Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$. Let the points $E$ and $F$ be the feet of the perpendiculars drawn from point $A$ on the lines $A_1D$ and $A_1C$, respectively, and the points $P$ and $Q$ be the feet of the perpendiculars drawn from point $B_1$ on the lines $A_1C_1$ and $A_1C$, respectively. Prove that $\angle EFA = \angle PQB_1$

A wooden beam $EFGH$ $ABCD$ is with three cuts in $8$ smaller ones sawn beams. Each cut is parallel to one of the three pair of opposit sides. Each pair of saw cuts is shown perpendicular to each other. The smaller bars at the corners $A, C, F$ and $H$ have a capacity of $9, 12, 8, 24$ respectively.(The proportions in the picture are not correct!!). Calculate content of the entire bar. [asy] unitsize (0.5 cm); pair A, B, C, D, E, F, G, H; pair x, y, z; x = (1,0.5); y = (-0.8,0.8); z = (0,1); B = (0,0); C = 5*x; A = 3*y; F = 4*z; E = A + F; G = C + F; H = A + C + F; fill(y--3*y--(3*y + z)--(y + z)--cycle, gray(0.8)); fill(2*x--5*x--(5*x + z)--(2*x + z)--cycle, gray(0.8)); fill((y + z)--(y + 4*z)--(y + 4*z + 2*x)--(4*z + 2*x)--(2*x + z)--z--cycle, gray(0.8)); fill((2*x + y + 4*z)--(2*x + 3*y + 4*z)--(5*x + 3*y + 4*z)--(5*x + y + 4*z)--cycle, gray(0.8)); draw(B--C--G--H--E--A--cycle); draw(B--F); draw(E--F); draw(G--F); draw(y--(y + 4*z)--(y + 4*z + 5*x)); draw(2*x--(2*x + 4*z)--(2*x + 4*z + 3*y)); draw((3*y + z)--z--(5*x + z)); label("$A$", A, SW); label("$B$", B, S); label("$C$", C, SE); label("$E$", E, NW); label("$F$", F, SE); label("$G$", G, NE); label("$H$", H, N); [/asy]

There are $100$ boxes, each containing either a red cube or a blue cube. Alex has a sum of money initially, and places bets on the colour of the cube in each box in turn. The bet can be anywhere from $0$ up to everything he has at the time. After the bet has been placed, the box is opened. If Alex loses, his bet will be taken away. If he wins, he will get his bet back, plus a sum equal to the bet. Then he moves onto the next box, until he has bet on the last one, or until he runs out of money. What is the maximum factor by which he can guarantee to increase his amount of money, if he knows that the exact number of blue cubes is [list][b](a)[/b] $1$; [b](b)[/b] some integer $k$, $1 < k \leq 100$.[/list]

Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?

How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?

A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?

A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?

Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2?$ $\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }20$

The pyramid $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.

A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \[\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.\]

Consider two spheres $\Sigma_1$ and $\Sigma_2$ and a line $\Delta$ not meeting them. Let $C_i$ and $r_i$ be the center and radius of $\Sigma_i$, and let $H_i$ and $d_i$ be the orthogonal projection of $C_i$ onto $\Delta$ and the distance of $C_i$ from $\Delta~(i=1,2)$. For a point $M$ on $\Delta$, let $\delta_i(M)$ be the length of a tangent $MT_i$ to $\Sigma_i$, where $T_i\in\Sigma_i~(i=1,2)$. Find $M$ on $\Delta$ for which $\delta_1(M)+\delta_2(M)$ is minimal.

Suppose that $ K$ is a compact Hausdorff space and $ K\equal{} \cup_{n\equal{}0}^{\infty}A_n$, where $ A_n$ is metrizable and $ A_n \subset A_m$ for $ n<m$. Prove that $ K$ is metrizable. [i]Z. Balogh[/i]

Sphere $S$ with radius $100$ has diameter $\overline{AB}$ and center $C$. Four small spheres all with radius $17$ have centers that lie in a plane perpendicular to $\overline{AB}$ such that each of the four spheres is internally tangent to $S$ and externally tangent to two of the other small spheres. Find the radius of the smallest sphere that is both externally tangent to two of the four spheres with radius $17$ and internally tangent to $S$ at a point in the plane perpendicular to $\overline{AB}$ at $C$.

