Five distinct points and five distinct lines are given in the plane. Prove that one can select two of the points and two of the lines so that none of the selected lines contains any of the selected points.

Some squares of the infinite sheet of cross-lined paper are red. Each $2\times 3$ rectangle (of $6$ squares) contains exactly two red squares. How many red squares can be in the $9\times 11$ rectangle of $99$ squares?

Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$. Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.

How many colorings of an $n$-gon in $p \ge 2$ colors are there such that no two neighboring vertices have the same color?

We have $n$ points in the plane, no three on a line. We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon. Suppose that for a fixed $k$ the number of $k$ good points is $c_k$. Show that the following sum is independent of the structure of points and only depends on $n$ : \[ \sum_{i=3}^n (-1)^i c_i \]

With $21$ pieces, some white and some black, a rectangle is formed of $3 \times 7$. Prove that there are always four pieces of the same color located at the vertices of a rectangle.

A strip of width $1$ is to be divided by rectangular panels of common width $1$ and denominations long $a_1$, $a_2$, $a_3$, $. . .$ be paved without gaps ($a_1 \ne 1$). From the second panel on, each panel is similar but not congruent to the already paved part of the strip. When the first $n$ slabs are laid, the length of the paved part of the strip is $sn$. Given $a_1$, is there a number that is not surpassed by any $s_n$? The accuracy answer has to be proven.

As the figure is shown, place a $2\times 5$ grid table in horizontal or vertical direction, and then remove arbitrary one $1\times 1$ square on its four corners. The eight different shapes consisting of the remaining nine small squares are called [i]banners[/i]. [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--C--H--J--N^^B--I^^D--N^^E--M^^F--L^^G--K); draw(Aa--Ca--Ha--Ja--Aa^^Ba--Ia^^Da--Na^^Ea--Ma^^Fa--La^^Ga--Ka); [/asy] [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--Ca--Ea--M--N^^B--O^^C--E^^Aa--Ma^^Ba--Oa^^Da--N); draw(L--Fa--Ha--J--L^^Ga--K^^P--I^^F--H^^Ja--La^^Pa--Ia); [/asy] Here is a fixed $9\times 18$ grid table. Find the number of ways to cover the grid table completely with 18 [i]banners[/i].

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

Pinocchio claims that he can divide an isoceles triangle into three triangles, any two of which can be put together to form a new isosceles triangle. Is Pinocchio lying? (A Shapovalov)

Every point of the plane is painted with one of three colors. Can we always find two points a distance $1$ cm apart which are of the same color?

Six equal disks are placed on a plane so that their centers lie at the vertices of a regular hexagon with sides equal to the diameter of the disks. How many revolutions will a seventh disk of the same size make when rolling in the same plane externally over the disks before returning to its initial position?

The figure $F$ is the intersection of $N$ circles (they may have different radii). Find the maximal number of curvilinear “sides” which $F$ can have. Curvilinear sides of $F$ are the arcs (of the given circumferences) that constitute the boundary of $F$. (Their ends are the “vertices” of $F$ - the points of intersection of given circumferences that lie on the boundary of $F$.) (N Brodsky)

Coins of diameter $1$ have been placed on a square of side $11$, without overlapping or protruding from the square. Can there be $126$ coins? and $127$? and $128$?

Each vertex of a convex polygon has integer coordinates, and no side of this polygon is horizontal or vertical. Prove that the sum of the lengths of the segments of lines of the form $x = m$, $m$ an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form $y = n$, $n$ an integer, that lie within the polygon. (G Galperin)

Decide if it is possible to consider $2011$ points in a plane such that the distance between every two of these points is different from $1$ and each unit circle centered at one of these points leaves exactly $1005$ points outside the circle.

A $5779$-dimensional polytope is call a [b]$k$-tope[/b] if it has exactly $k$ $5778$-dimensional faces. Find all sequences $b_{5780}, b_{5781}, \dots, b_{11558}$ of nonnegative integers, not all $0$, such that the following condition holds: It is possible to tesselate every $5779$-dimensional polytope with [u]convex[/u] $5779$-dimensional polytopes, such that the number of $k$-topes in the tessellation is proportional to $b_k$, while there are no $k$-topes in the tessellation if $k\notin \{5780, 5781, \dots, 11558\}$.

There are given $20$ points in the plane, no three of which lie on a single line. Prove that there exist at least $969$ quadrilaterals with vertices from the given points.

In four-dimensional space, the $24$-cell of sidelength $\sqrt{2}$ is the convex hull of (smallest convex set containing) the $24$ points $(\pm 1, \pm 1, 0, 0)$ and its permutations. Find the four-dimensional volume of this region.

On the plane there is a finite set of points $Z$ with the property that no two distances of the points of the set $Z$ are equal. We connect the points $ A, B $ belonging to $ Z $ if and only if $ A $ is the point closest to $ B $ or $ B $ is the point closest to $ A $. Prove that no point in the set $Z$ will be connected to more than five others.

Given $100$ points on the plane. Prove that you can cover them with a family of circles with the sum of their diameters less than $100$ and the distance between any two of the circles more than one.

A rectangle $ABCD$ is given. On the side $AB$, n different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$ between $A$ and $D$. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way? An example of a particular rectangle $ABCD$ is shown with a shaded one rectangle that may be formed in this way. [img]https://cdn.artofproblemsolving.com/attachments/e/4/f7a04300f0c846fb6418d12dc23f5c74b54242.png[/img]

It is known that diagonals of a square, as well as a regular pentagon, are all equal. Find the bigeest natural $n$ such that a convex $n$-gon has all it's diagonals equal.

Find the smallest positive number $\lambda$ such that for an arbitrary $12$ points on the plane $P_1,P_2,...P_{12}$ (points may coincide), with distance between arbitrary two of them does not exceeds $1$, holds the inequality $\sum_{1\le i\le j\le 12} P_iP_j^2 \le \lambda$

Prove that in each polyhedron there exist two faces with the same number of edges.