A $20 \times 20 \times 20$ cube is composed of $2000$ bricks of size $2 \times 2 \times 1$ . Prove that it is possible to pierce the cube with a needle so that the needle passes through the cube without passing through a brick . (A . Andjans , Riga)

$4026$ points were noted on the plane, not three of which lie on a straight line. The $2013$ points are the vertices of a convex polygon, and the other $2013$ vertices are inside this polygon. It is allowed to paint each point in one of two colors. Coloring will be good if some pairs of dots can be combined segments with the following conditions: $\bullet$ Each segment connects dots of the same color. $\bullet$ No two drawn segments intersect at their inner points. $\bullet$ For an arbitrary pair of dots of the same color, there is a path along the lines from one point to another. Please note that the sides of the convex $2013$ rectangle are not automatically drawn segments, although some (or all) can be drawn as needed. Prove that the total number of good colors does not depend on the specific locations of the points and find that number.

Cut a rectangle into $18$ rectangles so that no two adjacent ones form a rectangle.

Several (at least three) turtles are crawling along the plane, the velocities of which are constant in magnitude and direction (all are equal in magnitude, but pairwise different in direction). Prove that regardless of the initial location, after some time all the turtles will be at the vertices of some convex polygon.

On a large chessboard $2n$ of its $1 \times 1$ squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles. (A Shapovalov)

A [i]magic[/i] octagon is an octagon whose sides follow the lines of the checkerboard's checkers and the side lengths are $1, 2, 3, 4, 5, 6, 7, 8$ (in any order). What is the largest possible area of the magic octagon? [hide=original wording]Burvju astoņstūris ar astoņstūris, kura malas iet pa rūtiņu lapas rūtiņu līnijām un malu garumi ir 1, 2,3, 4, 5, 6, 7, 8 (jebkādā secībā). Kāds ir lielākais iespējamais burvju astoņstūra laukums?[/hide]

Given a trapezium with two parallel sides of lengths $m$ and $n$, where $m$, $n$ are integers, prove that it is possible to divide the trapezium into several congruent triangles.

$99$ straight lines divide a plane into $n$ parts. Find all possible values of $n$ less than $199$.

Find all finite sets $S$ of points in the plane with the following property: for any three distinct points $A,B,$ and $C$ in $S,$ there is a fourth point $D$ in $S$ such that $A,B,C,$ and $D$ are the vertices of a parallelogram (in some order).

We have $4n + 5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the $n$ triangles have the same color.

Show that amongst any $ 8$ points in the interior of a $7 \times 12$ rectangle, there exists a pair whose distance is less than $5$. Note: The interior of a rectangle excludes points lying on the sides of the rectangle.

An equilateral triangle of side $n$ is divided into $n^2$ equilateral triangles of side $1$. A path is drawn along the sides of the triangles which passes through each vertex just once. Prove that the path makes an acute angle at at least $n$ vertices.

Let $P_i(x_i,y_i)\ (i=1,2,\cdots,2023)$ be $2023$ distinct points on a plane equipped with rectangular coordinate system. For $i\neq j$, define $d(P_i,P_j) = |x_i - x_j| + |y_i - y_j|$. Define $$\lambda = \frac{\max_{i\neq j}d(P_i,P_j)}{\min_{i\neq j}d(P_i,P_j)}$$. Prove that $\lambda \geq 44$ and provide an example in which the equality holds.

For a given natural number $k{}$, a convex polygon is called $k{}$[i]-triangular[/i] if it is the intersection of some $k{}$ triangles. [list=a] [*]What is the largest $n{}$ for which there exist a $k{}$-triangular $n{}$-gon? [*]What is the largest $n{}$ for which any convex $n{}$-gon is $k{}$-triangular? [/list] [i]Proposed by P. Kozhevnikov[/i]

There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?

There are some ink-blots on a white paper square with side length $a$. The area of each blot is not greater than $1$ and every line parallel to any one of the sides of the square intersects no more than one blot. Prove that the total area of the blots is not greater than $a$. (A. Razborov, Moscow)

There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or $\textit{vice versa}$. Suppose that Turbo’s path entirely covers all circles. Prove that $n$ must be odd. [i]Proposed by Tejaswi Navilarekallu, India[/i]

Let $n$ distinct points in the plane be given. Prove that fewer than $2 n^{3 \slash 2}$ pairs of them are a unit distance apart.

Given two points $p_1=(x_1, y_1)$ and $p_2=(x_2, y_2)$ on the plane, denote by $\mathcal{R}(p_1,p_2)$ the rectangle with sides parallel to the coordinate axes and with $p_1$ and $p_2$ as opposite corners, that is, \[\{(x,y)\in \mathbb{R}^2:\min\{x_1, x_2\}\leq x\leq \max\{x_1, x_2\},\min\{y_1, y_2\}\leq y\leq \max\{y_1, y_2\}\}.\] Find the largest value of $k$ for which the following statement is true: for all sets $\mathcal{S}\subset\mathbb{R}^2$ with $|\mathcal{S}|=2024$, there exist two points $p_1, p_2\in\mathcal{S}$ such that $|\mathcal{S}\cap\mathcal{R}(p_1, p_2)|\geq k$.

We wish to lay down boards on a floor with width $B$ in the direction across the boards. We have $n$ boards of width $b$, and $B/b$ is an integer, and $nb \le B$. There are enough boards to cover the floor, but the boards may have different lengths. Show that we can cut the boards in such a way that every board length on the floor has at most one join where two boards meet end to end. [img]https://cdn.artofproblemsolving.com/attachments/f/f/24ce8ae05d85fd522da0e18c0bb8017ca3c8e8.png[/img]

There is an infinite set of points $S$ on the plane, and any $1\times 1$ square contains a finite number of points from the set $S$. Prove that there are two different points $A$ and $B$ from $S$ such that for any other point $X$ from $S$ the following inequalities hold: $$|XA|, |XB| \ge 0.999|AB|.$$

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

For which pairs $(m,n)$ does there exist an injective function $f:\mathbb{R}^2\to\mathbb{R}^2$ under which the image of every regular $m$-gon is a regular $n$-gon. (Note that $m,n\geq 3$, and that by a regular $N$-gon we mean the union of the boundary segments, not the closed polygonal region.) [i]Proposed by Sutanay Bhattacharya, Bishnupur, India[/i]

A circular target with a radius of $12$ cm was hit by $19$ shots. Prove that the distance between two hits is less than $7$ cm.

Consider five segments $AB_1, AB_2, AB_3, AB_4, AB_5$. From each point $B_i$ there can exit either $5$ segments or no segments at all, so that the endpoints of any two segments of the resulting graph (system of segments) do not coincide. Can the number of free endpoints of the segments thus constructed be equal to $1001$? (A free endpoint is an endpoint from which no segment begins.)