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.

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Found problems: 414

1984 Tournament Of Towns, (075) T1
Tags: hexagon , equal areas , areas , parallel , geometry
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 .

2013 Dutch BxMO/EGMO TST, 1
Tags: geometry , areas , parallel , equal 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 $DM // BC$.

Kvant 2022, M2720
Tags: Kvant , geometry , areas
Let $\Omega$ be the circumcircle of the triangle $ABC$. The points $M_a,M_b$ and $M_c$ are the midpoints of the sides $BC, CA$ and $AB{}$, respectively. Let $A_l, B_l$ and $C_l$ be the intersection points of $\Omega$ with the rays $M_cM_b, M_aM_c$ and $M_bM_a$ respectively. Similarly, let $A_r, B_r$ and $C_r$ be the intersection points of $\Omega$ with the rays $M_bM_c, M_cM_a$ and $M_aM_b$ respectively. Prove that the mean of the areas of the ​​triangles $A_lB_lC_l$ and $A_rB_rC_r$ is not less than the area of the ​​triangle $ABC$. [i]Proposed by L. Shatunov and T. Kazantseva[/i]

2011 Oral Moscow Geometry Olympiad, 4
Tags: geometry , area of a triangle , areas , 3D geometry
Prove that any rigid flat triangle $T$ of area less than $4$ can be inserted through a triangular hole $Q$ with area $3$.

2006 May Olympiad, 4
Tags: geometry , trapezoid , areas
Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$. If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$, calculate the area of the triangle $BCO$.

Durer Math Competition CD Finals - geometry, 2009.D1
Tags: geometry , areas
Fencing Ferdinand wants to fence three rectangular areas. there are fences in three types, with $4$ amount of fences of each type. You will notice that there is always at least as much area it manages to enclose a total of three by enclosing three square areas (i.e., each area fencing elements of the same size to enclose it) as if it were three different, rectangular would encircle an area (i.e., use two different elements for each of the three areas). Why is this is so? When does it not matter how he fences the rectangles, in terms of the sum of the areas?

Kvant 2020, M2629
Tags: geometry , areas , polygon , Kvant
The figure shows an arbitrary (green) triangle in the center. White squares were built on its sides to the outside. Some of their vertices were connected by segments, white squares were built on them again to the outside, and so on. In the spaces between the squares, triangles and quadrilaterals were formed, which were painted in different colors. Prove that [list=a] [*]all colored quadrilaterals are trapezoids; [*]the areas of all polygons of the same color are equal; [*]the ratios of the bases of one-color trapezoids are equal; [*]if $S_0=1$ is the area of the original triangle, and $S_i$ is the area of the colored polygons at the $i^{\text{th}}$ step, then $S_1=1$, $S_2=5$ and for $n\geqslant 3$ the equality $S_n=5S_{n-1}-S_{n-2}$ is satisfied. [/list] [i]Proposed by F. Nilov[/i] [center][img width="40"]https://i.ibb.co/n8gt0pV/Screenshot-2023-03-09-174624.png[/img][/center]

2019 Ecuador NMO (OMEC), 6
Tags: geometry , rational , areas
Let $n\ge 3$ be a positive integer. Danielle draws a math flower on the plane Cartesian as follows: first draw a unit circle centered on the origin, then draw a polygon of $n$ vertices with both rational coordinates on the circumference so that it has two diametrically opposite vertices, on each side draw a circumference that has the diameter of that side, and finally paints the area inside the $n$ small circles but outside the unit circle. If it is known that the painted area is rational, find all possible polygons drawn by Danielle.

1992 Tournament Of Towns, (336) 4
Tags: combinatorics , geometry , combinatorial geometry , areas
Three triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$ are given such that their centres of gravity (intersection points of their medians) lie on a straight line, but no three of the $9$ vertices of the triangles lie on a straight line. Consider the set of $27$ triangles $A_iB_jC_k$ (where $i$, $j$, $k$ take the values $1$, $2$, $3$ independently). Prove that this set of triangles can be divided into two parts of the same total area. (A. Andjans, Riga)

1972 All Soviet Union Mathematical Olympiad, 164
Tags: geometry , combinatorial geometry , areas , Squares
Given several squares with the total area $1$. Prove that you can pose them in the square of the area $2$ without any intersections.

2016 Auckland Mathematical Olympiad, 3
Tags: geometry , areas , square
Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that: $\bullet$ The area of square $KLMN$ is $1$. $\bullet$ The vertices of the triangle divide three sides of the square up into these ratios: $KX : XL = 3 : 2$ $KY : YN = 4 : 1$ $NZ : ZM = 2 : 3$ What is the area of the triangle $XYZ$? (Note that the sketch is not drawn to scale). [img]https://cdn.artofproblemsolving.com/attachments/8/0/38e76709373ba02346515f9949ce4507ed4f8f.png[/img]

2015 Peru MO (ONEM), 2
Tags: geometry , area of a triangle , areas , hexagon
Let $ABCDEF$ be a convex hexagon. The diagonal $AC$ is cut by $BF$ and $BD$ at points $P$ and $Q$, respectively. The diagonal $CE$ is cut by $DB$ and $DF$ at points $R$ and $S$, respectively. The diagonal $EA$ is cut by $FD$ and $FB$ at points $T$ and $U$, respectively. It is known that each of the seven triangles $APB, PBQ, QBC, CRD, DRS, DSE$ and $AUF$ has area $1$. Find the area of the hexagon $ABCDEF$.

