Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.

The tetrahedrons $ ABCD $ and $ A_1B_1C_1D_1 $ are situated so that the midpoints of the segments $ AA_1 $, $ BB_1 $, $ CC_1 $, $ DD_1 $ are the centroids of the triangles $BCD$, $ ACD $, $ A B D $ and $ ABC $, respectively. What is the ratio of the volumes of these tetrahedrons?

The net of 20 triangles shown below can be folded to form a regular icosahedron. Inside each of the triangular faces, write a number from 1 to 20 with each number used exactly once. Any pair of numbers that are consecutive must be written on faces sharing an edge in the folded icosahedron, and additionally, 1 and 20 must also be on faces sharing an edge. Some numbers have been given to you. (No proof is necessary.) [asy] unitsize(1cm); pair c(int a, int b){return (a-b/2,sqrt(3)*b/2);} draw(c(0,0)--c(0,1)--c(-1,1)--c(1,3)--c(1,1)--c(2,2)--c(3,2)--c(4,3)--c(4,2)--c(3,1)--c(2,1)--c(2,-1)--c(1,-1)--c(1,-2)--c(0,-3)--c(0,-2)--c(-1,-2)--c(1,0)--cycle); draw(c(0,0)--c(1,1)--c(0,1)--c(1,2)--c(0,2)--c(0,1),linetype("4 4")); draw(c(4,2)--c(3,2)--c(3,1),linetype("4 4")); draw(c(3,2)--c(1,0)--c(1,1)--c(2,1)--c(2,2),linetype("4 4")); draw(c(1,-2)--c(0,-2)--c(0,-1)--c(1,-1)--c(1,0)--c(2,0)--c(0,-2),linetype("4 4")); label("2",(c(0,2)+c(1,2))/2,S); label("15",(c(1,1)+c(2,1))/2,S); label("6",(c(0,1)+c(1,1))/2,N); label("14",(c(0,0)+c(1,0))/2,N);[/asy]

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

Two planes, $ \alpha $ and $ \beta, $ form a dihedral angle of $ 30^{\circ} , $ and their intersection is the line $ d. $ A point $ A $ situated at the exterior of this angle projects itself in $ P\not\in d $ on $ \alpha , $ and in $ Q\not\in d $ on $ \beta $ such that $ AQ<AP. $ Name $ B $ the projection of $ A $ upon $ d. $ [b]a)[/b] Are $ A,B,P,Q, $ coplanar? [b]b)[/b] Knowing that a perpendicular to $ \beta $ make with $ AB $ an angle of $ 60^{\circ} , $ and $ AB=4, $ find the area of $ BPQ. $

Is there exist four points on plane, such that distance between any two of them is integer odd number? May be it is geometry or number theory or combinatoric, I don't know, so it here :blush:

In a pyramid, the base is a right-angled triangle with integer sides. The height of the pyramid is also integer. Show that the volume of the pyramid is even.

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

Let $\mathsf{S}_1,\mathsf{S}_2$ be concentric spheres with radii $a,b$ respectively, where $a<b.$ Denote $ABCDA'B'C'D'$ a square cuboid ($ABCD,A'B'C'D$ are the squares and $AA'\parallel BB'\parallel CC'\parallel DD'$) such that $A,B,C,D\in\mathsf{S}_2$ and the plane $A'B'C'D'$ is tangent to $\mathsf{S}_1.$ Finally assume that \[\frac{AB}{AA'}=\frac ab.\] Compute the lengths $AB,AA'.$ How many of such cuboids exist (up to a congruence)?

Prove that for each tetrahedron, the three products of pairs of opposite edges are sides of a triangle.

Find the greatest positive integer $n$ for which $n$ is divisible by all positive integers whose cube is not greater than $n.$

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.

Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.

