Found problems: 33
1987 Mexico National Olympiad, 8
(a) Three lines $l,m,n$ in space pass through point $S$. A plane perpendicular to $m$ intersects $l,m,n $ at $A,B,C$ respectively. Suppose that $\angle ASB = \angle BSC = 45^o$ and $\angle ABC = 90^o$. Compute $\angle ASC$.
(b) Furthermore, if a plane perpendicular to $l$ intersects $l,m,n$ at $P,Q,R$ respectively and $SP = 1$, find the sides of triangle $PQR$.
2015 Sharygin Geometry Olympiad, P24
The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and $D_1$.
a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$.
Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $ P$.
b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I $ with plane $B_1C_1D_1$. Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$.
1991 Mexico National Olympiad, 3
Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?
1977 Vietnam National Olympiad, 6
The planes $p$ and $p'$ are parallel. A polygon $P$ on $p$ has $m$ sides and a polygon $P'$ on $p'$ has $n$ sides. Find the largest and smallest distances between a vertex of $P$ and a vertex of $P'$.
2015 Tournament of Towns, 7
It is well-known that if a quadrilateral has the circumcircle and the incircle with the same centre then it is a square. Is the similar statement true in 3 dimensions: namely, if a cuboid is inscribed into a sphere and circumscribed around a sphere and the centres of the spheres coincide, does it imply that the cuboid is a cube? (A cuboid is a polyhedron with 6 quadrilateral faces such that each vertex belongs to $3$ edges.)
[i]($10$ points)[/i]
1997 Rioplatense Mathematical Olympiad, Level 3, 2
Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$. A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA_1$ , $BB _1$ and $DD_ 1$ of the prism. Calculate $r$ .
2015 Sharygin Geometry Olympiad, P23
A tetrahedron $ABCD$ is given. The incircles of triangles $ ABC$ and $ABD$ with centers $O_1, O_2$, touch $AB$ at points $T_1, T_2$. The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$. The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.
2017 Yasinsky Geometry Olympiad, 2
In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.
2017 Yasinsky Geometry Olympiad, 5
$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .
KoMaL A Problems 2018/2019, A. 737
$100$ points are given in space such that no four of them lie in the same plane. Consider those convex polyhedra with five vertices that have all vertices from the given set. Prove that the number of such polyhedra is even.
2015 Caucasus Mathematical Olympiad, 4
The midpoint of the edge $SA$ of the triangular pyramid of $SABC$ has equal distances from all the vertices of the pyramid. Let $SH$ be the height of the pyramid. Prove that $BA^2 + BH^2 = C A^2 + CH^2$.
1992 Mexico National Olympiad, 1
The tetrahedron $OPQR$ has the $\angle POQ = \angle POR = \angle QOR = 90^o$. $X, Y, Z$ are the midpoints of $PQ, QR$ and $RP.$ Show that the four faces of the tetrahedron $OXYZ$ have equal area.
2017 Yasinsky Geometry Olympiad, 5
Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$, when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$.
1978 Vietnam National Olympiad, 6
Given a rectangular parallelepiped $ABCDA'B'C'D'$ with the bases $ABCD, A'B'C'D'$, the edges $AA',BB', CC',DD'$ and $AB = a,AD = b,AA' = c$. Show that there exists a triangle with the sides equal to the distances from $A,A',D$ to the diagonal $BD'$ of the parallelepiped. Denote those distances by $m_1,m_2,m_3$. Find the relationship between $a, b, c,m_1,m_2,m_3$.
1976 Vietnam National Olympiad, 5
$L, L'$ are two skew lines in space and $p$ is a plane not containing either line. $M$ is a variable line parallel to $p$ which meets $L$ at $X$ and $L'$ at $Y$. Find the position of $M$ which minimises the distance $XY$. $L''$ is another fixed line. Find the line $M$ which is also perpendicular to $L''$ .
2014 Sharygin Geometry Olympiad, 22
Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?
2000 Austrian-Polish Competition, 2
In a unit cube, $CG$ is the edge perpendicular to the face $ABCD$. Let $O_1$ be the incircle of square $ABCD$ and $O_2$ be the circumcircle of triangle $BDG$. Determine min$\{XY|X\in O_1,Y\in O_2\}$.
1998 Mexico National Olympiad, 6
A plane in space is equidistant from a set of points if its distances from the points in the set are equal. What is the largest possible number of equidistant planes from five points, no four of which are coplanar?
2014 Sharygin Geometry Olympiad, 24
A circumscribed pyramid $ABCDS$ is given. The opposite sidelines of its base meet at points $P$ and $Q$ in such a way that $A$ and $B$ lie on segments $PD$ and $PC$ respectively. The inscribed sphere touches faces $ABS$ and $BCS$ at points $K$ and $L$. Prove that if $PK$ and $QL$ are complanar then the touching point of the sphere with the base lies on $BD$.
1979 Vietnam National Olympiad, 6
$ABCD$ is a rectangle with $BC / AB = \sqrt2$. $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.
1964 Vietnam National Olympiad, 3
Let $P$ be a plane and two points $A \in (P),O \notin (P)$. For each line in $(P)$ through $A$, let $H$ be the foot of the perpendicular from $O$ to the line. Find the locus $(c)$ of $H$.
Denote by $(C)$ the oblique cone with peak $O$ and base $(c)$. Prove that all planes, either parallel to $(P)$ or perpendicular to $OA$, intersect $(C)$ by circles.
Consider the two symmetric faces of $(C)$ that intersect $(C)$ by the angles $\alpha$ and $\beta$ respectively. Find a relation between $\alpha$ and $\beta$.
2017 Yasinsky Geometry Olympiad, 2
Prove that if all the edges of the tetrahedron are equal triangles (such a tetrahedron is called equilateral), then its projection on the plane of a face is a triangle.
2007 Sharygin Geometry Olympiad, 5
Each edge of a convex polyhedron is shifted such that the obtained edges form the frame of another convex polyhedron. Are these two polyhedra necessarily congruent?
2018 Yasinsky Geometry Olympiad, 3
In the tetrahedron $SABC$, points $E, F, K, L$ are the midpoints of the sides $SA , BC, AC, SB$ respectively, . The lengths of the segments $EF$ and $KL$ are equal to $11 cm$ and $13 cm$ respectively, and the length of the segment $AB$ equals to $18 cm$. Find the length of the side $SC$ of the tetrahedron.
2020 Polish Junior MO First Round, 7.
Consider the right prism with the rhombus with side $a$ and acute angle $60^{\circ}$ as a base. This prism was intersected by some plane intersecting its side edges, such that the cross-section of the prism and the plane is a square. Determine all possible lengths of the side of this square.