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

2006 May Olympiad, 4
Tags: geometry , trapezoid , area
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 $BCO$.

2012 AMC 8, 6
Tags: geometry , area
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches? $\textbf{(A)}\hspace{.05in}36 \qquad \textbf{(B)}\hspace{.05in}40 \qquad \textbf{(C)}\hspace{.05in}64 \qquad \textbf{(D)}\hspace{.05in}72 \qquad \textbf{(E)}\hspace{.05in}88 $

2008 Abels Math Contest (Norwegian MO) Final, 4a
Tags: geometry , area , angle , circles
Three distinct points $A, B$, and $C$ lie on a circle with centre at $O$. The triangles $AOB, BOC$ , and $COA$ have equal area. What are the possible measures of the angles of the triangle $ABC$ ?

2021 China Team Selection Test, 5
Tags: combinatorial geometry , combinatorics , area , convex hull , convex polygon , symmetry , geometry
Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.

Estonia Open Junior - geometry, 2000.1.5
Tags: geometry , equal circles , area of a circle , area
Find the total area of the shaded area in the figure if all circles have an equal radius $R$ and the centers of the outer circles divide into six equal parts of the middle circle. [img]http://3.bp.blogspot.com/-Ax0QJ38poYU/XovXkdaM-3I/AAAAAAAALvM/DAZGVV7TQjEnSf2y1mbnse8lL6YIg-BQgCK4BGAYYCw/s400/estonia%2B2000%2Bo.j.1.5.png[/img]

1986 Tournament Of Towns, (126) 1
Tags: geometry , trapezoid , tangential , inradius , area
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .

2017-IMOC, G2
Tags: geometry , area of a triangle , geometric inequality , area
Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$

1955 Poland - Second Round, 5
Tags: rectangle , geometry , geometric inequalities , area
Given a triangle $ ABC $. Find the rectangle of smallest area containing the triangle.

2003 Estonia National Olympiad, 1
Tags: geometry , regular , pentagon , circles , area
The picture shows $10$ equal regular pentagons where each two neighbouring pentagons have a common side. The smaller circle is tangent to one side of each pentagon and the larger circle passes through the opposite vertices of these sides. Find the area of the larger circle if the area of the smaller circle is $1$. [img]https://cdn.artofproblemsolving.com/attachments/0/6/84fe98370868a5cf28d92d4b207ccb00e6eaa3.png[/img]

Estonia Open Senior - geometry, 2000.1.3
Tags: equal area , geometry , locus , area of a triangle , area
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 $CD$.

Kvant 2022, M2720
Tags: geometry , area
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]

1956 Moscow Mathematical Olympiad, 340
Tags: combinatorics , combinatorial geometry , geometry , rectangle , area
a) * In a rectangle of area $5$ sq. units, $9$ rectangles of area $1$ are arranged. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$ b) In a rectangle of area $5$ sq. units, lie $9$ arbitrary polygons each of area $1$. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$

2000 District Olympiad (Hunedoara), 4
Tags: geometry , 3d geometry , pyramid , perimeter , area
Consider the pyramid $ VABCD, $ where $ V $ is the top and $ ABCD $ is a rectangular base. If $ \angle BVD = \angle AVC, $ then prove that the triangles $ VAC $ and $ VBD $ share the same perimeter and area.

2018 Hanoi Open Mathematics Competitions, 9
Tags: geometry , combinatorial geometry , geometric inequality , area
There are three polygons and the area of each one is $3$. They are drawn inside a square of area $6$. Find the greatest value of $m$ such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than $m$.

2018 Polish Junior MO Second Round, 2
Tags: geometry , area , plane geometry , quadrilateral , circumcircle
Let $ABC$ be an acute traingle with $AC \neq BC$. Point $K$ is a foot of altitude through vertex $C$. Point $O$ is a circumcenter of $ABC$. Prove that areas of quadrilaterals $AKOC$ and $BKOC$ are equal.

1989 IMO Longlists, 3
Tags: geometry , area , rectangle , equation , algebra
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

2019 Costa Rica - Final Round, 2
Tags: geometry , parallelogram , area
Consider the parallelogram $ABCD$, with $\angle ABC = 60$ and sides $AB =\sqrt3$, $BC = 1$. Let $\omega$ be the circle of center $B$ and radius $BA$, and let $\tau$ be the circle of center $D$ and radius $DA$. Determine the area of the region between the circumferences $\omega$ and $\tau$, within the parallelogram $ABCD$ (the area of the shaded region). [img]https://cdn.artofproblemsolving.com/attachments/5/a/02b17ec644289d95b6fce78cb5f1ecb3d3ba5b.png[/img]

2011 Abels Math Contest (Norwegian MO), 2b
Tags: geometry , hexagon , area of a triangle , diagonal , area , concurrency
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point. Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$, where $a(KLM)$ is the area of the triangle $KLM$. [img]https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png[/img]

Durer Math Competition CD 1st Round - geometry, 2019.C3
Tags: geometry , area
The best parts of grandma’s $30$ cm $ \times 30$ cm square shaped pie are the edges. For this reason grandma’s three grandchildren would like to split the pie between each other so that everyone gets the same amount (of the area) of the pie, but also of the edges. Can they cut the pie into three connected pieces like that?

May Olympiad L2 - geometry, 2006.4
Tags: geometry , trapezoid , area
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 $BCO$.

1989 Tournament Of Towns, (239) 3
Tags: geometry , area
Choose a point $A$ inside a circle of radius $R$. Construct a pair of perpendicular lines through $A$. Then rotate these lines through the same angle $V$ about $A$. The figure formed inside the circle, as the lines move from their initial to their final position, is in the form of a cross with its centre at $A$. Find the area of this cross. (Problem from Latvia)

1998 Belarus Team Selection Test, 3
Tags: geometry , parallelogram , hexagon , area , equilateral
Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$

Denmark (Mohr) - geometry, 2002.1
Tags: geometry , rectangle , area
An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$. [img]https://1.bp.blogspot.com/-BipDNHELjJI/XzcCa68P3HI/AAAAAAAAMXY/H2Iqya9VItMLXrRqsdyxHLTXCAZ02nEtgCLcBGAsYHQ/s0/2002%2BMohr%2Bp1.png[/img]

1993 IMO, 4
Tags: geometry , area , inequalities
For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane, \[ m(ABC) \leq m(ABX) + m(AXC) + m(XBC). \]

1962 Polish MO Finals, 2
Tags: geometry , construction , area
Inside a given convex quadrilateral, find a point such that the segments connecting this point with the midpoints of the quadrilateral's sides divide the quadrilateral into four parts with equal areas.