Found problems: 86
1956 Moscow Mathematical Olympiad, 339
Find the union of all projections of a given line segment $AB$ to all lines passing through a given point $O$.
2006 Estonia Team Selection Test, 2
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 $F$.
2019 Tournament Of Towns, 5
The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area $1$, which has only one pair of parallel sides.
a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area $1$?
b) The same question for a square of area $1/2019$.
(Mikhail Evdokimov)
2015 Thailand TSTST, 2
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 $MP = MQ$
2016 Stars of Mathematics, 3
Let $ ABC $ be a triangle, $ M_A $ be the midpoint of the side $ BC, $ and $ P_A $ be the orthogonal projection of $ A $ on $ BC. $ Similarly, define $ M_B,M_C,P_B,P_C. M_BM_C $ intersects $ P_BP_C $ at $ S_A, $ and the tangent of the circumcircle of $ ABC $ at $ A $ meets $ BC $ at $ T_A. $ Similarly, define $ S_B,S_C,T_B,T_C. $
Show that the perpendiculars through $ A,B,C, $ to $ S_AT_A,S_BT_B, $ respectively, $ S_CT_C, $ are concurent.
[i]Flavian Georgescu[/i]
2012 Czech-Polish-Slovak Junior Match, 3
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 $BC$
2021-IMOC qualification, G1
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 $AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2$.
1985 Bundeswettbewerb Mathematik, 2
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.
2010 Saudi Arabia BMO TST, 4
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 $$QB \cdot BC + SD \cdot DA = \frac12(AB ^2 + CD^2)$$
Swiss NMO - geometry, 2013.3
Let $ABCD$ be a cyclic quadrilateral with $\angle ADC = \angle DBA$. Furthermore, let $E$ be the projection of $A$ on $BD$. Show that $BC = DE - BE$ .
1983 Tournament Of Towns, (042) O5
A point is chosen inside a regular $k$-gon in such a way that its orthogonal projections on to the sides all meet the respective sides at interior points. These points divide the sides into $2k$ segments. Let these segments be enumerated consecutively by the numbers $1,2, 3, ... ,2k$. Prove that the sum of the lengths of the segments having even numbers equals the sum of the segments having odd numbers.
(A Andjans, Riga)
2006 Sharygin Geometry Olympiad, 18
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.
1983 All Soviet Union Mathematical Olympiad, 358
The points $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of lines $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ .
1988 ITAMO, 5
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?
2010 Thailand Mathematical Olympiad, 3
Let $\vartriangle ABC$ be a scalene triangle with $AB < BC < CA$. Let $D$ be the projection of $A$ onto the angle bisector of $\angle ABC$, and let $E$ be the projection of $A$ onto the angle bisector of $\angle ACB$. The line $DE$ cuts sides $AB$ and AC at $M$ and $N$, respectively. Prove that $$\frac{AB+AC}{BC} =\frac{DE}{MN} + 1$$
2008 Oral Moscow Geometry Olympiad, 4
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)
1997 ITAMO, 4
Let $ABCD$ be a tetrahedron. Let $a$ be the length of $AB$ and let $S$ be the area of the projection of the tetrahedron onto a plane perpendicular to $AB$. Determine the volume of the tetrahedron in terms of $a$ and $S$.
IV Soros Olympiad 1997 - 98 (Russia), 11.4
Find the largest value of the area of the projection of the cylinder onto the plane if its radius is $r$ and its height is $h$ (orthogonal projection).
1989 All Soviet Union Mathematical Olympiad, 505
$S$ and $S'$ are two intersecting spheres. The line $BXB'$ is parallel to the line of centers, where $B$ is a point on $S, B'$ is a point on $S'$ and $X$ lies on both spheres. $A$ is another point on $S$, and $A'$ is another point on S' such that the line $AA'$ has a point on both spheres. Show that the segments $AB$ and $A'B'$ have equal projections on the line $AA'$.
2010 Sharygin Geometry Olympiad, 4
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.
2017 Sharygin Geometry Olympiad, P22
Let $P$ be an arbitrary point on the diagonal $AC$ of cyclic quadrilateral $ABCD$, and $PK, PL, PM, PN, PO$ be the perpendiculars from $P$ to $AB, BC, CD, DA, BD$ respectively. Prove that the distance from $P$ to $KN$ is equal to the distance from $O$ to $ML$.
2009 Balkan MO Shortlist, G3
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 $BD$.
1997 Romania National Olympiad, 3
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 $n \ge N.$
2006 Estonia Team Selection Test, 2
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 $F$.
2002 Moldova Team Selection Test, 4
Let $C$ be the circle with center $O(0,0)$ and radius $1$, and $A(1,0), B(0,1)$ be points on the circle. Distinct points $A_1,A_2, ....,A_{n-1}$ on $C$ divide the smaller arc $AB$ into $n$ equal parts ($n \ge 2$). If $P_i$ is the orthogonal projection of $A_i$ on $OA$ ($i =1, ... ,n-1$), find all values of $n$ such that $P_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1}$ is an integer for every positive integer $p$.