Found problems: 58
1953 Moscow Mathematical Olympiad, 242
Let $A$ be a vertex of a regular star-shaped pentagon, the angle at $A$ being less than $180^o$ and the broken line $AA_1BB_1CC_1DD_1EE_1$ being its contour. Lines $AB$ and $DE$ meet at $F$. Prove that polygon $ABB_1CC_1DED_1$ has the same area as the quadrilateral $AD_1EF$.
Note: A regular star pentagon is a figure formed along the diagonals of a regular pentagon.
1986 Tournament Of Towns, (107) 1
Through vertices $A$ and $B$ of triangle $ABC$ are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral .
(G . Galperin , A . Savin, Moscow)
Ukrainian TYM Qualifying - geometry, 2015.18
Is it possible to divide a circle by three chords, different from diameters, into several equal parts?
1990 Tournament Of Towns, (248) 2
If a square is intersected by another square equal to it but rotated by $45^o$ around its centre, each side is divided into three parts in a certain ratio $a : b : a$ (which one can compute). Make the following construction for an arbitrary convex quadrilateral: divide each of its sides into three parts in this same ratio $a : b : a$, and draw a line through the two division points neighbouring each vertex. Prove that the new quadrilateral bounded by the four drawn lines has the same area as the original one.
(A. Savin, Moscow)
1989 Tournament Of Towns, (237) 1
Is it possible to choose a sphere, a triangular pyramid and a plane so that every plane, parallel to the chosen one, intersects the sphere and the pyramid in sections of equal area?
(Problem from Latvia)
2021 Israel National Olympiad, P7
Triangle $ABC$ is given.
The circle $\omega$ with center $I$ is tangent at points $D,E,F$ to segments $BC,AC,AB$ respectively.
When $ABC$ is rotated $180$ degrees about point $I$, triangle $A'B'C'$ results.
Lines $AD, B'C'$ meet at $U$, lines $BE, A'C'$ meet at $V$, and lines $CF, A'B'$ meet at $W$.
Line $BC$ meets $A'C', A'B'$ at points $D_1, D_2$ respectively.
Line $AC$ meets $A'B', B'C'$ at $E_1, E_2$ respectively.
Line $AB$ meets $B'C', A'C'$ at $F_1,F_2$ respectively.
Six (not necessarily convex) quadrilaterals were colored orange:
\[AUIF_2 , C'FIF_2 , BVID_1 , A'DID_2 , CWIE_1 , B'EIE_2\]
Six other quadrilaterals were colored green:
\[AUIE_2 , C'FIF_1 , BVIF_2 , A'DID_1 , CWID_2 , B'EIE_1\]
Prove that the sum of the green areas equals the sum of the orange areas.
Durer Math Competition CD Finals - geometry, 2020.D2
Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.
1989 Tournament Of Towns, (232) 6
A regular hexagon is cut up into $N$ parallelograms of equal area. Prove that $N$ is divisible by three.
(V. Prasolov, I. Sharygin, Moscow)