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: 698

1988 All Soviet Union Mathematical Olympiad, 464
Tags: equal area , geometry , convex , area
$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area.

Denmark (Mohr) - geometry, 1997.2
Tags: geometry , square , area
Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

Cono Sur Shortlist - geometry, 1993.9
Tags: geometry , perimeter , bisector , incenter , area
Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.

2023 Bulgaria National Olympiad, 4
Tags: area , convex quadrilateral , geometry
Prove that there exists a unique point $M$ on the side $AD$ of a convex quadrilateral $ABCD$ such that \[\sqrt{S_{ABM}}+\sqrt{S_{CDM}} = \sqrt{S_{ABCD}}\] if and only if $AB\parallel CD$.

1989 Chile National Olympiad, 3
Tags: geometry , right triangle , semicircle , area
In a right triangle with legs $a$, $b$ and hypotenuse $c$, draw semicircles with diameters on the sides of the triangle as indicated in the figure. The purple areas have values $X,Y$ . Calculate $X + Y$. [img]https://cdn.artofproblemsolving.com/attachments/1/a/5086dc7172516b0a986ef1af192c15eba4d6fc.png[/img]

2006 IMO, 6
Tags: geometry , area , geometric inequality
Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at least twice the area of $P$.

1983 All Soviet Union Mathematical Olympiad, 355
Tags: geometry , geometric inequality , inequalities , area
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 $BDF$.

2012 Thailand Mathematical Olympiad, 4
Tags: geometry , right triangle , geometric inequality , incenter , area
Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.

2007 Singapore Junior Math Olympiad, 1
Tags: trapezium , geometry , trapezoid , area
Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$. Suppose the area of $\vartriangle DOC$ is $2S/9$. Find the value of $a/b$.

2009 IMAC Arhimede, 2
Tags: hexagon , incircle , area , geometry
In the triangle $ABC$, the circle with the center at the point $O$ touches the pages $AB, BC$ and $CA$ in the points $C_1, A_1$ and $B_1$, respectively. Lines $AO, BO$ and $CO$ cut the inscribed circle at points $A_2, B_2$ and $C_2,$ respectively. Prove that it is the area of the triangle $A_2B_2C_2$ is double from the surface of the hexagon $B_1A_2C_1B_2A_1C_2$. (Moldova)

1980 Spain Mathematical Olympiad, 7
Tags: analytic geometry , locus , area , geometry
The point $M$ varies on the segment $AB$ that measures $2$ m. a) Find the equation and the graphical representation of the locus of the points of the plane whose coordinates, $x$, and $y$, are, respectively, the areas of the squares of sides $AM$ and $MB$ . b) Find out what kind of curve it is. (Suggestion: make a $45^o$ axis rotation). c) Find the area of the enclosure between the curve obtained and the coordinate axes.

2022 Durer Math Competition Finals, 5
Tags: geometry , circles , area
Benedek draws circles with the same center in the following way. The first circle he draws has radius $1$. Next, he draws a second circle such that the ring between the first and second circles has twice the area of the first circle. Next, he draws a third circle such that the ring between the second and third circles is three times the area of the first circle, and so on (see the diagram). What is the smallest $n$ fow which the radius of the $n$-th circle is an integer greater than $1$? [img]https://cdn.artofproblemsolving.com/attachments/e/2/afa6d5ead6f2252aa821028370a3768912e674.png[/img]

Denmark (Mohr) - geometry, 2013.2
Tags: geometry , circumcircle , semicircle , rectangle , area
The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle. [img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]

2013 Oral Moscow Geometry Olympiad, 3
Tags: polyhedron , area , ratio , 3d geometry , geometry
Is there a polyhedron whose area ratio of any two faces is at least $2$ ?

Ukrainian TYM Qualifying - geometry, I.7
Tags: ratio , geometry , polygon , inscribed , area
Given a natural number $n> 3$. On the plane are considered convex $n$ - gons $F_1$ and $F_2$ such that on each side of $F_1$ lies one vertex of $F_2$ and no two vertices $F_1$ and $F_2$ coincide. For each $n$, determine the limits of the ratio of the areas of the polygons $F_1$ and $F_2$. For each $n$, determine the range of the areas of the polygons $F_1$ and $F_2$, if $F_1$ is a regular $n$-gon. Determine the set of values of this value for other partial cases of the polygon $F_1$.

1974 All Soviet Union Mathematical Olympiad, 202
Tags: convex polygon , geometry , area
Given a convex polygon. You can put no triangle with area $1$ inside it. Prove that you can put the polygon inside a triangle with the area $4$.

2018 Auckland Mathematical Olympiad, 3
Tags: geometry , paper folding , folding , rectangle , area
A rectangular sheet of paper whose dimensions are $12 \times 18$ is folded along a diagonal, creating the $M$-shaped region drawn in the picture (see below). Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/4/7/d82cde3e91ab83fa14cd6cefe9bba28462dde1.png[/img]

1989 All Soviet Union Mathematical Olympiad, 497
Tags: geometry , geometric inequality , equal ratio , area
$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.

1981 IMO Shortlist, 18
Tags: sphere , 3d geometry , area , geometry
Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.

2020 Denmark MO - Mohr Contest, 2
Tags: geometry , area
A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes. The grey stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral. [img]https://1.bp.blogspot.com/-ia13b4RsNs0/XzP0cepAcEI/AAAAAAAAMT8/0UuCogTRyj4yMJPhfSK3OQihRqfUT7uSgCLcBGAsYHQ/s0/2020%2Bmohr%2Bp2.png[/img]

2002 Junior Balkan Team Selection Tests - Romania, 3
Tags: geometry , max , area , midpoint , sidelengths
Let $ABC$ be a triangle and $a = BC, b = CA$ and $c = AB$ be the lengths of its sides. Points $D$ and $E$ lie in the same halfplane determined by $BC$ as $A$. Suppose that $DB = c, CE = b$ and that the area of $DECB$ is maximal. Let $F$ be the midpoint of $DE$ and let $FB = x$. Prove that $FC = x$ and $4x^3 = (a^2+b^2 + c^2)x + abc$.

2020 BMT Fall, 16
Tags: geometry , area of a triangle , area
The triangle with side lengths $3, 5$, and $k$ has area $6$ for two distinct values of $k$: $x$ and $y$. Compute $ |x^2 -y^2|$.

2010 Malaysia National Olympiad, 5
Tags: ratio , geometry , area
A circle and a square overlap such that the overlapping area is $50\%$ of the area of the circle, and is $25\%$ of the area of the square, as shown in the figure. Find the ratio of the area of the square outside the circle to the area of the whole figure. [img]https://cdn.artofproblemsolving.com/attachments/e/2/c209a95f457dbf3c46f66f82c0a45cc4b5c1c8.png[/img]

2006 Sharygin Geometry Olympiad, 9.5
Tags: geometry , area of a triangle , area , perimeter , bisects , circumcenter , orthocenter
A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.

2004 Bosnia and Herzegovina Team Selection Test, 2
Tags: number theory , geometry , area
Determine whether does exists a triangle with area $2004$ with his sides positive integers.