An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions: [list=] [*]The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.) [*]No two beams have intersecting interiors. [*]The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam. [/list] What is the smallest positive number of beams that can be placed to satisfy these conditions? [i]Proposed by Alex Zhai[/i]

Consider an infinite strip of unit squares. The squares are numbered "1", "2", "3", ... A pawn starts on one of the squares and it can move according to the following rules: (1) from the square numbered "$n$" to the square numbered "$2n$", and vice versa; (2) from the square numbered "$n$" to the square numbered "$3n + 1$", and vice versa. Show that the pawn can reach the square numbered "$1$" in a finite number of moves.

Prove that there are infinitely many positive integers $n$ such that $\Delta = nr^2$, where $\Delta$ and $r$ are respectively the area and the inradius of a triangle with integer sides.

Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$. (a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point. (b) Find the the locus of point $ E$.

In $\triangle ABC$ with a right angle at $C,$ point $D$ lies in the interior of $\overline{AB}$ and point $E$ lies in the interior of $\overline{BC}$ so that $AC=CD,$ $DE=EB,$ and the ratio $AC:DE=4:3.$ What is the ratio $AD:DB?$ $\textbf{(A) } 2:3 \qquad\textbf{(B) } 2:\sqrt{5} \qquad\textbf{(C) } 1:1 \qquad\textbf{(D) } 3:\sqrt{5} \qquad\textbf{(E) } 3:2$

Proof that there exist constant $\lambda$, so that for any positive integer $m(\ge 2)$, and any lattice triangle $T$ in the Cartesian coordinate plane, if $T$ contains exactly one $m$-lattice point in its interior(not containing boundary), then $T$ has area $\le \lambda m^3$. PS. lattice triangles are triangles whose vertex are lattice points; $m$-lattice points are lattice points whose both coordinates are divisible by $m$.

The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$. $M$ is the midpoint of the radium $OC$. Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.

In a regular polygon with 100 vertices, 10 vertices are painted blue, and 10 other vertices are painted red. 1. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( R_1 \) is equal to the distance between \( A_2 \) and \( R_2 \). 2. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( A_2 \) is equal to the distance between \( R_1 \) and \( R_2 \).

In a given trapezium $ ABCD $ with $ AB$ parallel to $ CD $ and $ AB > CD $, the line $ BD $ bisects the angle $ \angle ADC $. The line through $ C $ parallel to $ AD $ meets the segments $ BD $ and $ AB $ in $ E $ and $ F $, respectively. Let $ O $ be the circumcenter of the triangle $ BEF $. Suppose that $ \angle ACO = 60^{\circ} $. Prove the equality \[ CF = AF + FO .\]

Given a triangle $ABC$ and a point $P_0$ on the side $AB$. Construct points $P_i, Q_i, R_i $ as follows. $Q_i$ is the foot of the perpendicular from $P_i$ to $BC, R_i$ is the foot of the perpendicular from $Q_i$ to $AC$ and $P_i$ is the foot of the perpendicular from $R_{i-1}$ to $AB$. Show that the points $P_i$ converge to a point $P$ on $AB$ and show how to construct $P$.

Let $A_1A_2A_3\dots A_{20}$ be a $20$-sided polygon $P$ in the plane, where all of the side lengths of $P$ are equal, the interior angle at $A_i$ measures $108$ degrees for all odd $i$, and the interior angle $A_i$ measures $216$ degrees for all even $i$. Prove that the lines $A_2A_8$, $A_4A_{10}$, $A_5A_{13}$, $A_6A_{16}$, and $A_7A_{19}$ all intersect at the same point. [asy] import graph; size(10cm); pair temp= (-1,0); pair A01 = (0,0); pair A02 = rotate(306,A01)*temp; pair A03 = rotate(144,A02)*A01; pair A04 = rotate(252,A03)*A02; pair A05 = rotate(144,A04)*A03; pair A06 = rotate(252,A05)*A04; pair A07 = rotate(144,A06)*A05; pair A08 = rotate(252,A07)*A06; pair A09 = rotate(144,A08)*A07; pair A10 = rotate(252,A09)*A08; pair A11 = rotate(144,A10)*A09; pair A12 = rotate(252,A11)*A10; pair A13 = rotate(144,A12)*A11; pair A14 = rotate(252,A13)*A12; pair A15 = rotate(144,A14)*A13; pair A16 = rotate(252,A15)*A14; pair A17 = rotate(144,A16)*A15; pair A18 = rotate(252,A17)*A16; pair A19 = rotate(144,A18)*A17; pair A20 = rotate(252,A19)*A18; dot(A01); dot(A02); dot(A03); dot(A04); dot(A05); dot(A06); dot(A07); dot(A08); dot(A09); dot(A10); dot(A11); dot(A12); dot(A13); dot(A14); dot(A15); dot(A16); dot(A17); dot(A18); dot(A19); dot(A20); draw(A01--A02--A03--A04--A05--A06--A07--A08--A09--A10--A11--A12--A13--A14--A15--A16--A17--A18--A19--A20--cycle); label("$A_{1}$",A01,E); label("$A_{2}$",A02,W); label("$A_{3}$",A03,NE); label("$A_{4}$",A04,SW); label("$A_{5}$",A05,N); label("$A_{6}$",A06,S); label("$A_{7}$",A07,N); label("$A_{8}$",A08,SE); label("$A_{9}$",A09,NW); label("$A_{10}$",A10,E); label("$A_{11}$",A11,W); label("$A_{12}$",A12,E); label("$A_{13}$",A13,SW); label("$A_{14}$",A14,NE); label("$A_{15}$",A15,S); label("$A_{16}$",A16,N); label("$A_{17}$",A17,S); label("$A_{18}$",A18,NW); label("$A_{19}$",A19,SE); label("$A_{20}$",A20,W);[/asy]

