The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area $1$, which has only one pair of parallel sides. a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area $1$? b) The same question for a square of area $1/2019$. (Mikhail Evdokimov)

There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?

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.

A tetrahedron $ABCD$ satisfies $AB=6$, $CD=8$, and $BC=DA=5$. Let $V$ be the maximum value of $ABCD$ possible. If we can write $V^4=2^n3^m$ for some integers $m$ and $n$, find $mn$.

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

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 $\theta=-\arccos(\cot\alpha\cdot\cot\beta)$.

(a) Prove that the area of a given convex quadrilateral is at least twice the area of an arbitrary convex quadrilateral inscribed in it whose sides are parallel to the diagonals of the original one. (b) A tetrahedron with surface area $S$ is intersected by a plane perpendicular to two opposite edges. If the area of the cross-section is $Q$, prove that $S>4Q$.

Let $O$ be an interior point of a tetrahedron $A_1A_2A_3A_4$. Let $ S_1, S_2, S_3, S_4$ be spheres with centers $A_1,A_2,A_3,A_4$, respectively, and let $U, V$ be spheres with centers at $O$. Suppose that for $i, j = 1, 2, 3, 4, i \neq j$, the spheres $S_i$ and $S_j$ are tangent to each other at a point $B_{ij}$ lying on $A_iA_j$ . Suppose also that $U $ is tangent to all edges $A_iA_j$ and $V$ is tangent to the spheres $ S_1, S_2, S_3, S_4$. Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.

Prove that the product of the sines of two opposite dihedrals in a tetrahedron is proportional to the product of the lengths of the edges of these dihedrals.

In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$

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]

Let $ ABCD $ be a tetrahedron with $ \angle ABC = \angle ABD = \angle CBD = 90^\circ $ and $ AB = BC $. Let $ E, F, G $ be points on $ \overline{AD} $, $ BD $, and $ \overline{CD} $, respectively, such that each of the quadrilaterals $ AEFB $, $ BFGC $, and $ CGEA $ have an inscribed circle. Let $ r $ be the smallest real number such that $ \dfrac{[\triangle EFG]}{[\triangle ABC]} \leq r $ for all such configurations $ A, B, C, D, E, F, G $. If $ r $ can be expressed as $ \dfrac{\sqrt{a - b\sqrt{c}}}{d} $ where $ a, b, c, d $ are positive integers with $ \gcd(a, b) $ squarefree and $ c $ squarefree, find $ a + b + c + d $. Note: Here, $ [P] $ denotes the area of polygon $ P $. (This wasn't in the original test; instead they used the notation $ \text{area}(P) $, which is clear but frankly cumbersome. :P)

(a) What is the maximum area of a triangle with vertices in a given square (or on its boundary)? (b) What is the maximum volume of a tetrahedron with vertices in a given cube (or on its boundary)?

Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$. [i]Ray Li[/i]