Olympiad problems

1979 Bulgaria National Olympiad, Problem 2

Points $P,Q,R,S$ are taken on respective edges $AC$, $AB$, $BD$, and $CD$ of a tetrahedron $ABCD$ so that $PR$ and $QS$ intersect at point $N$ and $PS$ and $QR$ intersect at point $M$. The line $MN$ meets the plane $ABC$ at point $L$. Prove that the lines $AL$, $BP$, and $CQ$ are concurrent.

2012 Online Math Open Problems, 50

Tags: geometry , 3d geometry , tetrahedron , circumcircle , sphere , incenter

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be the largest possible value of the circumradius of face $ABC$. Given that $R$ can be expressed in the form $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Author: Alex Zhu[/i]

1956 Putnam, B3

Tags: geometry , 3d geometry , sphere , tetrahedron

A sphere is inscribed in a tetrahedron and each point of contact of the sphere with the four faces is joined to the vertices of the face containing the point. Show that the four sets of three angles so formed are identical.

2007 Romania National Olympiad, 3

Tags: inequalities , geometry , 3d geometry , tetrahedron

a) In a triangle $ MNP$, the lenghts of the sides are less than $ 2$. Prove that the lenght of the altitude corresponding to the side $ MN$ is less than $ \sqrt {4 \minus{} \frac {MN^2}{4}}$. b) In a tetrahedron $ ABCD$, at least $ 5$ edges have their lenghts less than $ 2$.Prove that the volume of the tetrahedron is less than $ 1$.

1996 Romania National Olympiad, 2

Tags: geometry , 3d geometry , fixed , tetrahedron

Let $ABCD$ a tetrahedron and $M$ a variable point on the face $BCD$. The line perpendicular to $(BCD)$ in $M$ . intersects the planes$ (ABC)$, $(ACD)$, and $(ADB)$ in $M_1$, $M_2$, and $M_3$. Show that the sum $MM_1 + MM_2 + MM_3$ is constant if and only if the perpendicular dropped from $A$ to $(BCD)$ passes through the centroid of triangle $BCD$.

1972 Dutch Mathematical Olympiad, 3

Tags: geometry , tetrahedron , geometric inequality , 3d geometry

$ABCD$ is a regular tetrahedron. The points $P,Q,R$ and $S$ lie outside this tetrahedron in such a way that $ABCP$, $ABDQ$, $ACDR$ and $BCDS$ are regular tetrahedra. Prove that the volume of the tetrahedron $PQRS$ is less than the sum of the volumes of $ABCP$,$ABDQ$,$ACDR$, $BCDS$ and $ABCD$.

1979 Vietnam National Olympiad, 6

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

1983 Bulgaria National Olympiad, Problem 3

Tags: geometry , 3d geometry , tetrahedron , ratio

A regular triangular pyramid $ABCD$ with the base side $AB=a$ and the lateral edge $AD=b$ is given. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively. A line $\alpha$ through $MN$ intersects the edges $AD$ and $BC$ at $P$ and $Q$, respectively. (a) Prove that $AP/AD=BQ/BC$. (b) Find the ratio $AP/AD$ which minimizes the area of $MQNP$.

2003 Austrian-Polish Competition, 6

Tags: geometry , 3d geometry , sphere , tetrahedron

$ABCD$ is a tetrahedron such that we can find a sphere $k(A,B,C)$ through $A, B, C$ which meets the plane $BCD$ in the circle diameter $BC$, meets the plane $ACD$ in the circle diameter $AC$, and meets the plane $ABD$ in the circle diameter $AB$. Show that there exist spheres $k(A,B,D)$, $k(B,C,D)$ and $k(C,A,D)$ with analogous properties.

1970 IMO, 2

Tags: geometry , 3d geometry , tetrahedron , geometric inequality

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?

2006 China Team Selection Test, 1

Tags: geometry , circumcircle , parallelogram , 3d geometry , tetrahedron , geometric transformation , homothety

Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.

1956 Moscow Mathematical Olympiad, 329

Tags: geometry , 3d geometry , tetrahedron , locus , altitude , area

Consider positive numbers $h, s_1, s_2$, and a spatial triangle $\vartriangle ABC$. How many ways are there to select a point $D$ so that the height of tetrahedron $ABCD$ drawn from $D$ equals $h$, and the areas of faces $ACD$ and $BCD$ equal $s_1$ and $s_2$, respectively?

