Olympiad problems

2009 Romanian Master of Mathematics, 3

Tags: rhombus , geometry , circumcircle , 3d geometry , tetrahedron , prism

Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that \[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, \] denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel. [i]Nikolai Ivanov Beluhov, Bulgaria[/i]

Kyiv City MO Seniors 2003+ geometry, 2003.11.3

Tags: inequalities , 3d geometry , geometrical inequality , tetrahedron , geometry

Let $x_1, x_2, x_3, x_4$ be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let $h_1, h_2, h_3, h_4$ be the corresponding heights of the tetrahedron. Prove that $$\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$ (Dmitry Nomirovsky)

2022/2023 Tournament of Towns, P6

Tags: geometry , tetrahedron , 3d geometry

The midpoints of all heights of a certain tetrahedron lie on its inscribed sphere. Is this tetrahedron necessarily regular then?

1988 Balkan MO, 3

Tags: geometry , 3d geometry , tetrahedron

Let $ABCD$ be a tetrahedron and let $d$ be the sum of squares of its edges' lengths. Prove that the tetrahedron can be included in a region bounded by two parallel planes, the distances between the planes being at most $\frac{\sqrt{d}}{2\sqrt{3}}$

2003 Romania National Olympiad, 4

Tags: geometry , 3d geometry , tetrahedron

In tetrahedron $ ABCD$, $ G_1,G_2$ and $ G_3$ are barycenters of the faces $ ACD,ABD$ and $ BCD$ respectively. (a) Prove that the straight lines $ BG_1,CG_2$ and $ AG_3$ are concurrent. (b) Knowing that $ AG_3\equal{}8,BG_1\equal{}12$ and $ CG_2\equal{}20$ compute the maximum possible value of the volume of $ ABCD$.

1986 ITAMO, 5

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

Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.

1980 IMO, 3

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

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

1987 Traian Lălescu, 1.4

Tags: geometry , perimeter , 3d geometry , tetrahedron

Let $ ABCD $ be a regular tetahedron and $ M,N $ be middlepoints for $ AD, $ respectively, $ BC. $ Through a point $ P $ that is on segment $ MN, $ passes a plane perpendicular on $ MN, $ and meets the sides $ AB,AC,CD,BD $ of the tetahedron at $ E,F,G, $ respectively, $ H. $ [b]a)[/b] Prove that the perimeter of the quadrilateral $ EFGH $ doesn't depend on $ P. $ [b]b)[/b] Determine the maximum area of $ EFGH $ (depending on a side of the tetahedron).

II Soros Olympiad 1995 - 96 (Russia), 11.2

Tags: 3d geometry , tetrahedron , geometry

Is it possible that the heights of a tetrahedron (that is, a triangular pyramid) would be equal to the numbers $1$, $2$, $3$ and $6$?

1987 Traian Lălescu, 1.3

Tags: geometry , 3d geometry , tetrahedron

Let $ ABCD $ be a tetahedron and $ M,N $ the middlepoints of $ AB, $ respectively, $ CD. $ Show that any plane that contains $ M $ and $ N $ cuts the tetrahedron in two polihedra that have same volume.

1995 Poland - Second Round, 5

Tags: 3d geometry , tetrahedron , concyclic , incircle , geometry

The incircles of the faces $ABC$ and $ABD$ of a tetrahedron $ABCD$ are tangent to the edge $AB$ in the same point. Prove that the points of tangency of these incircles to the edges $AC,BC,AD,BD$ are concyclic.

1977 Bulgaria National Olympiad, Problem 2

Tags: 3d geometry , tetrahedron , geometry

In the space are given $n$ points and no four of them belongs to a common plane. Some of the points are connected with segments. It is known that four of the given points are vertices of tetrahedron which edges belong to the segments given. It is also known that common number of the segments, passing through vertices of tetrahedron is $2n$. Prove that there exists at least two tetrahedrons every one of which have a common face with the first (initial) tetrahedron. [i]N. Nenov, N. Hadzhiivanov[/i]

1986 China Team Selection Test, 2

Tags: function , inequalities , perimeter , geometry , 3d geometry , tetrahedron

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

2000 Harvard-MIT Mathematics Tournament, 7

Tags: 3d geometry , tetrahedron , geometry

A regular tetrahedron of volume $1$ is filled with water of total volume $\frac{7}{16}$. Is it possible that the center of the tetrahedron lies on the surface of the water? How about in a cube of volume $1$?

2006 Federal Math Competition of S&M, Problem 3

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

Show that for an arbitrary tetrahedron there are two planes such that the ratio of the areas of the projections of the tetrahedron onto the two planes is not less than $\sqrt2$.

1984 IMO Longlists, 65

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

A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.

1949 Putnam, A4

Tags: geometry , 3d geometry , tetrahedron

Given that $P$ is a point inside a tetrahedron with vertices at $A, B, C$ and $D$, such that the sum of the distances $PA+PB+PC+PD$ is a minimum, show that the two angles $\angle APB$ and $\angle CPD$ are equal and are bisected by the same straight line. What other pair of angles must be equal?

1984 Bulgaria National Olympiad, Problem 3

Tags: geometry , tetrahedron , combinatorial geometry , combinatorics , 3d geometry

Points $P_1,P_2,\ldots,P_n,Q$ are given in space $(n\ge4)$, no four of which are in a plane. Prove that if for any three distinct points $P_\alpha,P_\beta,P_\gamma$ there is a point $P_\delta$ such that the tetrahedron $P_\alpha P_\beta P_\gamma P_\delta$ contains the point $Q$, then $n$ is an even number.

1969 IMO Longlists, 12

Tags: tetrahedron , geometry , 3d geometry , locus , locus problems

$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.

2006 China Team Selection Test, 1

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

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.

2000 IMC, 4

Tags: tetrahedron , 3d geometry , inequalities

Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$. Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.

1937 Moscow Mathematical Olympiad, 034

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

Two segments slide along two skew lines. On each straight line there is a segment. Consider the tetrahedron with vertices at the endpoints of the segments. Prove that the volume of the tetrahedron does not depend on the position of the segments

1977 IMO Longlists, 44

Tags: geometry , 3d geometry , tetrahedron

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 $B,C,D.$

2022 AMC 12/AHSME, 12

Tags: geometry , trigonometry , tetrahedron

Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. What is $\cos({\angle CMD})$? $\textbf{(A)} ~\frac{1}{4} \qquad\textbf{(B)} ~\frac{1}{3} \qquad\textbf{(C)} ~\frac{2}{5} \qquad\textbf{(D)} ~\frac{1}{2} \qquad\textbf{(E)} ~\frac{\sqrt{3}}{2} $

1982 National High School Mathematics League, 9

Tags: geometry , tetrahedron , 3d geometry

In tetrahedron $SABC$, $\angle ASB=\frac{\pi}{2}, \angle ASC=\alpha(0<\alpha<\frac{\pi}{2}), \angle BSC=\beta(0<\beta<\frac{\pi}{2})$. Let $\theta=A-SC-B$, prove that $\theta=-\arccos(\cot\alpha\cdot\cot\beta)$.