Olympiad problems

Estonia Open Junior - geometry, 2002.1.4

Consider a point $M$ inside triangle $ABC$ such that triangles $ABM, BCM$ and $CAM$ have equal areas. Prove that $M$ is the intersection point of the medians of triangle $ABC$.

1993 Romania Team Selection Test, 3

Tags: symmetry , hexagon , geometry , diagonals , equal areas

Suppose that each of the diagonals $AD,BE,CF$ divides the hexagon $ABCDEF$ into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?

2004 Estonia National Olympiad, 1

Tags: geometry , circle , equal areas , chord

Inside a circle, point $K$ is taken such that the ray drawn from $K$ through the centre $O$ of the circle and the chord perpendicular to this ray passing through $K$ divide the circle into three pieces with equal area. Let $L$ be one of the endpoints of the chord mentioned. Does the inequality $\angle KOL < 75^o$ hold?

1988 Tournament Of Towns, (165) 2

Tags: geometry , areas , midpoint , equal areas

We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .

2006 Peru MO (ONEM), 2

Tags: geometry , equal areas , equal area , combinatorial geometry

Find all values of $k$ by which it is possible to divide any triangular region in $k$ quadrilaterals of equal area.

Indonesia MO Shortlist - geometry, g9

Tags: floor function , equal areas , geometry

Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$,$ ...$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor$$

2005 Cuba MO, 1

Tags: geometry , perimeter , equal areas

Determine all the quadrilaterals that can be divided by a diagonal into two triangles of equal area and equal perimeter.

2013 APMO, 1

Tags: geometry , trigonometry , circumcircle , parallelogram , analytic geometry , equal areas , ratios

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.

Estonia Open Senior - geometry, 1999.2.5

Tags: Squares , areas , equal areas , geometry

Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.

2020 Durer Math Competition Finals, 1

Tags: geometry , equal area , equal areas

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.

1987 Swedish Mathematical Competition, 2

Tags: arc , Geometric Inequalities , geometry , equal areas

A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.

2022 Sharygin Geometry Olympiad, 8.4

Tags: geometry , equal areas , areas , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral, $O$ be its circumcenter, $P$ be a common points of its diagonals, and $M , N$ be the midpoints of $AB$ and $CD$ respectively. A circle $OPM$ meets for the second time segments $AP$ and $BP$ at points $A_1$ and $B_1$ respectively and a circle $OPN$ meets for the second time segments $CP$ and $DP$ at points $C_1$ and $D_1$ respectively. Prove that the areas of quadrilaterals $AA_1B_1B$ and $CC_1D_1D$ are equal.

1983 All Soviet Union Mathematical Olympiad, 363

Tags: geometry , areas , area , equal areas

The points $A_1,B_1,C_1$ belong to $[BC],[CA],[AB]$ sides of the $ABC$ triangle respectively. The $[AA_1], [BB_1], [CC_1]$ segments split the $ABC$ onto $4$ smaller triangles and $3$ quadrangles. It is known, that the smaller triangles have the same area. Prove that the quadrangles have equal areas. What is the quadrangle area, it the small triangle has the unit area?

1936 Eotvos Mathematical Competition, 2

Tags: geometry , equal areas , Centroid

$S$ is a point inside triangle $ABC$ such that the areas of the triangles $ABS$, $BCS$ and $CAS$ are all equal. Prove that $S$ is the centroid of $ABC$.

1949-56 Chisinau City MO, 42

Tags: geometry , inscribed , equal areas , trapezoid , isosceles

A trapezoid and an isosceles triangle are inscribed in a circle. The larger base of the trapezoid is the diameter of the circle, and the sides of the triangle are parallel to the sides of the trapezoid. Show that the trapezoid and the triangle have equal areas.

2013 Brazil Team Selection Test, 1

Tags: geometry , trigonometry , circumcircle , parallelogram , analytic geometry , equal areas , ratios

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.

2020 Yasinsky Geometry Olympiad, 3

Tags: geometry , trapezoid , equal areas

A trapezoid $ABCD$ with bases $BC$ and $AD$ is given. The points $K$ and $L$ are chosen on the sides $AB$ and $CD$, respectively, so that $KL \parallel AD$. It turned out that the areas of the quadrilaterals $AKLD$ and $KBCL$ are equal. Find the length $KL$ if $BC = 3, AD = 5$.

2004 Olympic Revenge, 1

Tags: geometry , equal angles , equal areas , Equilateral , Circumcenter

$ABC$ is a triangle and $D$ is an internal point such that $\angle DAB=\angle DBC =\angle DCA$. $O_a$ is the circumcenter of $DBC$. $O_b$ is the circumcenter of $DAC$. $O_c$ is the circumcenter of $DAB$. Show that if the area of $ABC$ and $O_aO_bO_c$ are equal then $ABC$ is equilateral.

2020 Regional Olympiad of Mexico West, 2

Tags: geometry , equal areas

Let $ L $, $ M $ and $ N $ be the midpoints on the sides $ BC $, $ AC $ and $ AB$ of a triangle $ ABC $. Points $ D $, $ E $ and $ F $ are taken on the circle circumscribed to the triangle $ LMN $ so that the segments $ LD $, $ ME $ and $ NF $ are diameters of said circumference. Prove that the area of the hexagon $ LENDMF $ is equal to half the area of the triangle $ ABC $

2007 Estonia Math Open Junior Contests, 2

Tags: equal areas , midpoints , geometry , parallelogram

The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.

1994 Czech And Slovak Olympiad IIIA, 5

Tags: geometry , Equilateral , triangle area , equal areas , altitudes

In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?

2000 Singapore Senior Math Olympiad, 1

Tags: triangle area , equal areas , areas , geometry

In $\vartriangle ABC$, the points $D, E$ and $F$ lie on $AB, BC$ and $CA$ respectively. The line segments $AE, BF$ and $CD$ meet at the point $G$. Suppose that the area of each of $\vartriangle BGD, \vartriangle ECG$ and $\vartriangle GFA$ is $1$ cm$^2$. Prove that the area of each of $\vartriangle BEG, \vartriangle GCF$ and $\vartriangle ADG$ is also $1$ cm$^2$. [img]https://cdn.artofproblemsolving.com/attachments/e/7/ec090135bd2e47a9681d767bb984797d87218c.png[/img]

Estonia Open Senior - geometry, 2000.1.3

Tags: geometry , Locus , area of a triangle , equal areas , areas

In the plane, the segments $AB$ and $CD$ are given, while the lines $AB$ and $CD$ intersect. Prove that the set of all points $P$ in the plane such that triangles $ABP$ and $CDP$ have equal areas , form two lines intersecting at the intersection of the lines $AB$ and $CD$.

Denmark (Mohr) - geometry, 1994.5

Tags: geometry , equal areas , areas , right triangle , isosceles

In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment

Estonia Open Junior - geometry, 2007.1.2

Tags: equal areas , midpoints , geometry , parallelogram

The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.