A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?

In the plane we are given $3 \cdot n$ points ($n>$1) no three collinear, and the distance between any two of them is $\leq 1$. Prove that we can construct $n$ pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the area of these triangles $< 1/2$.

Prove that for all $n \ge 3$ there are an infinite number of $n$-sided polygonal numbers which are also the sum of two other (not necessarily different) $n$-sided polygonal numbers! The first $n$-sided polygonal number is $1$. The kth n-sided polygonal number for $k \ge 2$ is the number of different points in a figure that consists of all of the regular $n$-sided polygons which have one common vertex, are oriented in the same direction from that vertex and their sides are $\ell$ cm long where $1 \le \ell \le k - 1$ cm and $\ell$ is an integer. [i]In this figure, what we call points are the vertices of the polygons and the points that break up the sides of the polygons into exactly $1$ cm long segments. For example, the first four pentagonal numbers are 1,5,12, and 22, like it is shown in the figure.[/i] [img]https://cdn.artofproblemsolving.com/attachments/1/4/290745d4be1888813678127e6d63b331adaa3d.png[/img]

Prove that any triangle can be divided into $22$ triangles, each of which has an angle of $22^o$, and another $23$ triangles, each of which has an angle of $23^o$.

Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$.

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

Cucumber river in the Flower city has parallel banks with the distance between them $1$ metre. It has some islands with the total perimeter $8$ metres. Mr. Know-All claims that it is possible to cross the river in a boat from the arbitrary point, and the trajectory will not exceed $3$ metres. Is he right?

We consider the ways to divide a $1$ by $1$ square into rectangles (of which the sides are parallel to those of the square). All rectangles must have the same circumference, but not necessarily the same shape. a) Is it possible to divide the square into 20 rectangles, each having a circumference of $2:5$? b) Is it possible to divide the square into 30 rectangles, each having a circumference of $2$?

You have an unlimited supply of square tiles with side length $ 1$ and equilateral triangle tiles with side length $ 1$. For which n can you use these tiles to create a convex $n$-sided polygon? The tiles must fit together without gaps and may not overlap.

There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?

Consider an $m\times n$ board where $m, n \ge 3$ are positive integers, divided into unit squares. Initially all the squares are white. What is the minimum number of squares that need to be painted red such that each $3\times 3$ square contains at least two red squares? Andrei Eckstein and Alexandru Mihalcu

The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?

In a billiard with shape of a rectangle $ABCD$ with $AB=2013$ and $AD=1000$, a ball is launched along the line of the bisector of $\angle BAD$. Supposing that the ball is reflected on the sides with the same angle at the impact point as the angle shot , examine if it shall ever reach at vertex B.

What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear?

For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$? (NB Vassiliev)

2. A unit square paper has a triangle-shaped hole (vertices of the hole are not on the border of the paper). Prove that a triangle with area of $1 / 6$ can be cut from the remaining paper. Alexandr Yuran

Let $D$ be a regular dodecagon in the plane. How many squares are there in the plane at least two vertices in common with the vertices of $D$?

Each face of a cube is of the same size as each square of a chessboard. The cube is coloured black and white, placed on one of the squares of the chessboard and rolled so that each square of the chessboard is visited exactly once. Can this be done in such a way that the colour of the visited square and the colour of the bottom face of the cube are always the same? (A Shapovalov)

$S_n$ is a square side $\frac{1}{n}$. Find the smallest $k$ such that the squares $S_1, S_2,S_3, ...$ can be put into a square side $k$ without overlapping.

$N> 4$ points move around the circle, each with a constant speed. For Any four of them have a moment in time when they all meet. Prove that is the moment when all the points meet.

A satellite is considered accessible from the point $A{}$ of the planet's surface if it is located relative to the tangent plane drawn at point $A{}$, strictly on the other side than the planet. What is the smallest number of satellites that need to be launched over a spherical planet so that at some point the signals of at least two satellites are available from each point on the planet's surface? [i]Proposed by S. Volchenkov[/i]

Can it happen that $6$ parallelepipeds, no two of which have common points, are placed in space so that there is a point outside of them from which no vertex of a parallelepiped is visible? (The parallelepipeds are not transparent.) (V Proizvolov)

A square board with dimensions $100 \times 100$ is divided into $10 000 $unit squares. One of the squares is cut out. Is it possible to cover the rest of the board by isosceles right angled triangles which have hypotenuses of length $2$, and in such a way that their hypotenuses lie on sides of the squares and their other two sides lie on diagonals? The triangles must not overlap each other or extend beyond the edges of the board. (S Fomin, Leningrad)

Let $n$ points lie on the circle. Exactly half of triangles formed by these points are acute-angled. Find all possible $n$. (B.Frenkin)

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