Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points. It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$ to the sixth point? (Alexey Panasenko)

Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$. [i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.

Which of the followings is congruent (in $\bmod{25}$) to the sum in of integers $0\leq x < 25$ such that $x^3+3x^2-2x+4 \equiv 0 \pmod{25}$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ \text{None of the preceding} $

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

The insphere of any tetrahedron has radius $r$. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are $r_1, r_2, r_3$ and $r_4$. Prove that $$r_1 + r_2 + r_3 + r_4 = 2r$$

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

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.)

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

Let $ABCDA'B'C'D'$ be the rectangular parallelepiped. Let $M, N, P$ be midpoints of the edges $[AB], [BC],[BB']$ respectively . Let $\{O\} = A'N \cap C'M$. a) Prove that the points $D, O, P$ are collinear. b) Prove that $MC' \perp (A'PN)$ if and only if $ABCDA'B'C'D'$ is a cube.

A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds $\frac32$ m. What can be the cross-sectional area of a cube drawn by a plane?

A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$. Find $100r^2$. [asy] // dragon96, replacing // [img]http://i.imgur.com/08FbQs.png[/img] size(140); defaultpen(linewidth(.7)); real alpha=10, x=-0.12, y=0.025, r=1/sqrt(3); path hex=rotate(alpha)*polygon(6); pair A = shift(x,y)*(r*dir(330+alpha)), B = shift(x,y)*(r*dir(90+alpha)), C = shift(x,y)*(r*dir(210+alpha)); pair X = (-A.x, -A.y), Y = (-B.x, -B.y), Z = (-C.x, -C.y); int i; pair[] H; for(i=0; i<6; i=i+1) { H[i] = dir(alpha+60*i);} fill(X--Y--Z--cycle, rgb(204,255,255)); fill(H[5]--Y--Z--H[0]--cycle^^H[2]--H[3]--X--cycle, rgb(203,153,255)); fill(H[1]--Z--X--H[2]--cycle^^H[4]--H[5]--Y--cycle, rgb(255,203,153)); fill(H[3]--X--Y--H[4]--cycle^^H[0]--H[1]--Z--cycle, rgb(153,203,255)); draw(hex^^X--Y--Z--cycle); draw(H[1]--B--H[2]^^H[3]--C--H[4]^^H[5]--A--H[0]^^A--B--C--cycle, linewidth(0.6)+linetype("5 5")); draw(H[0]--Z--H[1]^^H[2]--X--H[3]^^H[4]--Y--H[5]);[/asy]

A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$

Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles. [b]a.)[/b] What is the volume of this polyhedron ? [b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?

A right prism \(ABCA_1B_1C_1\) is given. It is known that triangles \(A_1BC\), \(AB_1C\), \(ABC_1\), and \(ABC\) are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle \(ABC\), lie on the same sphere.

If an integer $a > 1$ is given such that $\left(a-1\right)^3+a^3+\left(a+1\right)^3$ is the cube of an integer, then show that $4\mid a$.