Let $n$ be a positive integer. The Georgian folk dance team consists of $2n$ dancers, with $n$ males and $n$ females. Each dancer, both male and female, is assigned a number from 1 to $n$. During one of their dances, all the dancers line up in a single line. Their wish is that, for every integer $k$ from 1 to $n$, there are exactly $k$ dancers positioned between the $k$th numbered male and the $k$th numbered female. Prove the following statements: a) If $n \equiv 1 \text{ or } 2 \mod{4}$, then the dancers cannot fulfill their wish. b) If $n \equiv 0 \text{ or } 3 \mod{4}$, then the dancers can fulfill their wish. [i]Proposed by Giorgi Arabidze, Georgia[/i]

Prove that infinitely many positive integers can be represented as $(a-1)/b + (b-1)/c + (c-1)/a$, where $a$, $b$ and $c$ are pairwise distinct positive integers greater than 1.

Construct a triangle $ABC$ with the right angle at vertex $C$ given lengths of its medians $m_a$, $m_b$. Discuss conditions of solvability.

On a straight line $4$ points are successively set , $A, P, Q,W $, which are the points of intersection of the bisector $AL $ of the triangle $ABC$ with the circumscribed and inscribed circle. Knowing only these points, construct a triangle $ABC $.

Tanya has cut out a triangle from checkered paper as shown in the picture. The lines of the grid have faded. Can Tanya restore them without any instruments only folding the triangle (she remembers the triangle sidelengths)? (T. Kazitsyna)

Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.

What should the angle at the vertex of an isosceles triangle be so that it is possible to construct a triangle with sides equal to the height, base, and one of the other sides of the isosceles triangle?

The triangle $ABC$ is drawn on the board such that $AB + AC = 2BC$. The bisectors $AL_1, BL_2, CL_3$ were drawn in this triangle, after which everything except the points $L_1, L_2, L_3$ was erased. Use a compass and a ruler to reconstruct triangle $ABC$.

Construct triangle $ABC$, given $h_a$, $h_b$ (the altitudes from $A$ and $B$), and $m_a$, the median from vertex $A$.

Given the base, height and the difference between the angles at the base of a triangle, construct the triangle.

Let $m, n \ge 2$ be integers. Carl is given $n$ marked points in the plane and wishes to mark their centroid.* He has no standard compass or straightedge. Instead, he has a device which, given marked points $A$ and $B$, marks the $m-1$ points that divide segment $\overline{AB}$ into $m$ congruent parts (but does not draw the segment). For which pairs $(m,n)$ can Carl necessarily accomplish his task, regardless of which $n$ points he is given? *Here, the [i]centroid[/i] of $n$ points with coordinates $(x_1, y_1), \dots, (x_n, y_n)$ is the point with coordinates $\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)$. [i]Proposed by Holden Mui and Carl Schildkraut[/i]

Two chords are drawn in a circle of radius equal to unit, $AB$ and $AC$ of equal length. a) Describe how you can construct a third chord $DE$ that is divided into three equal parts by the intersections with $AB$ and $AC$. b) If $AB = AC =\sqrt2$, what are the lengths of the two segments that the chord $DE$ determines in $AB$?

Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.

Of a rhombus $ABCD$ we know the circumradius $R$ of $\Delta ABC$ and $r$ of $\Delta BCD$. Construct the rhombus.

Given an angle less than $180^o$, and a point $M$ outside the angle. Draw a line through $M$ so that the triangle, whose vertices are the vertex of the angle and the intersection points of its legs with the line drawn, has a given perimeter.