Olympiad problems

1999 USAMTS Problems, 4

Tags: geometry , 3d geometry , tetrahedron , symmetry , ratio , pythagorean theorem

There are $8436$ steel balls, each with radius $1$ centimeter, stacked in a tetrahedral pile, with one ball on top, $3$ balls in the second layer, $6$ in the third layer, $10$ in the fourth, and so on. Determine the height of the pile in centimeters.

1970 IMO Longlists, 20

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

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

1980 AMC 12/AHSME, 16

Tags: geometry , ratio , 3d geometry , tetrahedron , congruent triangles

Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron. $\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ \sqrt 3 \qquad \text{(C)} \ \sqrt{\frac{3}{2}} \qquad \text{(D)} \ \frac{2}{\sqrt{3}} \qquad \text{(E)} \ 2$

1968 Bulgaria National Olympiad, Problem 5

Tags: geometry , 3d geometry , tetrahedron , ratio

The point $M$ is inside the tetrahedron $ABCD$ and the intersection points of the lines $AM,BM,CM$ and $DM$ with the opposite walls are denoted with $A_1,B_1,C_1,D_1$ respectively. It is given also that the ratios $\frac{MA}{MA_1}$, $\frac{MB}{MB_1}$, $\frac{MC}{MC_1}$, and $\frac{MD}{MD_1}$ are equal to the same number $k$. Find all possible values of $k$. [i]K. Petrov[/i]

Kyiv City MO 1984-93 - geometry, 1988.10.2

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

Given an arbitrary tetrahedron. Prove that its six edges can be divided into two triplets so that from each triple it was possible to form a triangle.

2000 Belarusian National Olympiad, 6

Tags: geometry , 3d geometry , tetrahedron

A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have?

1977 Bulgaria National Olympiad, Problem 2

Tags: geometry , 3d geometry , tetrahedron

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]

1989 Brazil National Olympiad, 5

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

A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron. Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.

1984 National High School Mathematics League, 8

Tags: function , geometry , 3d geometry , tetrahedron

Lengths of five edges of a tetrahedron are $1$, while the last one is $x$. Its volume is $F(x)$. On its domain of definition, we have $\text{(A)}$ $F(x)$ is an increasing function, it has no maximum value. $\text{(B)}$ $F(x)$ is an increasing function, it has maximum value. $\text{(C)}$ $F(x)$ is not an increasing function, it has no maximum value. $\text{(D)}$ $F(x)$ is an increasing function, it has maximum value.

1982 All Soviet Union Mathematical Olympiad, 334

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

Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)

2002 Austrian-Polish Competition, 3

Tags: 3d geometry , tetrahedron , geometry

Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$. Prove that \[\frac{1}{3}\leq \frac{KS}{LS}\leq 3\]

1984 Austrian-Polish Competition, 1

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

Prove that if the feet of the altitudes of a tetrahedron are the incenters of the corresponding faces, then the tetrahedron is regular.

2015 Sharygin Geometry Olympiad, P24

Tags: sphere , tetrahedron , 3-dimensional geometry , 3d geometry , 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 $A_2B_2C_2D_2$.

2019 Jozsef Wildt International Math Competition, W. 68

Tags: tetrahedron , 3d geometry , inequalities , exradius , inradius

In all tetrahedron $ABCD$ holds [list=1] [*] $\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}$ [*] $\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}$ [/list] for all $t\in [0,1]$

2015 CCA Math Bonanza, I13

Tags: geometry , 3d geometry , tetrahedron , rectangle

Let $ABCD$ be a tetrahedron such that $AD \perp BD$, $BD \perp CD$, $CD \perp AD$ and $AD=10$, $BD=15$, $CD=20$. Let $E$ and $F$ be points such that $E$ lies on $BC$, $DE \perp BC$, and $ADEF$ is a rectangle. If $S$ is the solid consisting of all points inside $ABCD$ but outside $FBCD$, compute the volume of $S$. [i]2015 CCA Math Bonanza Individual Round #13[/i]

1988 Czech And Slovak Olympiad IIIA, 3

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

Given a tetrahedron $ABCD$ with edges $|AD|=|BC|= a$, $|AC|=|BD|=b$, $AB=c$ and $|CD| = d$. Determine the smallest value of the sum $|AX|+|BX|+|CX|+|DX|$, where $X$ is any point in space.

1986 Polish MO Finals, 2

Tags: volume , tetrahedron , 3d geometry , geometry

Find the maximum possible volume of a tetrahedron which has three faces with area $1$.

1951 Moscow Mathematical Olympiad, 205

Tags: geometry , 3d geometry , tetrahedron , maximum , area , projection

Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.

1972 USAMO, 2

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

A given tetrahedron $ ABCD$ is isoceles, that is, $ AB\equal{}CD$, $ AC\equal{}BD$, $ AD\equal{}BC$. Show that the faces of the tetrahedron are acute-angled triangles.

2014 Online Math Open Problems, 29

Tags: geometry , 3d geometry , tetrahedron , circumcircle , calculus , integration

Let $ABCD$ be a tetrahedron whose six side lengths are all integers, and let $N$ denote the sum of these side lengths. There exists a point $P$ inside $ABCD$ such that the feet from $P$ onto the faces of the tetrahedron are the orthocenter of $\triangle ABC$, centroid of $\triangle BCD$, circumcenter of $\triangle CDA$, and orthocenter of $\triangle DAB$. If $CD = 3$ and $N < 100{,}000$, determine the maximum possible value of $N$. [i]Proposed by Sammy Luo and Evan Chen[/i]

1989 Swedish Mathematical Competition, 4

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

Let $ABCD$ be a regular tetrahedron. Find the positions of point $P$ on the edge $BD$ such that the edge $CD$ is tangent to the sphere with diameter $AP$.

2011 Pre-Preparation Course Examination, 3

Tags: geometry , 3d geometry , tetrahedron , combinatorics

Calculate number of the hamiltonian cycles of the graph below: (15 points)

1984 Bulgaria National Olympiad, Problem 3

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

1991 Romania Team Selection Test, 2

Tags: plane , concurrent planes , tetrahedron , concurrency

Let $A_1A_2A_3A_4$ be a tetrahedron. For any permutation $(i, j,k,h)$ of $1,2,3,4$ denote: - $P_i$ – the orthogonal projection of $A_i$ on $A_jA_kA_h$; - $B_{ij}$ – the midpoint of the edge $A_iAj$, - $C_{ij}$ – the midpoint of segment $P_iP_j$ - $\beta_{ij}$– the plane $B_{ij}P_hP_k$ - $\delta_{ij}$ – the plane $B_{ij}P_iP_j$ - $\alpha_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_kA_h$ - $\gamma_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_iA_j$. Prove that if the points $P_i$ are not in a plane, then the following sets of planes are concurrent: (a) $\alpha_{ij}$, (b) $\beta_{ij}$, (c) $\gamma_{ij}$, (d) $\delta_{ij}$.

1952 Miklós Schweitzer, 1

Tags: 3d geometry , pyramid , tetrahedron , convex , geometry

Find all convex polyhedra which have no diagonals (that is, for which every segment connecting two vertices lies on the boundary of the polyhedron).