A cubical box with sides of length $7$ has vertices at $(0,0,0)$, $(7,0,0)$, $(0,7,0)$, $(7,7,0)$, $(0,0,7)$, $(7,0,7)$, $(0,7,7)$, $(7,7,7)$. The inside of the box is lined with mirrors and from the point $(0,1,2)$, a beam of light is directed to the point $(1,3,4)$. The light then reflects repeatedly off the mirrors on the inside of the box. Determine how far the beam of light travels before it first returns to its starting point at $(0,1,2)$.

Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

Square corners, $5$ units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is [asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy] $\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000$

Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$. Prove that $L,T,K$ lie on one line.

In a unit cube, $CG$ is the edge perpendicular to the face $ABCD$. Let $O_1$ be the incircle of square $ABCD$ and $O_2$ be the circumcircle of triangle $BDG$. Determine min$\{XY|X\in O_1,Y\in O_2\}$.

In $\mathbb{R}^3$ some $n$ points are coloured. In every step, if four coloured points lie on the same line, Vojtěch can colour any other point on this line. He observes that he can colour any point $P \in \mathbb{R}^3$ in a finite number of steps (possibly depending on $P$). Find the minimal value of $n$ for which this could happen.

Three points of a triangle are among 8 vertex of a cube. So the number of such acute triangles is $\text{(A)}0\qquad\text{(B)}6\qquad\text{(C)}8\qquad\text{(D)}24$

A sphere sits inside a cubic box, tangent on all $6$ sides of the box. If a side of the box is $5$, and the volume of the sphere is $x\pi$ , find $x$.

Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.

We want to paint some identically-sized cubes so that each face of each cube is painted a solid color and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce?

The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that perimeter of $KLMN$ is less than $4/3$ perimeter of $ABCD$.

P6. Determine all integer pairs $(x, y)$ satisfying the following system of equations. \[ \begin{cases} x + y - 6 &= \sqrt{2x + y + 1} \\ x^2 - x &= 3y + 5 \end{cases} \] P7. Determine the sum of all (positive) integers $n \leq 2019$ such that $1^2 + 2^2 + 3^2 + \cdots + n^2$ is an odd number and $1^1 + 2^2 + 3^3 + \cdots + n^n$ is also an odd number. P8. Two quadrilateral-based pyramids where the length of all its edges are the same, have their bases coincide, forming a new 3D figure called "8-plane" (octahedron). If the volume of such "8-plane" (octahedron) is $a^3\sqrt{2}$ cm$^3$, determine the volume of the largest sphere that can be fit inside such "8-plane" (octahedron). P9. Six-digit numbers $\overline{ABCDEF}$ with distinct digits are arranged from the digits 1, 2, 3, 4, 5, 6, 7, 8 with the rule that the sum of the first three numbers and the sum of the last three numbers are the same. Determine the probability that such arranged number has the property that either the first or last three digits (might be both) form an arithmetic sequence or a geometric sequence. [hide=Remarks (Answer spoiled)]It's a bit ambiguous whether the first or last three digits mentioned should be in that order, or not. If it should be in that order, the answer to this problem would be $\frac{1}{9}$, whereas if not, it would be $\frac{1}{3}$. Some of us agree that the correct interpretation should be the latter (which means that it's not in order) and the answer should be $\frac{1}{3}$. However since this is an essay problem, your interpretation can be written in your solution as well and it's left to the judges' discretion to accept your interpretation, or not. This problem is very bashy.[/hide] P10. $X_n$ denotes the number which is arranged by the digit $X$ written (concatenated) $n$ times. As an example, $2_{(3)} = 222$ and $5_{(2)} = 55$. For $A, B, C \in \{1, 2, \ldots, 9\}$ and $1 \leq n \leq 2019$, determine the number of ordered quadruples $(A, B, C, n)$ satisfying: \[ A_{(2n)} = 2 \left ( B_{(n)} \right ) + \left ( C_{(n)} \right )^2. \]

Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible? $\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$

$605$ spheres of same radius are divided in two parts. From one part, upright "pyramid" is made with square base. From the other part, upright "pyramid" is made with equilateral triangle base. Both "pyramids" are put together from equal numbers of sphere rows. Find number of spheres in every "pyramid"

a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections. b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.

Show that for any tetrahedron, the condition that opposite edges have the same length is equivalent to the condition that the segment drawn between the midpoints of any two opposite edges is perpendicular to the two edges.

Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.

There is a wooden $3 \times 3 \times 3$ cube and 18 rectangular $3 \times 1$ paper strips. Each strip has two dotted lines dividing it into three unit squares. The full surface of the cube is covered with the given strips, flat or bent. Each flat strip is on one face of the cube. Each bent strip (bent at one of its dotted lines) is on two adjacent faces of the cube. What is the greatest possible number of bent strips? Justify your answer.

We are given a tetrahedron with two edges of length $a$ and the remaining four edges of length $b$ where $a$ and $b$ are positive real numbers. What is the range of possible values for the ratio $v=a/b$?

A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.

Let $A,B,C$ be variable points on edges $OX,OY,OZ$ of a trihedral angle $OXYZ$, respectively. Let $OA = a, OB = b, OC = c$ and $R$ be the radius of the circumsphere $S$ of $OABC$. Prove that if points $A,B,C$ vary so that $a+b+c = R+l$, then the sphere $S$ remains tangent to a fixed sphere.

Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ? [i]Proposed by Hungary.[/i] [hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]

$(a)$ A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. We define $p, q, r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$. Prove that \[p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2 = 2a^2,\] where $a$ is the length of an edge of a tetrahedron. $(b)$ Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane. [u]Note:[/u] Part $(b)$ is [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=60825&start=0]IMO 1972 Problem 6[/url]

(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular. (b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?

Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.