2005 Swedish Mathematical Competition, 6

Tags: geometry , 3d geometry , tetrahedron

A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.

2008 Sharygin Geometry Olympiad, 24

Tags: inequalities , 3d geometry , tetrahedron , geometry

(I.Bogdanov, 11) Let $ h$ be the least altitude of a tetrahedron, and $ d$ the least distance between its opposite edges. For what values of $ t$ the inequality $ d>th$ is possible?

2024 All-Russian Olympiad, 1

Tags: 3d geometry , tetrahedron , geometry

We are given an infinite cylinder in space (i.e. the locus of points of a given distance $R>0$ from a given straight line). Can six straight lines containing the edges of a tetrahedron all have exactly one common point with this cylinder? [i]Proposed by A. Kuznetsov[/i]

2013 Sharygin Geometry Olympiad, 3

Tags: incenter , 3d geometry , tetrahedron , sphere , geometry

Let $X$ be a point inside triangle $ABC$ such that $XA.BC=XB.AC=XC.AC$. Let $I_1, I_2, I_3$ be the incenters of $XBC, XCA, XAB$. Prove that $AI_1, BI_2, CI_3$ are concurrent. [hide]Of course, the most natural way to solve this is the Ceva sin theorem, but there is an another approach that may surprise you;), try not to use the Ceva theorem :))[/hide]

1983 Polish MO Finals, 6

Tags: tetrahedron , dihedral angle , faces , 3d geometry , geometry

Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.

1997 Vietnam Team Selection Test, 1

Tags: 3d geometry , tetrahedron , circumcircle , inequalities , analytic geometry , geometry

Let $ ABCD$ be a given tetrahedron, with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$, $ DA \equal{} a_1$, $ DB \equal{} b_1$, $ DC \equal{} c_1$. Prove that there is a unique point $ P$ satisfying \[ PA^2 \plus{} a_1^2 \plus{} b^2 \plus{} c^2 \equal{} PB^2 \plus{} b_1^2 \plus{} c^2 \plus{} a^2 \equal{} PC^2 \plus{} c_1^2 \plus{} a^2 \plus{} b^2 \equal{} PD^2 \plus{} a_1^2 \plus{} b_1^2 \plus{} c_1^2 \] and for this point $ P$ we have $ PA^2 \plus{} PB^2 \plus{} PC^2 \plus{} PD^2 \ge 4R^2$, where $ R$ is the circumradius of the tetrahedron $ ABCD$. Find the necessary and sufficient condition so that this inequality is an equality.

1977 IMO Longlists, 48

Tags: 3d geometry , tetrahedron , perimeter , geometry

The intersection of a plane with a regular tetrahedron with edge $a$ is a quadrilateral with perimeter $P.$ Prove that $2a \leq P \leq 3a.$

1964 IMO, 6

Tags: geometry , 3d geometry , tetrahedron , analytic geometry

In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?

2017 Sharygin Geometry Olympiad, P24

Tags: geometry , tetrahedron , similarity , 3d geometry

Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?

1977 IMO Longlists, 44

Tags: 3d geometry , tetrahedron , geometry

Let $E$ be a finite set of points in space such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that $E$ contains the vertices of a tetrahedron $T = ABCD$ such that $T \cap E = \{A,B,C,D\}$ (including interior points of $T$ ) and such that the projection of $A$ onto the plane $BCD$ is inside a triangle that is similar to the triangle $BCD$ and whose sides have midpoints $B,C,D.$

1998 Tuymaada Olympiad, 4

Tags: plane , geometry , tetrahedron , dissecting plane , 3d geometry , solid geometry

Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.

1962 IMO Shortlist, 7

Tags: geometry , 3d geometry , tetrahedron , sphere , incenter

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB,$ or to their extensions. a) Prove that the tetrahedron $SABC$ is regular. b) Prove conversely that for every regular tetrahedron five such spheres exist.

1967 Bulgaria National Olympiad, Problem 4

Tags: geometry , 3d geometry , tetrahedron

Outside of the plane of the triangle $ABC$ is given point $D$. (a) prove that if the segment $DA$ is perpendicular to the plane $ABC$ then orthogonal projection of the orthocenter of the triangle $ABC$ on the plane $BCD$ coincides with the orthocenter of the triangle $BCD$. (b) for all tetrahedrons $ABCD$ with base, the triangle $ABC$ with smallest of the four heights that from the vertex $D$, find the locus of the foot of that height.