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

2011 Abels Math Contest (Norwegian MO), 2b
Tags: concurrency , diagonal , area of a triangle , hexagon , area , geometry
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]

Cono Sur Shortlist - geometry, 1993.9
Tags: bisector , perimeter , incenter , area , geometry
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.

1952 Poland - Second Round, 2
Tags: area , geometry
Prove that if $ a $, $ b $, $ c $, $ d $ are the sides of a quadrilateral in which a circle can be circumscribed and a circle can be inscribed in it, then the area $ S $ of the quadrilateral is given by $$S = \sqrt{abcd}.$$

1991 Austrian-Polish Competition, 6
Tags: diagonal , convex , bisects , area of a triangle , area , geometry
Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$.

2016 Swedish Mathematical Competition, 1
Tags: geometry , max , area
In a garden there is an $L$-shaped fence, see figure. You also have at your disposal two finished straight fence sections that are $13$ m and $14$ m long respectively. From point $A$ you want to delimit a part of the garden with an area of at least $200$ m$^2$ . Is it possible to do this? [img]https://1.bp.blogspot.com/-VLWIImY7HBA/X0yZq5BrkTI/AAAAAAAAMbg/8CyP6DzfZTE5iX01Qab3HVrTmaUQ7PvcwCK4BGAYYCw/s400/sweden%2B16p1.png[/img]

Novosibirsk Oral Geo Oly IX, 2019.3
Tags: geometry , area , square
The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square. (Distance in the picture is measured horizontally from the midpoint of the side of the square.) [img]https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png[/img]

2016 AMC 10, 23
Tags: area , geometry
In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$? $\textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{10}{27}\qquad\textbf{(C)}\ \frac{11}{27}\qquad\textbf{(D)}\ \frac{4}{9}\qquad\textbf{(E)}\ \frac{13}{27}$

2007 Sharygin Geometry Olympiad, 5
Tags: area , cut , polygon , geometry
A non-convex $n$-gon is cut into three parts by a straight line, and two parts are put together so that the resulting polygon is equal to the third part. Can $n$ be equal to: a) five? b) four?

1994 Denmark MO - Mohr Contest, 5
Tags: right triangle , area , isosceles , equal area , geometry
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

1946 Moscow Mathematical Olympiad, 122
Tags: ratio , area , sum , locus , geometry
On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn. Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB$, $\vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB$, $\vartriangle S_0CD$, $\vartriangle S0EF$. Consider separately the case $$\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}.$$

1978 Putnam, B1
Tags: geometry , area
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length $3$ and the remaining four sides of length $2$. Give the answer in the form $r+s\sqrt{t}$ with $r,s, t$ positive integers.

1980 Tournament Of Towns, (005) 5
Tags: combinatorial geometry , area , combinatorics , square , geometry
A finite set of line segments, of total length $18$, belongs to a square of unit side length (we assume that the square includes its boundary and that a line segment includes its end points). The line segments are parallel to the sides of the square and may intersect one another. Prove that among the regions into which the square is divided by the line segments, at least one of these must have area not less than $0.01$. (A Berzinsh, Riga)

2012 NZMOC Camp Selection Problems, 1
Tags: regular , octagon , area , square , geometry
From a square of side length $1$, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?

1989 IMO Longlists, 3
Tags: area , equation , geometry , rectangle , 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?

1992 Czech And Slovak Olympiad IIIA, 2
Tags: geometric inequalities , 3d geometry , tetrahedron , geometry , area , inequalities
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$

1979 All Soviet Union Mathematical Olympiad, 277
Tags: area , square , combinatorial geometry , geometry
Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.

1980 IMO Longlists, 8
Tags: triangle , function , geometry , area
Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$. Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$.

2008 Mathcenter Contest, 7
Tags: area of a triangle , geometric inequality , area , geometry
$ABC$ is a triangle with an area of $1$ square meter. Given the point $D$ on $BC$, point $E$ on $CA$, point $F$ on $AB$, such that quadrilateral $AFDE$ is cyclic. Prove that the area of $DEF \le \frac{EF^2}{4 AD^2}$. [i](holmes)[/i]

2015 Peru MO (ONEM), 2
Tags: area , geometry , area of a triangle , 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 May Olympiad, 3
Tags: geometry , area of a triangle , rectangle , area
In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ . [img]https://1.bp.blogspot.com/-4w6e729AF9o/XNY9hqHaBaI/AAAAAAAAKL0/eCaNnWmgc7Yj9uV4z29JAvTcWCe21NIMgCK4BGAYYCw/s400/may%2B2011%2Bl1.png[/img]

1974 Poland - Second Round, 4
Tags: area , geometry
In a convex quadrilateral $ ABCD $ with area $ S $, each side was divided into 3 equal parts and segments were drawn connecting the appropriate points of division of the opposite sides in such a way that the quadrilateral was divided into 9 quadrilaterals. Prove that the sum of the areas of the following three quadrilaterals resulting from the division: the one containing the vertex $ A $, the middle one and the one containing the vertex $ C $ is equal to $ \frac{S}{3} $.

Estonia Open Senior - geometry, 2010.2.1
Tags: area , trapezoid , geometric inequality , geometry
The diagonals of trapezoid $ABCD$ with bases $AB$ and $CD$ meet at $P$. Prove the inequality $S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA}$, where $S_{XYZ}$ denotes the area of triangle $XYZ$.

1955 Moscow Mathematical Olympiad, 288
Tags: right triangle , geometry , area , bisects segment , angle
We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.

2015 Singapore Junior Math Olympiad, 2
Tags: area , equal area , triangle area , hexagon , parallel , geometry
In a convex hexagon $ABCDEF, AB$ is parallel to $DE, BC$ is parallel to $EF$ and $CD$ is parallel to $FA$. Prove that the triangles $ACE$ and $BDF$ have the same area.

2017 Adygea Teachers' Geometry Olympiad, 1
Tags: trapezoid , geometry , area
Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.