In triangle $ABC$ we have $AB = 2, AC = 6$ and $\angle A = 120^o$ . The bisector of angle $A$ intersects the side BC at the point $D$. Determine the length of $AD$. The answer must be given as a fraction with integer numerator and denominator.

Prove that for any triangle the circumscribed circle divides the line segment connecting the center of its inscribed circle with the center of one of the exscribed circles in halves.

Let $ a$, $ b$, $ c$ be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if $ S_a \circ S_b \circ S_c$ is a reflection in a line, where $ S_x$ denotes the reflection in line $ x$.

$ABC$ is acuteangled triangle. Variable point $X$ lies on segment $AC$, and variable point $Y$ lies on the ray $BC$ but not segment $BC$, such that $\angle ABX+\angle CXY =90$. $T$ is projection of $B$ on the $XY$. Prove that all points $T$ lies on the line.

$ABC$ is a triangle. $X$, $Y$, $Z$ lie on $BC$, $CA$, $AB$ respectively. Show that area $XYZ$ cannot be smaller than each of area $AYZ$, area $BZX$, area $CXY$.

A chord $AB$ is drawn in a circle, with center $M$ and radius $r$, that the two diameters which divide the largest arc $AB$ into three equal parts also divide the chord $AB$ into three equal parts. Express the length of that chord in terms of $r$.

In the triangle $ ABC $, the bisectors of the internal and external angles are drawn at the vertices $ A $ and $ B $. Prove that the orthogonal projections of the point $ C $ on these bisectors lie on one straight line.

Let $P$ be a point in the interior of the triangle $ABC$. The lines $AP$, $BP$, and $CP$ intersect the sides $BC$, $CA$, and $AB$ at $D,E$, and $F$, respectively. A point $Q$ is taken on the ray $[BE$ such that $E\in [BQ]$ and $m(\widehat{EDQ})=m(\widehat{BDF})$. If $BE$ and $AD$ are perpendicular, and $|DQ|=2|BD|$, prove that $m(\widehat{FDE})=60^\circ$.

A treasure is buried at some point on a round island with a radius of $1$ km. On the coast of the island there is a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed $500$ m. In addition, the mathematician has a map of the island, on which he can record all his movements, perform measurements and geometric constructions. The mathematician claims that he has an algorithm for how to get to the treasure after walking less than $4$ km. Could this be true? (B. Frenkin)

It is known that in the triangle $ABC$, $ 2 \angle BAC + 3 \angle ABC= 180^o$. Prove that $4(BC + CA)< 5AB$.

Can the touchpoints of the inscribed circle of a triangle with the triangle form an obtuse triangle?

On the side $AC$ of triangle $ABC$ point $D$ is chosen. Let $I_1, I_2, I$ be the incenters of triangles $ABD, BCD, ABC$ respectively. It turned out that $I$ is the orthocentre of triangle $I_1I_2B$. Prove that $BD$ is an altitude of triangle $ABC$.

Let $P_0 = (3,1)$ and define $P_{n+1} = (x_n, y_n)$ for $n \ge 0$ by $$x_{n+1} = - \frac{3x_n - y_n}{2}, y_{n+1} = - \frac{x_n + y_n}{2}$$Find the area of the quadrilateral formed by the points $P_{96}, P_{97}, P_{98}, P_{99}$.

Point $M$ is the midpoint of the side $CD$ of the trapezoid $ABCD$, point $K$ is the foot of the perpendicular drawn from point $M$ to the side $AB$. Give that $3BK \le AK$. Prove that $BC + AD\ge 2BM$.