Prove that if the cross-section of a cube cut by a plane is a pentagon, as shown in the figure below, then there are two adjacent sides of the pentagon such that the sum of the lengths of those two sides is greater than the sum of the lengths of the other three sides. (For ease of grading, please use the names of the points from the figure below in your solution.) [asy] import three; defaultpen(linewidth(0.8)); currentprojection=orthographic(1,3/5,1/2); draw(unitcube, white, thick(), nolight); draw(O--(1,0,0)^^O--(0,1,0)^^O--(0,0,1), linetype("4 4")+linewidth(0.7)); triple A=(1/3, 1, 1), B=(2/3, 1, 0), C=(1, 1/2, 0), D=(1, 0, 1/2), E=(2/3, 0, 1); draw(E--A--B^^C--D); draw(B--C^^D--E, linetype("4 4")+linewidth(0.7)); label("$A$", A, dir(85)); label("$B$", B, SE); label("$C$", C, S); label("$D$", D, W); label("$E$", E, NW);[/asy]

Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger? $ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $

What is the total surface area of an ice cream cone, radius $R$, height $H$, with a spherical scoop of ice cream of radius $r$ on top? (Given $R<r$)

A cube with an edge of $ n $ cm is divided in $ n^3 $ mini-cubes with edges of legth $ 1 $ cm. Only the exterior of the cube is colored. [b]a)[/b] How many of the mini-cubes haven't any colored face? [b]b)[/b] How many of the mini-cubes have only one colored face? [b]c)[/b] How many of the mini-cubes have, at least, two colored faces? [b]d)[/b] If we draw with blue all the diagonals of all the faces of the cube, upon how many mini-cubes do we find blue segments?

Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.

If $ x$ varies as the cube of $ y$, and $ y$ varies as the fifth root of $ z$, then $ x$ varies as the $ n$th power of $ z$, where $ n$ is: $ \textbf{(A)}\ \frac{1}{15} \qquad \textbf{(B)}\ \frac{5}{3} \qquad \textbf{(C)}\ \frac{3}{5} \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 8$

Is it possible to partition three-dimensional space into tetrahedra (not necessarily regular) such that there exists a plane that intersects the edges of each tetrahedron at exactly 4 or 0 points? Proposed by Arvin Taheri

Viktor and Natalia play a colouring game with a 3-dimensional cube taking turns alternatingly. Viktor goes first, and on each of his turns, he selects an unpainted edge, and paints it violet. On each of Natalia's turns, she selects an unpainted edge, or at most once during the game a face diagonal, and paints it neon green. If the player on turn cannot make a legal move, then the turn switches to the other player. The game ends when nobody can make any more legal moves. Natalia wins if at the end of the game every vertex of the cube can be reached from every other vertex by traveling only along neon green segments (edges or diagonal), otherwise Viktor wins. Who has a winning strategy? (Prove your answer.) [i]Authored by Viktor Simjanoski[/i]

The height $SO$ of a regular quadrangular pyramid $SABCD$ forms an angle $60^o$ with a side edge , the volume of this pyramid is equal to $18$ cm$^3$ . The vertex of the second regular quadrangular pyramid is at point $S$, the center of the base is at point $C$, and one of the vertices of the base lies on the line $SO$. Find the volume of the common part of these pyramids. (The common part of the pyramids is the set of all such points in space that lie inside or on the surface of both pyramids).

Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?

The pairs of values of $ x$ and $ y$ that are the common solutions of the equations $ y\equal{}(x\plus{}1)^2$ and $ xy\plus{}y\equal{}1$ are: $ \textbf{(A)}\ \text{3 real pairs} \qquad \textbf{(B)}\ \text{4 real pairs} \qquad \textbf{(C)}\ \text{4 imaginary pairs} \\ \textbf{(D)}\ \text{2 real and 2 imaginary pairs} \qquad \textbf{(E)}\ \text{1 real and 2 imaginary pairs}$

The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm. [img]https://cdn.artofproblemsolving.com/attachments/4/2/cda98a00f8586132fe519855df123534516b50.png[/img] a) What is the height of the liquid when it lies as shown in figure $2$? b) What is the height of the liquid when it lies as shown in figure$ 3$?