In the rectangular parallelepiped in the figure, the lengths of the segments $EH$, $HG$, and $EG$ are consecutive integers. The height of the parallelepiped is $12$. Find the volume of the parallelepiped. [img]https://cdn.artofproblemsolving.com/attachments/6/4/f74e7fed38c815bff5539613f76b0c4ca9171b.png[/img]

[asy] import olympiad; import cse5; size(5cm); pointpen = black; pair A = Drawing((10,17.32)); pair B = Drawing((0,0)); pair C = Drawing((20,0)); draw(A--B--C--cycle); pair X = 0.85*A + 0.15*B; pair Y = 0.82*A + 0.18*C; pair W = (-11,0) + X; pair Z = (19, 9); draw(W--X, EndArrow); draw(X--Y, EndArrow); draw(Y--Z, EndArrow); anglepen=black; anglefontpen=black; MarkAngle("\theta", C,Y,Z, 3); [/asy] The cross-section of a prism with index of refraction $1.5$ is an equilateral triangle, as shown above. A ray of light comes in horizontally from air into the prism, and has the opportunity to leave the prism, at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number. [i](Ahaan Rungta, 5 points)[/i]

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 $PQRS$).

A variable tetrahedron $ABCD$ has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$; and the vertices $A,B,C$ lie respectively on three fixed spheres with the same center $P$ and radii $3, 4, 12$. What is the maximal length of $PD$?

A tetrahedron $ABCD$ has all edges of length less than $100$, and contains two nonintersecting spheres of diameter $1$. Prove that it contains a sphere of diameter $1.01$.

The right prism $[A_1A_2A_3\ldots A_nA_1'A_2'A_3'\ldots A_n'],n\in\mathbb{N},n\ge 3$, has a convex polygon as its base. It is known that $A_1A_2'\perp A_2A_3',A_2A_3'\perp A_3A_4',$$\ldots A_{n-1}A_n'\perp A_nA_1', A_nA_1'\perp A_1A_2'$. Show that: $a)$ $n=3$; $b)$ the prism is regular.

Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$. [i]Proposed by Isabella Li[/i]

Consider the pyramid $OABC$. Let the equilateral triangle $ABC$ with side length $6$ be the base. Also $9=OA=OB=OC$. Let $M$ be the midpoint of $AB$. Find the square of the distance from $M$ to $OC$.

[b](a)[/b] [Problem for class 11] Let r be the inradius and $r_a$, $r_b$, $r_c$ the exradii of a triangle ABC. Prove that $\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$. [b](b)[/b] [Problem for classes 12/13] Let r be the radius of the insphere and let $r_a$, $r_b$, $r_c$, $r_d$ the radii of the four exspheres of a tetrahedron ABCD. (An [i]exsphere[/i] of a tetrahedron is a sphere touching one sideface and the extensions of the three other sidefaces.) Prove that $\frac{2}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}+\frac{1}{r_d}$. I am really sorry for posting these, but else, Orl will probably post them. This time, we really did not have any challenging problem on the DeMO. But at least, the problems were simple enough that I solved all of them. ;) Darij

A cube has edge length $2$. Suppose that we glue a cube of edge length $1$ on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to [asy] draw((0,0)--(2,0)--(3,1)--(3,3)--(2,2)--(0,2)--cycle); draw((2,0)--(2,2)); draw((0,2)--(1,3)); draw((1,7/3)--(1,10/3)--(2,10/3)--(2,7/3)--cycle); draw((2,7/3)--(5/2,17/6)--(5/2,23/6)--(3/2,23/6)--(1,10/3)); draw((2,10/3)--(5/2,23/6)); draw((3,3)--(5/2,3)); [/asy] $\text{(A)}\ 10 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 21 \qquad \text{(E)}\ 25$

Let $A,B,C,D,E,F$ be points in space such that the quadrilaterals $ABDE,BCEF, CDFA$ are parallelograms. Prove that the six midpoints of the sides $AB,BC,CD,DE,EF,FA$ are coplanar

A fly and $k$ spiders are placed in some vertices of $2012 \times 2012$ lattice. One move consists of following: firstly, fly goes to some adjacent vertex or stays where it is and then every spider goes to some adjacent vertex or stays where it is (more than one spider can be in the same vertex). Spiders and fly knows where are the others all the time. a) Find the smallest $k$ so that spiders can catch the fly in finite number of moves, regardless of their initial position. b) Answer the same question for three-dimensional lattice $2012\times 2012\times 2012$. (Vertices in lattice are adjacent if exactly one coordinate of one vertex is different from the same coordinate of the other vertex, and their difference is equal to $1$. Spider catches a fly if they are in the same vertex.)