Fix a positive integer $n\ge 2$. For any cyclic $2n$-gon $P_1 P_2\cdots P_{2n}$ in this order, define its score as the maximal possible value of $$\angle P_iXP_{i+1} + \angle P_{i+n}XP_{i+n+1}$$ across all $1\le i\le n$ (indices modulo $n$), and over all points $X$ inside the $2n$-gon including its boundary. Prove that there exist a real number $r$ such that a cyclic $2n$-gon is regular if and only if it has score $r$. [i]Proposed by Wong Jer Ren[/i]

A square is given. Lines divide it into $n$ polygons. What is he the largest possible sum of the internal angles of all polygons?

For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$. For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if \[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\] Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.

Determine all integers $m$ for which the $m \times m$ square can be dissected into five rectangles, the side lengths of which are the integers $1,2,3,\ldots,10$ in some order.

A $7 \times 7$ square is divided into unit squares by lines parallel to its sides. Some Swiss crosses (obtained by removing corner unit squares from a square of side $3$) are to be put on the large square, with the edges along division lines. Find the smallest number of unit squares that need to be marked in such a way that every cross covers at least one marked square.

Is it possible to mark four points on the plane so that the distances between any point and three other points form an arithmetic progression? (V. Brayman)

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 $m$.

Find the maximum number of parts into which the $Oxy$-plane can be divided by $100$ graphs of different quadratic functions of the form $y = ax^2 + bx + c$. (N.B. Vasiliev, Moscow)

An $8 \times 8$ board is divided into unit squares. Ten of these squares have their centers marked. Prove that either there exist two marked points on the distance at most $\sqrt2$, or there is a point on the distance $1/2$ from the edge of the board.

Prove that one can always mark $50$ points inside of any convex $100$-gon, so that each its vertix is on a straight line connecting some two marked points. (4)

Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.

An $8 \times 8$ chessboard is covered completely and without overlaps by $32$ dominoes of size $1 \times 2$. Show that there are two dominoes forming a $2 \times 2$ square.

A $3\times 3\times 3$ cube is made of $1\times 1\times 1$ cubes glued together. What is the maximal number of small cubes one can remove so the remaining solid has the following features: 1) Projection of this solid on each face of the original cube is a $3\times 3$ square, 2) The resulting solid remains face-connected (from each small cube one can reach any other small cube along a chain of consecutive cubes with common faces).

Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.

The segments connecting the interior point of a convex non-sided $n$-gon with its vertices divide the $n$-gon into $n$ congruent triangles. For what is the smallest $n$ that is possible?

Given are the points $A$ and $B$ in the plane. If $x$ is a straight line is in that plane, and $x$ does not coincide with the perpendicular bisectror of $AB$, then denote the number of points $C$ located at $x$ such that $\vartriangle ABC$ is isosceles, as the "weight of the line $x$”. Prove that the weight of any line $x$ is at most $5$ and determine the set of points $P$ which has a line with weight $1$, but none with weight $0$.

Let $S$ denote a square of side-length $7$, and let eight squares with side-length $3$ be given. Show that it is impossible to cover $S$ by those eight small squares with the condition: an arbitrary side of those (eight) squares is either coincided, parallel, or perpendicular to others of $S$ .

We will say that a set $ A $ of points is [i]disastrous [/i] if it meets the following conditions: $\bullet$ There are no $ 3 $ collinear points $\bullet$ There is not a trio of mutually equal distances between points. If $ P $ and $ Q $ are points in $ A $, then there are $ M $, $ N $, $ R $ and $ T $ in $ A $ such that: $$ d (P, Q) = \frac {d (M, N) + d (R, T)} {2} $$ Show that all disastrous sets are infinite. [hide=original wording of second condition]No existe ni un trío de distancias entre puntos mutuamente iguales. [/hide]

There are $n > 1$ points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?

We color all the points of the plane with two colors. Prove that there are (at least) two points of the plane having the same color and at distance $1$ among them.

Ten square cardboards of $3$ centimeters on a side are cut by a line, as indicated in the figure. After the cuts, there are $20$ pieces: $10$ triangles and $10$ trapezoids. Assemble a square that uses all $20$ pieces without overlaps or gaps. [img]https://cdn.artofproblemsolving.com/attachments/7/9/ec2242cca617305b02eef7a5409e6a6b482d66.gif[/img]

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.

$S$ is a $(k+1) \times (k+1)$ array of lattice points. How many squares have their vertices in $S$?

There are 100 points on the plane so that any 10 of them are vertices of a convex polygon. Does it follow from this that all these points are the vertices of a convex 100-gon? [i]From the folklore[/i]