2011 Saudi Arabia BMO TST, 3
Tags: areas , equal areas , geometry , circumcircle
In an acute triangle $ABC$ the angle bisector $AL$, $L \in BC$, intersects its circumcircle at $N$. Let $K$ and $M$ be the projections of $L$ onto sides $AB$ and $AC$. Prove that triangle $ABC$ and quadrilateral $A K N M$ have equal areas.

Durer Math Competition CD 1st Round - geometry, 2021.C3
Tags: geometry , areas , Durer
Csenge has a yellow and a red foil on her rectangular window which look beautiful in the morning light. Where the two foils overlap, they look orange. The window is $80$ cm tall, $120$ cm wide and its corners are denoted by $A, B, C$ and $D$ in the figure. The two foils are triangular and both have two of their vertices at the two bottom corners of the window, A and $B$. The third vertex of the yellow foil is $S$, the trisecting point of side $DC$ closer to $D$, whereas the third vertex of the red foil is $P$, which is one fourth on the way on segment $SC$, closer to $C$. The red region (i.e. triangle $BPE$) is of area $16$ dm$^2$. What is the total area of the regions not covered by foil? [img]https://cdn.artofproblemsolving.com/attachments/b/c/ea371aeafde6968506da6f3456e88fa0bddc6d.png[/img]

2017 Greece Junior Math Olympiad, 1
Tags: square , geometry , area of a triangle , areas
Let $ABCD$ be a square of side $a$. On side $AD$ consider points $E$ and $Z$ such that $DE=a/3$ and $AZ=a/4$. If the lines $BZ$ and $CE$ intersect at point $H$, calculate the area of the triangle $BCH$ in terms of $a$.

2023 Novosibirsk Oral Olympiad in Geometry, 6
Tags: geometry , areas , area , construction , geometric construction
Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half. [img]https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png[/img]

Kvant 2023, M2750
Tags: geometry , areas
Let $D, E$ and $F{}$ be the midpoints of the sides $BC, CA$ and $AB{}$ of the acute-angled triangle $ABC$ and let $H_a, H_b$ and $H_c{}$ be the orthocenters of the triangles $ABD, BCE$ and $CAF{}$ respectively. Prove that the triangles $H_aH_bH_c$ and $DEF$ have equal areas. [i]Proposed by Tran Quang Hung[/i]

Novosibirsk Oral Geo Oly VII, 2023.2
Tags: geometry , areas , square
In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area? [img]https://cdn.artofproblemsolving.com/attachments/d/f/2be7bcda3fa04943687de9e043bd8baf40c98c.png[/img]

1982 Swedish Mathematical Competition, 3
Tags: equal areas , areas , geometry
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.

2009 Oral Moscow Geometry Olympiad, 2
Tags: geometry , trapezoid , areas , parallelogram
Trapezium $ABCD$ and parallelogram $MBDK$ are located so that the sides of the parallelogram are parallel to the diagonals of the trapezoid (see fig.). Prove that the area of the gray part is equal to the sum of the areas of the black part. (Yu. Blinkov) [img]https://cdn.artofproblemsolving.com/attachments/b/9/cfff83b1b85aea16b603995d4f3d327256b60b.png[/img]

Kyiv City MO 1984-93 - geometry, 1985.7.3
Tags: geometry , areas
$O$ is the point of intersection of the diagonals of the convex quadrilateral $ABCD$. It is known that the areas of triangles $AOB, BOC, COD$ and $DOA$ are expressed in natural numbers. Prove that the product of these areas cannot end in $1985$.

May Olympiad L1 - geometry, 2023.3
Tags: geometry , areas
On a straight line $\ell$ there are four points, $A$, $B$, $C$ and $D$ in that order, such that $AB=BC=CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed . Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn , which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.

1994 Denmark MO - Mohr Contest, 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

1983 All Soviet Union Mathematical Olympiad, 355
Tags: geometry , geometric inequality , inequalities , areas
The point $D$ is the midpoint of the side $[AB]$ of the triangle $ABC$ . The points $E$ and $F$ belong to $[AC]$ and $[BC]$ respectively. Prove that the area of triangle $DEF$ area does not exceed the sum of the areas of triangles $ADE$ and $BDF$.

2010 Hanoi Open Mathematics Competitions, 9
Tags: geometry , area of a triangle , areas
Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm$^2$, compute $S_{\vartriangle AMN}$?