle ABC$. Prove that in this case $\ell'$ passes through the center of its nine-points circle. [i]M. Marinov and N. Beluhov[/i]

Swiss NMO - geometry, 2012.10

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.

Durer Math Competition CD Finals - geometry, 2015.D4

Tags: geometry , rectangle , projections

The projection of the vertex $C$ of the rectangle $ABCD$ on the diagonal $BD$ is $E$. The projections of $E$ on $AB$ and $AD$ are $F$ and $G$ respectively. Prove that $$AF^{2/3} + AG^{2/3} = AC^{2/3}$$ .

2004 Oral Moscow Geometry Olympiad, 4

Tags: geometry , perpendicular , projections

Triangle $ABC$ is inscribed in a circle. Through points $A$ and $B$ tangents to this circle are drawn, which intersect at point $P$. Points $X$ and $Y$ are orthogonal projections of point $P$ onto lines $AC$ and $BC$. Prove that line $XY$ is perpendicular to the median of triangle $ABC$ from vertex $C$.

Estonia Open Senior - geometry, 1993.5

Tags: geometry , Equilateral , ratio , projections

Within an equilateral triangle $ABC$, take any point $P$. Let $L, M, N$ be the projections of $P$ on sides $AB, BC, CA$ respectively. Prove that $\frac{AP}{NL}=\frac{BP}{LM}=\frac{CP}{MN}$.

2018 Oral Moscow Geometry Olympiad, 1

Tags: right triangle , geometry , inradius , projections

In a right triangle $ABC$ with a right angle $C$, let $AK$ and $BN$ be the angle bisectors. Let $D,E$ be the projections of $C$ on $AK, BN$ respectively. Prove that the length of the segment $DE$ is equal to the radius of the inscribed circle.

2006 Sharygin Geometry Olympiad, 18

Tags: geometry , collinear , parallel , orthocenter , projections

Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$, one of which intersects $BC$ at point $X$, and the other intersects $AC$ at point $Y$. Lines $AZ, BZ$ are parallel, respectively with $HX$ and $HY$. Prove that the points $X, Y, Z$ lie on the same line.