A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.

Let's call a corner the figure that is obtained by removing one cell from a $2 \times 2$ square. Cut the $6 \times 6$ square into corners so that no two of them form a $2 \times 3$ or $3 \times 2$ rectangle together.

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

Let $S$ be a set of $n$ points $P_1,P_2,\ldots,P_n$ in a plane such that no three of the points are collinear. Let $\alpha$ be the smallest of the angles $\angle P_iP_jP_k$ ($i\ne j\ne k\ne i,i,j,k\in\{1,2,\ldots,n\}$). Find $\max_S\alpha$ and determine those sets $S$ for which this maximal value is attained.

Four lines are given in a plane so that any three of them determine a triangle. One of these lines is parallel to a median in the triangle determined by the other three lines. Prove that each of the other three lines also has this property.

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

The ellipsoid $E$ is contained in the simplex $S$, which is located in the unit ball B space $R^n$. Prove that the sum of the principal semi-axes of the ellipsoid $E$ is no more than units.

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 $X= \{A_1, A_2, A_3, A_4\}$ be a set of four distinct points in the plane. Show that there exists a subset $Y$ of $X$ with the property that there is no (closed) disk $K$ such that $K\cap X = Y$.

Can one divide a square into finitely many triangles such that no two triangles share a side? (The triangles have pairwise disjoint interiors and their union is the square.)

It is given regular polygon with $2n$ sides and center $S$. Consider every quadrilateral with vertices as vertices of polygon. Let $u$ be number of such quadrilaterals which contain point $S$ inside and $v$ number of remaining quadrilaterals. Find $u-v$

Sixteen points are placed in the centers of a $4 \times 4$ chess table in the following way: • • • • • • • • • • • • • • • • (a) Prove that one may choose $6$ points such that no isoceles triangle can be drawn with the vertices at these points. (b) Prove that one cannot choose $7$ points with the above property.

Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

Each vertex of a regular $17$-gon is colored red, blue, or green in such a way that no two adjacent vertices have the same color. Call a triangle “multicolored” if its vertices are colored red, blue, and green, in some order. Prove that the $17$-gon can be cut along nonintersecting diagonals to form at least two multicolored triangles. (A diagonal of a polygon is a a line segment connecting two nonadjacent vertices. Diagonals are called nonintersecting if each pair of them either intersect in a vertex or do not intersect at all.)

A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?

Cut an arbitrary triangle into $2019$ pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a $2019$-gon and one is a $2020$-gon. Polygons do not have to be convex.

A regular $2017$-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that among those triangles only one is acute-angled.

Given are $m\ge 3$ points in the plane. Prove that you can always choose three of these points $A,B,C$ such that $$\angle ABC \le \frac{180^o}{m}.$$

Prove that for every set of $2n$ lines in the plane, such that there are no two parallel lines, there are two lines that divide the plane into four quadrants such that in each quadrant the number of unbounded regions is equal to $n$. [asy] unitsize(1cm); pair[] A, B; pair P, Q, R, S; A[1] = (0,5.2); B[1] = (6.1,0); A[2] = (1.5,5.5); B[2] = (3.5,0); A[3] = (6.8,5.5); B[3] = (1,0); A[4] = (7,4.5); B[4] = (0,4); P = extension(A[2],B[2],A[4],B[4]); Q = extension(A[3],B[3],A[4],B[4]); R = extension(A[1],B[1],A[2],B[2]); S = extension(A[1],B[1],A[3],B[3]); fill(P--Q--S--R--cycle, palered); fill(A[4]--(7,0)--B[1]--S--Q--cycle, paleblue); draw(A[1]--B[1]); draw(A[2]--B[2]); draw(A[3]--B[3]); draw(A[4]--B[4]); label("Bounded region", (3.5,3.7), fontsize(8)); label("Unbounded region", (5.4,2.5), fontsize(8)); [/asy]

Prove that there are different points $A_0 \,\, ,A_1 \,\, , \cdots A_{2550}$ on the $XY$ plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If $ d(A_i,A_j)$ represents the distance between $A_i\,\, , A_j $ then $$ \sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008}$$ Note : $ \{x \}$ represents the decimal part of x e.g. $ \{ 3.16\} = 0.16$. [i] (passer-by)[/i]

Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$

In a square of area $1$ there are situated $2024$ polygons whose total area is greater than $2023$. Prove that they have a point in common.

Given a billiard in the form of a regular $1998$-gon $A_1A_2...A_{1998}$. A ball was released from the midpoint of side $A_1A_2$, which, reflected therefore from sides $A_2A_3$, $A_3A_4$, . . . , $A_{1998}A_1$ (according to the law, the angle of incidence is equal to the angle of reflection), returned to the starting point. Prove that the trajectory of the ball is a regular $1998$-gon.