This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 67

2007 Bulgarian Autumn Math Competition, Problem 8.2
Tags: geometry , locus of points , equal area
Let $ABCD$ be a convex quadrilateral. Determine all points $M$, which lie inside $ABCD$, such that the areas of $ABCM$ and $AMCD$ are equal.

2014 Belarusian National Olympiad, 6
Tags: equal area , geometry , median , aras
Points $C_1, A_1$ and $B_1$ are marked on the sides $AB, BC$ and $CA$ of a triangle $ABC$ so that the segments $AA_1, BB_1$, and $CC_1$ are concurrent (see the fig.). It is known that the area of the white part of the triangle $ABC$ is equal to the area of its black part. Prove that at least one of the segments $AA_1, BB_1, CC_1$ is a median of the triangle $ABC$. [img]https://1.bp.blogspot.com/-nVVhqdRdf0s/X-WVmt_gyqI/AAAAAAAAM40/943sCRGyCPwT-vqIilTCtXOXHByRLIvPwCLcBGAsYHQ/s0/2014%2Bbelarus%2B11.6.png[/img]

May Olympiad L1 - geometry, 1996.1
Tags: geometry , trapezoid , equal area
A terrain ( $ABCD$ ) has a rectangular trapezoidal shape. The angle in $A$ measures $90^o$. $AB$ measures $30$ m, $AD$ measures $20$ m and $DC$ measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the $AD$ side . At what distance from $D$ do we have to draw the parallel? [img]https://1.bp.blogspot.com/-DnyNY3x4XKE/XNYvRUrLVTI/AAAAAAAAKLE/gohd7_S9OeIi-CVUVw-iM63uXE5u-WmGwCK4BGAYYCw/s400/image002.gif[/img]

2013 Dutch BxMO/EGMO TST, 1
Tags: geometry , parallel , equal area , area
In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.

1982 Swedish Mathematical Competition, 3
Tags: equal area , geometry , area
Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.

1995 Singapore MO Open, 2
Tags: equal area , geometry , triangle area , centroid
Let $A_1A_2A_3$ be a triangle and $M$ an interior point. The straight lines $MA_1, MA_2, MA_3$ intersect the opposite sides at the points $B_1, B_2, B_3$ respectively (see Fig.). Show that if the areas of triangles $A_2B_1M, A_3B_2M$ and $A_1B_3M$ are equal, then $M$ coincides with the centroid of triangle $A_1A_2A_3$. [img]https://cdn.artofproblemsolving.com/attachments/1/7/b29bdbb1f2b103be1f3cb2650b3bfff352024a.png[/img]

1988 Tournament Of Towns, (165) 2
Tags: equal area , geometry , midpoint , area
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 .

1976 Bundeswettbewerb Mathematik, 2
Tags: equal area , geometry , geometric transformation , square , translation
Two congruent squares $Q$ and $Q'$ are given in the plane. Show that they can be divided into parts $T_1, T_2, \ldots , T_n$ and $T'_1 , T'_2 , \ldots , T'_n$, respectively, such that $T'_i$ is the image of $T_i$ under a translation for $i=1,2, \ldots, n.$

1958 Kurschak Competition, 3
Tags: equal area , geometry , hexagon
The hexagon $ABCDEF$ is convex and opposite sides are parallel. Show that the triangles $ACE$ and $BDF$ have equal area

2000 Singapore Senior Math Olympiad, 1
Tags: triangle area , geometry , equal area , area
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]

1985 Tournament Of Towns, (088) 4
Tags: equal area , geometry , rectangle , square
A square is divided into $5$ rectangles in such a way that its $4$ vertices belong to $4$ of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square. [img]https://3.bp.blogspot.com/-TQc1v_NODek/XWHHgmONboI/AAAAAAAAKi4/XES55OJS5jY9QpNmoURp4y80EkanNzmMwCK4BGAYYCw/s1600/TOT%2B1985%2BSpring%2BJ4.png[/img]

2010 Saudi Arabia BMO TST, 2
Tags: equal area , ratio , geometry , cevian
Consider a triangle $ABC$ and a point $P$ in its interior. Lines $PA$, $PB$, $PC$ intersect $BC$, $CA$, $AB$ at $A', B', C'$ , respectively. Prove that $$\frac{BA'}{BC}+ \frac{CB'}{CA}+ \frac{AC'}{AB}= \frac32$$ if and only if at least two of the triangles $PAB$, $PBC$, $PCA$ have the same area.

1990 Tournament Of Towns, (248) 2
Tags: square , equal area , ratio , geometry
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)

Estonia Open Junior - geometry, 2007.1.2
Tags: midpoint , equal area , 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 $EFGH$.

1984 Tournament Of Towns, (075) T1
Tags: equal area , hexagon , parallel , geometry , area
In convex hexagon $ABCDEF, AB$ is parallel to $CF, CD$ is parallel to $BE$ and $EF$ is parallel to $AD$. Prove that the areas of triangles $ACE$ and $BDF$ are equal .

2007 Swedish Mathematical Competition, 6
Tags: equal area , geometry , area
In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the vertices of the triangle.

2004 Estonia National Olympiad, 1
Tags: equal area , geometry , chord , circles
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?