Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$.

In a convex quadrangle $ ABCD $, the rays $ DA $ and $ CB $ intersect at point $ Q $, and the rays $ BA $ and $ CD $ at the point $ P $. It turned out that $ \angle AQB = \angle APD $. The bisectors of the angles $ \angle AQB $ and $ \angle APD $ intersect the sides quadrangle at points $ X $, $ Y $ and $ Z $, $ T $ respectively. Circumscribed circles of triangles $ ZQT $ and $ XPY $ intersect at $ K $ inside quadrangle. Prove that $ K $ lies on the diagonal $ AC $.

All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?

In the quadrilateral $ABCD$ the side $AB$ has length $7, BC$ length $14, CD$ length $26$, and $DA$ length $23$. Show that the diagonals are perpendicular. You may assume that the quadrilateral is convex (all internal angles are less than $180^o$).

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ meet at $O$. Let $m$ be the measure of the acute angle formed by these diagonals. A variable angle $xOy$ of measure $m$ intersects the quadrilateral by a convex quadrilateral of constant area. Prove that $ABCD$ is a square.

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

A circle $k$ and a point $A$ outside it are given in the plane. Prove that all trapezoids, whose non-parallel sides meet at $A$, have the same intersection of diagonals.

Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?

We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ . (V. Prasolov)

A piece is placed in the lower left-corner cell of the $15 \times 15$ board. It can move to the cells that are adjacent to the sides or the corners of its current cell. It must also alternate between horizontal and diagonal moves $($the first move must be diagonal$).$ What is the maximum number of moves it can make without stepping on the same cell twice$?$

A convex quadrilateral has equal diagonals. An equilateral triangle is constructed on the outside of each side of the quadrilateral. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular. [i]Alternative formulation.[/i] Given a convex quadrilateral $ ABCD$ with congruent diagonals $ AC \equal{} BD.$ Four regular triangles are errected externally on its sides. Prove that the segments joining the centroids of the triangles on the opposite sides are perpendicular to each other. [i]Original formulation:[/i] Let $ ABCD$ be a convex quadrilateral such that $ AC \equal{} BD.$ Equilateral triangles are constructed on the sides of the quadrilateral. Let $ O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $ AB,BC,CD,DA$ respectively. Show that $ O_1O_3$ is perpendicular to $ O_2O_4.$

In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.

Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.

Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$.

Quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$. Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$. [i]Nikolai Beluhov and Aleksey Zaslavsky[/i]