Let $U$ be the sum of lengths of sides of a tetrahedron (triangular pyramid) with vertices $O,A,B,C$. Let $V$ be the volume of the convex shape whose vertices are the midpoints of the sides of the tetrahedron. Show that $V\leq \frac{(U-|OA|-|BC| )(U-|OB|-|AC| )(U-|OC|-|AB| )}{(2^{7} \cdot 3)}$.

The midpoint of the edge $SA$ of the triangular pyramid of $SABC$ has equal distances from all the vertices of the pyramid. Let $SH$ be the height of the pyramid. Prove that $BA^2 + BH^2 = C A^2 + CH^2$.

Let $ABCD$ is a tetrahedron. Show that a)If $AB=CD,AC=DB,AD=BC$ then triangles $ABC,ABD,ACD,BCD$ are acute. b)If the triangles $ABC,ABD,ACD,BCD$ have the same area , then $AB=CD,AC=DB,AD=BC$.

Let $A_i$ and $A_i^{'}$ $(i=1,2,3,4)$ be diametrically opposite vertexes of a rectangular cuboid and $M{}$ a point inside it. Prove that $S\leq\sum_{i=1}^{4}MA_i\cdot MA_i^{'}$, where $S{}$ is the total surface area of the rectangular cuboid.

Let $ f(x)$ be a function with the two properties: [list=a] [*] for any two real numbers $ x$ and $ y$, $ f(x \plus{} y) \equal{} x \plus{} f(y)$, and [*] $ f(0) \equal{} 2$ [/list] What is the value of $ f(1998)$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 1996\qquad \textbf{(D)}\ 1998\qquad \textbf{(E)}\ 2000$

A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl? [asy] size(200); defaultpen(linewidth(0.8)); draw((342,-662) -- (600, -727) -- (757,-619) -- (967,-400) -- (1016,-300) -- (912,-116) -- (651,-46) -- (238,-90) -- (82,-204) -- (184, -388) -- (447,-458) -- (859,-410) -- (1016,-300)); draw((82,-204) -- (133,-490) -- (342, -662)); draw((652,-626) -- (600,-727)); draw((447,-458) -- (652,-626) -- (859,-410)); draw((133,-490) -- (184, -388)); draw((967,-400) -- (912,-116)^^(342,-662) -- (496, -545) -- (757,-619)^^(496, -545) -- (446, -262) -- (238, -90)^^(446, -262) -- (651, -46),linewidth(0.6)+linetype("5 5")+gray(0.4)); [/asy] $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }5+2\sqrt{2}\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

The faces of a convex polyhedron are congruent parallelograms. Prove that they are all rhombuses.

Two triangular pyramids have common base. One pyramid contains the other. Can the sum of the lengths of the edges of the inner pyramid be longer than that of the outer one?

Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$, when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$.

Let $A,B,C$ and $D$ be non-coplanar points such that $\angle ABC=\angle ADC$ and $\angle BAD=\angle BCD$. Show that $AB=CD$ and $AD=BC$.

The prism with the regular octagonal base and with all edges of the length equal to $1$ is given. The points $M_{1},M_{2},\cdots,M_{10}$ are the midpoints of all the faces of the prism. For the point $P$ from the inside of the prism denote by $P_{i}$ the intersection point (not equal to $M_{i}$) of the line $M_{i}P$ with the surface of the prism. Assume that the point $P$ is so chosen that all associated with $P$ points $P_{i}$ do not belong to any edge of the prism and on each face lies exactly one point $P_{i}$. Prove that \[\sum_{i=1}^{10}\frac{M_{i}P}{M_{i}P_{i}}=5\]

Using white cubes of side $1$, a prism (without holes) was assembled. The faces of the prism were painted black. It is known that the cubes left with exactly $4$ white faces are $20$ in total. Determine what the dimensions of the prism can be. Give all the possibilities.

Let $M$ be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. If the radii of these spheres are denoted by $r$ and $R$ respectively, find the possible values of $R/r$ over all tetrahedra from $M$ .

Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.

Let $ABCD$ be a regular tetrahedron and let $E$ be a point inside the face $ABC$. Denote by $s$ the sum of the distances from $E$ to the faces $DAB$, $DBC$, $DCA$, and by $S$ the sum of the distances from $E$ to the edges $AB$, $BC$, $CA$. Then $\dfrac sS$ equals $\textbf{(A) }\sqrt2\qquad\textbf{(B) }\dfrac{2\sqrt2}3\qquad\textbf{(C) }\dfrac{\sqrt6}2\qquad\textbf{(D) }2\qquad\textbf{(E) }3$

[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$ [b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $ W$? [asy]import three; size(200); defaultpen(linewidth(.8pt)+fontsize(10pt)); currentprojection=oblique; path3 p1=(0,2,2)--(0,2,0)--(2,2,0)--(2,2,2)--(0,2,2)--(0,0,2)--(2,0,2)--(2,2,2); path3 p2=(2,2,0)--(2,0,0)--(2,0,2); path3 p3=(0,0,2)--(0,2,1)--(2,2,1)--(2,0,2); path3 p4=(2,2,1)--(2,0,0); pen finedashed=linetype("4 4"); draw(p1^^p2^^p3^^p4); draw(shift((4,0,0))*p1); draw(shift((4,0,0))*p2); draw(shift((4,0,0))*p3); draw(shift((4,0,0))*p4); draw((4,0,2)--(5,2,2)--(6,0,2),finedashed); draw((5,2,2)--(5,2,0)--(6,0,0),finedashed); label("$W$",(3,0,2)); draw((2.7,.3,2)--(2.1,1.9,2),linewidth(.6pt)); draw((3.4,.3,2)--(5.9,1.9,2),linewidth(.6pt)); label("Figure 1",(1,-0.5,2)); label("Figure 2",(5,-0.5,2));[/asy]$ \textbf{(A)}\ \frac {1}{12}\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \frac {1}{8}\qquad \textbf{(D)}\ \frac {1}{6}\qquad \textbf{(E)}\ \frac {1}{4}$

Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

We are given $n$ identical cubes, each of size $1\times 1\times 1$. We arrange all of these $n$ cubes to produce one or more congruent rectangular solids, and let $B(n)$ be the number of ways to do this. For example, if $n=12$, then one arrangement is twelve $1\times1\times1$ cubes, another is one $3\times 2\times2$ solid, another is three $2\times 2\times1$ solids, another is three $4\times1\times1$ solids, etc. We do not consider, say, $2\times2\times1$ and $1\times2\times2$ to be different; these solids are congruent. You may wish to verify, for example, that $B(12) =11$. Find, with proof, the integer $m$ such that $10^m<B(2015^{100})<10^{m+1}$.

Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that \[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \] [i]Submitted by Viktor Vígh, Szeged[/i]

Each diagonal of the base and each lateral edge of a $9$-gonal pyramid is colored either green or red. Show that there must exist a triangle with the vertices at vertices of the pyramid having all three sides of the same color.

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]

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 $1$.

Perpendiculars at their centers of gravity (points of intersection of medians) are restored to the faces of the tetrahedron. Prove that the projections of the three perpendiculars to the fourth face intersect at one point.

Line $\ell $ intersects the plane $a$. It is known that in this plane there are $2011$ straight lines equidistant from $\ell$ and not intersecting $\ell$. Is it true that $\ell$ is perpendicular to $a$?