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

1976 Chisinau City MO, 122
Tags: Cyclic , geometry , areas , number theory , prime numbers
The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.

2014 Chile National Olympiad, 2
Tags: geometry , parallelogram , Centroid , areas , area
Consider an $ABCD$ parallelogram of area $1$. Let $E$ be the center of gravity of the triangle $ABC, F$ the center of gravity of the triangle $BCD, G$ the center of gravity of the triangle $CDA$ and $H$ the center of gravity of the triangle $DAB$. Calculate the area of quadrilateral $EFGH$.

Kyiv City MO 1984-93 - geometry, 1992.10.2
Tags: geometry , areas
In the triangle $ABC$, the median $BD$ is drawn and through its midpoint and vertex $A$ the line $\ell$. Thus the triangle $ABC$ is divided into three triangles and one quadrilateral. Determine the areas of these figures if the area of ​​triangle $ABC$ is equal to $S$.

2012 BMT Spring, 3
Tags: ratio , geometry , Squares , areas
Let $ABC$ be a triangle with side lengths $AB = 2011$, $BC = 2012$, $AC = 2013$. Create squares $S_1 =ABB'A''$, $S_2 = ACC''A'$ , and $S_3 = CBB''C'$ using the sides $AB$, $AC$, $BC$ respectively, so that the side $B'A''$ is on the opposite side of $AB$ from $C$, and so forth. Let square $S_4$ have side length $A''A' $, square $S_5$ have side length $C''C'$, and square $S_6$ have side length $B''B'$. Let $A(S_i)$ be the area of square $S_i$ . Compute $\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}$?

1997 Chile National Olympiad, 3
Tags: geometry , trapezoid , areas
Let $ ABCD $ be a quadrilateral, whose diagonals intersect at $ O $. The triangles $ \triangle AOB $, $ \triangle BOC $, $ \triangle COD $ have areas $1, 2, 4$, respectively. Find the area of $ \triangle AOD $ and prove that $ ABCD $ is a trapezoid.

1979 All Soviet Union Mathematical Olympiad, 269
Tags: geometry , areas , isosceles , inscribed
What is the least possible ratio of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?

II Soros Olympiad 1995 - 96 (Russia), 10.10
Tags: geometry , geometric inequality , areas
The Order "For Faithful Service" of the $7$th degree in shape is a combination of a semicircle with a diameter $AB = 2$ and a triangle $AM B$. The sides$ AM$ and $BM$ intersect the semicircle (the border of the semicircle). The part of the circle outside the triangle and the part of the triangle outside the circle are made of pure copper. What should the side of the triangle be equal to in order for the area of the copper part to be the smallest?

2020 Estonia Team Selection Test, 2
Tags: geometry , areas , geometric inequality
The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$. Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.

V Soros Olympiad 1998 - 99 (Russia), 10.9
Tags: geometry , geometric inequality , areas
A triangle of area $1$ is cut out of paper. Prove that it can be bent along a straight segment so that the area of the resulting figure is less than $s_0$, where $s_0=\frac{\sqrt5-1}{2}$. Note. The value $s_0$ specified in the condition can be reduced (the smallest value of$s_0$ is unknown to the authors of the problem). If you manage to do this (and justify it), write.

1936 Moscow Mathematical Olympiad, 029
Tags: geometry , rectangle , integer sides , areas
The lengths of a rectangle’s sides and of its diagonal are integers. Prove that the area of the rectangle is an integer multiple of $12$.

V Soros Olympiad 1998 - 99 (Russia), 11.3
Tags: analytic geometry , geometry , areas
Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$ and the graphs of the functions $y =\sqrt{x^3+ 1}$ and $y = - \sqrt[3]{x^2+ 2x}$.

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]

2004 Greece Junior Math Olympiad, 2
Tags: geometry , trapezoid , isosceles , areas
Let $ABCD$ be a rectangle. Let $K,L$ be the midpoints of $BC, AD$ respectively. From point $B$ the perpendicular line on $AK$, intersects $AK$ at point $E$ and $CL$ at point $Z$. a) Prove that the quadrilateral $AKZL$ is an isosceles trapezoid b) Prove that $2S_{ABKZ}=S_{ABCD}$ c) If quadrilateral $ABCD$ is a square of side $a$, calculate the area of the isosceles trapezoid $AKZL$ in terms of side $BC=a$

1971 All Soviet Union Mathematical Olympiad, 152
Tags: geometry , perimeter , areas , incenter
a) Prove that the line dividing the triangle onto two polygons with equal perimeters and equal areas passes through the centre of the inscribed circle. b) Prove the same statement for the arbitrary tangential polygon. c) Prove that all the lines halving its perimeter and area simultaneously, intersect in one point.

