Olympiad problems

2017 Yasinsky Geometry Olympiad, 2

Tags: 3-dimensional geometry , 3d geometry , geometry , equal angles , tetrahedron , perpendicular

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

1988 Mexico National Olympiad, 8

Tags: 3-dimensional geometry , octahedron , geometry , sphere

Compute the volume of a regular octahedron circumscribed about a sphere of radius $1$.

2015 Sharygin Geometry Olympiad, P24

Tags: 3d geometry , 3-dimensional geometry , sphere , tetrahedron , geometry

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 $A_2B_2C_2D_2$.

2017 Yasinsky Geometry Olympiad, 5

Tags: 3-dimensional geometry , 3d geometry , solid geometry , rectangle , geometry

$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .

1977 Vietnam National Olympiad, 6

Tags: 3-dimensional geometry , distance , geometry , maximum , minimum

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 $P'$.

2007 Sharygin Geometry Olympiad, 5

Tags: polyhedron , convex , 3-dimensional geometry , combinatorial geometry , geometry

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?

2015 Sharygin Geometry Olympiad, P23

Tags: plane , 3-dimensional geometry , 3d geometry , tetrahedron , geometry

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.

1998 Mexico National Olympiad, 6

Tags: geometry , combinatorial geometry , 3-dimensional geometry , distance

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?

2015 Tournament of Towns, 7

Tags: 3-dimensional geometry , geometry , circumcircle

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]

2014 Sharygin Geometry Olympiad, 22

Tags: geometry , polyhedron , convex , 3-dimensional geometry

Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?

2019 Azerbaijan Senior NMO, 4

Tags: plane geometry , 3-dimensional geometry , dimension

Is it possible to construct a equilateral triangle such that: $\text{a)}$ Coordinates of this triangle are integers in two dimensional plane? $\text{b)}$ Coordinates of this triangle are integers in three dimensional plane?

1997 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: geometry , rhombus , 3-dimensional geometry , prism , 3d geometry , sphere

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 $r$ .

1979 Vietnam National Olympiad, 6

Tags: 3-dimensional geometry , tetrahedron , 3d geometry , construction , geometry

$ABCD$ 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.

1987 Mexico National Olympiad, 8

Tags: geometry , 3-dimensional geometry , solid geometry , perpendicular

(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 $PQR$.

1992 Mexico National Olympiad, 1

Tags: tetrahedron , geometry , 3-dimensional geometry

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.

2000 Austrian-Polish Competition, 2

Tags: geometry , 3-dimensional geometry , distance , minimum , 3d geometry

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$\{XY|X\in O_1,Y\in O_2\}$.

1972 Vietnam National Olympiad, 4

Tags: geometry , angle , tetrahedron , 3-dimensional geometry , solid geometry , 3d geometry

Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.

1996 Spain Mathematical Olympiad, 6

Tags: geometry , regular polygon , volume , 3-dimensional geometry

A regular pentagon is constructed externally on each side of a regular pentagon of side $1$. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.

2020 Polish Junior MO First Round, 7.

Tags: 3-dimensional geometry , 3d geometry , prism , rhombus , geometry

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.

1991 Mexico National Olympiad, 3

Tags: geometry , sphere , 3-dimensional geometry

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?

1965 Vietnam National Olympiad, 2

Tags: tetrahedron , geometry , locus , 3-dimensional geometry , 3d geometry

$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$. Find the locus of $X$. What happens to $X$ as $M$ tends to (1) $D$, (2) $C$? Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.

2015 Caucasus Mathematical Olympiad, 4

Tags: geometry , 3d geometry , pyramid , 3-dimensional geometry

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 $BA^2 + BH^2 = C A^2 + CH^2$.

2017 Yasinsky Geometry Olympiad, 5

Tags: geometry , 3d geometry , 3-dimensional geometry , cube , area

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 $CC_1$.

1976 Vietnam National Olympiad, 5

Tags: geometry , 3d geometry , 3-dimensional geometry , line , distance

$L, L'$ 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 $L''$ .

1964 Vietnam National Olympiad, 3

Tags: 3-dimensional geometry , locus , geometry

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 $\beta$.