Olympiad problems

2003 Tournament Of Towns, 5

A paper tetrahedron is cut along some of so that it can be developed onto the plane. Could it happen that this development cannot be placed on the plane in one layer?

1967 IMO Longlists, 36

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

Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.

1983 All Soviet Union Mathematical Olympiad, 358

Tags: plane , 3d geometry , geometry , combinatorial geometry , parallel , projection , tetrahedron

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 $(D_1D_2)$ .

1961 Poland - Second Round, 2

Tags: geometry , tetrahedron , 3d geometry , concurrency

Prove that all the heights of a tetrahedron intersect at one point if and only if the sums of the squares of the opposite edges are equal.

2024 JHMT HS, 10

Tags: geometry , 3d geometry , sphere , tetrahedron

One triangular face $F$ of a tetrahedron $\mathcal{T}$ has side lengths $\sqrt{5}$, $\sqrt{65}$, and $2\sqrt{17}$. The other three faces of $\mathcal{T}$ are right triangles whose hypotenuses coincide with the sides of $F$. There exists a sphere inside $\mathcal{T}$ tangent to all four of its faces. Compute the radius of this sphere.

1993 French Mathematical Olympiad, Problem 5

Tags: geometry , 3d geometry , tetrahedron , inequalities

(a) Let there be two given points $A,B$ in the plane. i. Find the triangles $MAB$ with the given area and the minimal perimeter. ii. Find the triangles $MAB$ with a given perimeter and the maximal area. (b) In a tetrahedron of volume $V$, let $a,b,c,d$ be the lengths of its four edges, no three of which are coplanar, and let $L=a+b+c+d$. Determine the maximum value of $\frac V{L^3}$.

1960 Polish MO Finals, 2

Tags: geometry , 3d geometry , tetrahedron , trigonometry

A plane is drawn through the height of a regular tetrahedron, which intersects the planes of the lateral faces along $ 3 $ lines that form angles $ \alpha $, $ \beta $, $ \gamma $ with the plane of the tetrahedron's base. Prove that $$ tg^2 \alpha + tg^2 \beta + tg^2 \gamma =12.$$

1990 Vietnam National Olympiad, 3

Tags: 3d geometry , tetrahedron , geometry

A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron. (a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron? (b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.

2019 Tournament Of Towns, 5

Tags: orthogonal projection , projection , 3d geometry , geometry , tetrahedron , trapezoid

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)

1987 National High School Mathematics League, 8

Tags: 3d geometry , tetrahedron , geometry

We have two triangles that lengths of its sides are $3,4,5$, one triangle that lengths of its sides are $4,5,\sqrt{41}$, one triangle that lengths of its sides are $\frac{5}{6}\sqrt2,4,5$. The number of tetrahedrons with such four surfaces is________.

1973 Bulgaria National Olympiad, Problem 6

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

In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal. [i]H. Lesov[/i]

1993 Austrian-Polish Competition, 2

Tags: geometry , solid geometry , 3d geometry , tetrahedron , volume

Consider all tetrahedra $ABCD$ in which the sum of the areas of the faces $ABD, ACD, BCD$ does not exceed $1$. Among such tetrahedra, find those with the maximum volume.

2008 Flanders Math Olympiad, 3

Tags: geometry , 3d geometry , tetrahedron , pyramid

A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they have in common $1$ equilateral triangle . Prove that the resulting body has exactly $5$ sides.

PEN R Problems, 2

Tags: 3d geometry , tetrahedron , vector , modular arithmetic , the geometry of numbers , geometry

Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.

1969 IMO, 3

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

For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.

2008 District Olympiad, 1

Tags: rhombus , plane , 3d geometry , tetrahedron , square , geometry

A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.

2005 Harvard-MIT Mathematics Tournament, 2

Tags: geometry , 3d geometry , tetrahedron , symmetry

Let $ABCD$ be a regular tetrahedron with side length $2$. The plane parallel to edges $AB$ and $CD$ and lying halfway between them cuts $ABCD$ into two pieces. Find the surface area of one of these pieces.

1983 Canada National Olympiad, 3

Tags: 3d geometry , tetrahedron , area of a triangle , geometry

The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?

2006 Kazakhstan National Olympiad, 6

Tags: plane , geometry , 3d geometry , tetrahedron , perpendicular

In the tetrahedron $ ABCD $ from the vertex $ A $, the perpendiculars $ AB '$, $ AC' $ are drawn, $ AD '$ on planes dividing dihedral angles at edges $ CD $, $ BD $, $ BC $ in half. Prove that the plane $ (B'C'D ') $ is parallel to the plane $ (BCD) $.

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.

2024 All-Russian Olympiad Regional Round, 11.8

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

3 segments $AA_1$, $BB_1$, $CC_1$ in space share a common midpoint $M$. Turns out, the sphere circumscribed about the tetrahedron $MA_1B_1C_1$ is tangent to plane $ABC$ at point $D$. Point $O$ is the circumcenter of triangle $ABC$. Prove that $MO = MD$.

1977 IMO Longlists, 9

Tags: 3d geometry , tetrahedron , geometry

Let $ABCD$ be a regular tetrahedron and $\mathbf{Z}$ an isometry mapping $A,B,C,D$ into $B,C,D,A$, respectively. Find the set $M$ of all points $X$ of the face $ABC$ whose distance from $\mathbf{Z}(X)$ is equal to a given number $t$. Find necessary and sufficient conditions for the set $M$ to be nonempty.

2019 Jozsef Wildt International Math Competition, W. 58

Tags: tetrahedron , circumsphere , insphere , 3d geometry , inequalities , exsphere

In the $[ABCD]$ tetrahedron having all the faces acute angled triangles, is denoted by $r_X$, $R_X$ the radius lengths of the circle inscribed and circumscribed respectively on the face opposite to the $X \in \{A,B,C,D\}$ peak, and with $R$ the length of the radius of the sphere circumscribed to the tetrahedron. Show that inequality occurs$$8R^2 \geq (r_A + R_A)^2 + (r_B + R_B)^2 + (r_C + R_C)^2 + (r_D + R_D)^2$$

1948 Kurschak Competition, 2

Tags: geometry , 3d geometry , polyhedron , tetrahedron

A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.

2021 Abels Math Contest (Norwegian MO) Final, 4a

Tags: right angle , geometry , 3d geometry , tetrahedron

A tetrahedron $ABCD$ satisfies $\angle BAC=\angle CAD=\angle DAB=90^o$. Show that the areas of its faces satisfy the equation $area(BAC)^2 + area(CAD)^2 + area(DAB)^2 = area(BCD)^2$. .