2013 Sharygin Geometry Olympiad, 3
Tags: geometry , Inequality , areas
Each sidelength of a convex quadrilateral $ABCD$ is not less than $1$ and not greater than $2$. The diagonals of this quadrilateral meet at point $O$. Prove that $S_{AOB}+ S_{COD} \le 2(S_{AOD}+ S_{BOC})$.

1999 Estonia National Olympiad, 3
Tags: geometry , areas , incircle
The incircle of the triangle $ABC$, with the center $I$ , touches the sides $AB, AC$ and $BC$ in the points $K, L$ and $M$ respectively. Points $P$ and $Q$ are taken on the sides $AC$ and $BC$ respectively, such that $|AP| = |CL|$ and $|BQ| = |CM|$. Prove that the difference of areas of the figures $APIQB$ and $CPIQ$ is equal to the area of the quadrangle $CLIM$

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

2016 Chile National Olympiad, 2
Tags: geometry , areas
For a triangle $\vartriangle ABC$, determine whether or not there exists a point $P$ on the interior of $\vartriangle ABC$ in such a way that every straight line through $P$ divides the triangle $\vartriangle ABC$ in two polygons of equal area.

2016 Oral Moscow Geometry Olympiad, 3
Tags: geometry , Squares , square , areas , equal area
Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal. [img]https://3.bp.blogspot.com/-W50DOuizFvY/XT6wh3-L6sI/AAAAAAAAKaw/pIW2RKmttrwPAbrKK3bpahJz7hfIZwM8QCK4BGAYYCw/s400/Oral%2BSharygin%2B2016%2B10.11%2Bp3.png[/img]

2023 Chile Junior Math Olympiad, 3
Tags: geometry , areas , Equilateral
Let $\vartriangle ABC$ be an equilateral triangle with side $1$. Four points are marked $P_1$, $P_2$, $P_3$, $P_4$ on side $AC$ and four points $Q_1$, $Q_2$, $Q_3$, $Q_4$ on side $AB$ (see figure) of such a way to generate $9$ triangles of equal area. Find the length of segment $AP_4$. [img]https://cdn.artofproblemsolving.com/attachments/5/f/29243932262cb963b376244f4c981b1afe87c6.png[/img] PS. Easier version of [url=https://artofproblemsolving.com/community/c6h3323141p30741525]2023 Chile NMO L2 P3[/url]

2000 Tournament Of Towns, 1
Tags: areas , area , geometry , midpoint
The diagonals of a convex quadrilateral $ABCD$ meet at $P$. The sum of the areas of triangles $PAB$ and $PCD$ is equal to the sum of areas of triangles $PAD$ and $PCB$. Prove that $P$ is the midpoint of either $AC$ or $BD$. (Folklore)

2021 Durer Math Competition Finals, 3
Tags: areas , geometry
The figure shows a line intersecting a square lattice. The area of some arising quadrilaterals are also indicated. What is the area of the region with the question mark? [img]https://cdn.artofproblemsolving.com/attachments/0/d/4d5741a63d052e3f6971f87e60ca7df7302fb0.png[/img]

2018 Czech-Polish-Slovak Junior Match, 2
Tags: ratio , areas , geometry , symmetry , right triangle
Given a right triangle $ABC$ with the hypotenuse $AB$. Let $K$ be any interior point of triangle $ABC$ and points $L, M$ are symmetric of point $K$ wrt lines $BC, AC$ respectively. Specify all possible values for $S_{ABLM} / S_{ABC}$, where $S_{XY ... Z}$ indicates the area of the polygon $XY...Z$ .

Denmark (Mohr) - geometry, 2017.3
Tags: geometry , areas , arc
The figure shows an arc $\ell$ on the unit circle and two regions $A$ and $B$. Prove that the area of $A$ plus the area of $B$ equals the length of $\ell$. [img]https://1.bp.blogspot.com/-SYoSrFowZ30/XzRz0ygiOVI/AAAAAAAAMUs/0FCduUoxKGwq0gSR-b3dtb3SvDjZ89x_ACLcBGAsYHQ/s0/2017%2BMohr%2Bp3.png[/img]