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

Durer Math Competition CD 1st Round - geometry, 2013.C1
Tags: ratio , area
Each side of a triangle is extended in the same clockwise direction by the length of the given side as shown in the figure. How many times the area of the triangle, obtained by connecting the endpoints, is the area of the original triangle? [img]https://cdn.artofproblemsolving.com/attachments/1/c/a169d3ab99a894667caafee6dbf397632e57e0.png[/img]

2012 India Regional Mathematical Olympiad, 4
Tags: geometry , area of a triangle , orthocenter , area
$H$ is the orthocentre of an acute–angled triangle $ABC$. A point $E$ is taken on the line segment $CH$ such that $ABE$ is a right–angled triangle. Prove that the area of the triangle $ABE$ is the geometric mean of the areas of triangles $ABC$ and $ABH$.

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

1998 Denmark MO - Mohr Contest, 3
Tags: geometry , parallel , square , area
The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square. [img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]

1984 All Soviet Union Mathematical Olympiad, 394
Tags: combinatorial geometry , geometry , area , 3d geometry , cube
Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.

2018 Stanford Mathematics Tournament, 3
Tags: analytic geometry , number theory , area
Show that if $ A$ is a shape in the Cartesian coordinate plane with area greater than $ 1$, then there are distinct points $(a, b)$, $(c, d)$ in $A$ where $a - c = 2x + 5y$ and $b - d = x + 3y$ where $x, y$ are integers.

2017 Yasinsky Geometry Olympiad, 3
Tags: geometry , maximum value , area , circles
Given circle $\omega$ and point $D$ outside this circle. Find the following points $A, B$ and $C$ on the circle $\omega$ so that the $ABCD$ quadrilateral is convex and has the maximum possible area. Justify your answer.

2003 Greece JBMO TST, 4
Tags: ratio , circles , area of a triangle , geometry , area
Given are two points $B,C$. Consider point $A$ not lying on the line $BC$ and draw the circles $C_1(K_1,R_1)$ (with center $K_1$ and radius $R_1$) and $C_2(K_2,R_2)$ with chord $AB, AC$ respectively such that their centers lie on the interior of the triangle $ABC$ and also $R_1 \cdot AC= R_2 \cdot AB$. Let $T$ be the intersection point of the two circles, different from $A$, and M be a random pointof line $AT$, prove that $TC \cdot S_{(MBT)}=TB \cdot S_{(MCT)}$

2014 BMT Spring, 13
Tags: geometry , angle bisector , area
Let $ABC$ be a triangle with $AB = 16$, $AC = 10$, $BC = 18$. Let $D$ be a point on $AB$ such that $4AD = AB$ and let E be the foot of the angle bisector from $B$ onto $AC$. Let $P$ be the intersection of $CD$ and $BE$. Find the area of the quadrilateral $ADPE$.

1985 All Soviet Union Mathematical Olympiad, 415
Tags: pentagon , geometry , area
All the points situated more close than $1$ cm to ALL the vertices of the regular pentagon with $1$ cm side, are deleted from that pentagon. Find the area of the remained figure.

1969 IMO Shortlist, 20
Tags: polygon , lattice points , area , geometry
$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$

2013 Hanoi Open Mathematics Competitions, 10
Tags: geometry , perimeter , maximum , area
Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{16-p}{p^2+2p}$ where $p$ is the perimeter of the rectangle.

2002 Switzerland Team Selection Test, 7
Tags: geometry , linear algebra , area , triangle
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

Indonesia Regional MO OSP SMA - geometry, 2005.4
Tags: integer , geometry , perimeter , area
The lengths of the three sides $a, b, c$ with $a \le b \le c$, of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.

2020-21 IOQM India, 1
Tags: geometry , trapezoid , area
Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times [CDE]$, what is the value of $n$? (Here $[t]$ denotes the area of the geometrical figure$ t$.)

1992 ITAMO, 2
Tags: area , combinatorial geometry , area of a triangle , area of triangle , geometry
A convex quadrilateral of area $1$ is given. Prove that there exist four points in the interior or on the sides of the quadrilateral such that each triangle with the vertices in three of these four points has an area greater than or equal to $1/4$.

1946 Moscow Mathematical Olympiad, 122
Tags: sum , locus , ratio , geometry , area
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 $$\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}.$$

2004 Argentina National Olympiad, 5
Tags: geometry , area , pentagon , equal segments
The pentagon $ABCDE$ has $AB = BC, CD = DE, \angle ABC = 120^o, \angle CDE = 60^o$ and $BD = 2$. Calculate the area of the pentagon.

2020 Novosibirsk Oral Olympiad in Geometry, 1
Tags: geometry , dodecagon , area
All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in. [img]https://cdn.artofproblemsolving.com/attachments/9/0/9a6af9cef6a4bb03fb4d3eef715f3fd77c74b3.png[/img]

Geometry Mathley 2011-12, 2.1
Tags: area of triangle , geometric inequality , geometry , equilateral , circumcircle , area
Let $ABC$ be an equilateral triangle with circumcircle of center $O$ and radius $R$. Point $M$ is exterior to the triangle such that $S_bS_c = S_aS_b+S_aS_c$, where $S_a, S_b, S_c$ are the areas of triangles $MBC,MCA,MAB$ respectively. Prove that $OM \ge R$. Nguyễn Tiến Lâm

2014 Oral Moscow Geometry Olympiad, 5
Tags: geometry , circumcircle , area of a triangle , area , equilateral , equilateral triangle
Given a regular triangle $ABC$, whose area is $1$, and the point $P$ on its circumscribed circle. Lines $AP, BP, CP$ intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$. Find the area of the triangle $A'B'C'$.

1967 IMO Shortlist, 1
Tags: quadrilateral , area , maximization , geometry
Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.

1996 Tournament Of Towns, (520) 3
Tags: geometry , hexagon , area
Let $A', B', C', D', E'$ and $F'$ be the midpoints of the sides $AB$, $BC$, $CD$, $DE$, $EF$ and $FA$ of an arbitrary convex hexagon $ABCDEF$ respectively. Express the area of $ABCDEF$ in terms of the areas of the triangles $ABC$, $BCD'$, $CDS'$, $DEF'$, $EFA'$ and $FAB'$. (A Lopshi tz, NB Vassiliev)

1980 All Soviet Union Mathematical Olympiad, 289
Tags: geometry , chord , maximum , area , circles
Given a point $E$ on the diameter $AC$ of the certain circle. Draw a chord $BD$ to maximise the area of the quadrangle $ABCD$.

2003 JBMO Shortlist, 2
Tags: geometry , area
Is there a triangle with $12 \, cm^2$ area and $12$ cm perimeter?