To [i]clip[/i] a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$. Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$, and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$, for all $n \geq 6$.

We are given a convex polygon. Show that one can find a point $Q$ inside the polygon and three vertices $A_1,A_2,A_3$ (not necessarily consecutive) such that each ray $A_iQ$ ($i=1,2,3$) makes acute angles with the two sides emanating from $A_i$.

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.

(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords. Is it necessary that two of these chords are of equal length? (b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords. Prove that among these $10$ chords there are two chords of equal length. (VV Proizvolov, Moscow)

A set consisting of $n$ points of a plane is called an isosceles $n$-point if any three of its points are located in vertices of an isosceles triangle. Find all natural numbers for which there exist isosceles $n$-points.

A triangle is constructed with the lengths of the sides chosen from the set $\{2, 3, 5, 8, 13, 21, 34, 55, 89, 144\}$. Show that this triangle must be isosceles. (A triangle is isosceles if it has at least two sides the same length.)

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]

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

For non-coplanar points are given in space. A plane $\pi$ is called [i]equalizing [/i] if all four points have the same distance from $\pi$. Find the number of equilizing planes.

Given a square side $1$ and $2n$ positive reals $a_1, b_1, ... , a_n, b_n$ each $\le 1$ and satisfying $\sum a_ib_i \ge 100$. Show that the square can be covered with rectangles $R_i$ with sides length $(a_i, b_i)$ parallel to the square sides.

A rectangular $M \times N$ board is divided into $1 \times $ cells. There are also many domino pieces of size $1 \times 2$. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is (a) less than $\frac14 MN$, (b) less than $\frac15 MN$.

The points $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of lines $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ .

Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin. Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$. Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?

Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other. How many substantially different decompositions of a $2010 \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?

$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.

The vertices of a regular $2n$-gon (with $n > 2$ an integer) are labelled with the numbers $1,2,...,2n$ in some order. Assume that the sum of the labels at any two adjacent vertices equals the sum of the labels at the two diametrically opposite vertices. Prove that this is possible if and only if $n$ is odd.

Each point of a plane is colored in one of three colors: red, black and blue. Prove that there exists a rectangle in this plane whose vertices all have the same color.

Let $n \geq 2$ be an integer. There are $n$ distinct circles in general position, that is, any two of them meet in two distinct points and there are no three of them meeting at one point. Those circles divide the plane in limited regions by circular edges, that meet at vertices (note that each circle have exactly $2n-2$ vertices). For each circle, temporarily color its vertices alternately black and white (note that, doing this, each vertex is colored twice, one for each circle passing through it). If the two temporarily colouring of a vertex coincide, this vertex is definitely colored with this common color; otherwise, it will be colored with gray. Show that if a circle has more than $n-2 + \sqrt{n-2}$ gray points, all vertices of some region are grey. Observation: In this problem, a region cannot contain vertices or circular edges on its interior. Also, the outer region of all circles also counts as a region.

Prove that three discs of radius $1$ cannot cover entirely a square surface of side $2$, but they can cover more than $99.75\%$ of it.

We are given $100$ points. $N$ of these are vertices of a convex $N$-gon and the other $100 - N$ of these are inside this $N$-gon. The labels of these points make it impossible to tell whether or not they are vertices of the $N$-gon. It is known that no three points are collinear and that no $4$ points belong to two parallel lines. It has been decided to ask questions of the following type: What is the area of the triangle $XYZ$, where $X, Y$ and $Z$ are labels representing three of the $100$ given points? Prove that $300$ such questions are sufficient in order to clarify which points are vertices and to determine the area of the $N$-gon. (D. Fomin, Leningrad)

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.

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]

There are $n$ rectangles drawn on the rectangular sheet of paper with the sides of the rectangles parallel to the sheet sides. The rectangles do not have pairwise common interior points. Prove that after cutting out the rectangles the sheet will split into not more than $n+1$ part.

Ten vertices of a regular $20$-gon $A_1A_2....A_{20}$ are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal $A_1A_4$ and all other diagonals of the same length. 1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints. 2. Find all possible numbers of the diagonals with two black endpoints.

* On a circle, $20$ points are chosen. Ten non-intersecting chords without mutual endpoints connect some of the points chosen. How many distinct such arrangements are there?