Found problems: 414
1996 Cono Sur Olympiad, 1
In the following figure, the largest square is divided into two squares and three rectangles, as shown:
The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$. If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square.
[img]https://cdn.artofproblemsolving.com/attachments/f/a/0b424d9c293889b24d9f31b1531bed5081081f.png[/img]
Denmark (Mohr) - geometry, 2002.1
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]
2022 Abelkonkurransen Finale, 2b
Triangles $ABC$ and $DEF$ have pairwise parallel sides: $EF \| BC, FD \| CA$, and $DE \| AB$. The line $m_A$ is the reflection of $EF$ through $BC$, similarly $m_B$ is the reflection of $FD$ through $CA$, and $m_C$ the reflection of $DE$ through $AB$. Assume that the lines $m_A, m_B$, and $m_C$ meet in a common point. What is the ratio between the areas of triangles $ABC$ and $DEF$?
2002 Tuymaada Olympiad, 8
The circle with the center of $ O $ touches the sides of the angle $ A $ at the points of $ K $ and $ M $. The tangent to the circle intersects the segments $ AK $ and $ AM $ at points $ B $ and $ C $ respectively, and the line $ KM $ intersects the segments $ OB $ and $ OC $ at the points $ D $ and $ E $. Prove that the area of the triangle $ ODE $ is equal to a quarter of the area of a triangle $ BOC $ if and only if the angle $ A $ is $ 60^\circ $.
2013 Hanoi Open Mathematics Competitions, 8
Let $ABCDE$ be a convex pentagon and area of $\vartriangle ABC =$ area of $\vartriangle BCD =$ area of $\vartriangle CDE=$ area of $\vartriangle DEA =$ area of $\vartriangle EAB$. Given that area of $\vartriangle ABCDE = 2$. Evaluate the area of area of $\vartriangle ABC$.
2023 Auckland Mathematical Olympiad, 2
Triangle $ABC$ of area $1$ is given. Point $A'$ lies on the extension of side $BC$ beyond point $C$ with $BC = CA'$. Point $B'$ lies on extension of side $CA$ beyond $A$ and $CA = AB'$. $C'$ lies on extension of $AB$ beyond $B$ with $AB = BC'$. Find the area of triangle $A'B'C'$.
2020 Ukrainian Geometry Olympiad - December, 2
Let $ABCD$ be a cyclic quadrilateral such that $AC =56, BD = 65, BC>DA$ and $AB: BC =CD: DA$. Find the ratio of areas $S (ABC): S (ADC)$.
Durer Math Competition CD 1st Round - geometry, 2011.C4
Given a grid rectangle of size $2010 \times 1340$. A grid point is called [i]fair [/i] if the $2$ axis-parallel lines passing through it from the upper left and lower right corners of the large rectangle cut out a rectangle of equal area (such a point is shown in the figure). How many fair grid points lie inside the rectangle?
[img]https://cdn.artofproblemsolving.com/attachments/1/b/21d4fb47c94b774994ac1c3aae7690bb98c7ae.png[/img]
2003 May Olympiad, 2
The triangle $ABC$ is right in $A$ and $R$ is the midpoint of the hypotenuse $BC$ . On the major leg $AB$ the point $P$ is marked such that $CP = BP$ and on the segment $BP$ the point $Q$ is marked such that the triangle $PQR$ is equilateral. If the area of triangle $ABC$ is $27$, calculate the area of triangle $PQR$ .
1997 May Olympiad, 5
What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?
Brazil L2 Finals (OBM) - geometry, 2017.1
The points $X, Y,Z$ are marked on the sides $AB, BC,AC$ of the triangle $ABC$, respectively. Points $A',B', C'$ are on the $XZ, XY, YZ$ sides of the triangle $XYZ$, respectively, so that $\frac{AB}{A'B'} = \frac{AB}{A'B'} =\frac{BC}{B'C'}= 2$ and $ABB'A',BCC'B',ACC'A'$ are trapezoids in which the sides of the triangle $ABC$ are bases.
a) Determine the ratio between the area of the trapezium $ABB'A'$ and the area of the triangle $A'B'X$.
b) Determine the ratio between the area of the triangle $XYZ$ and the area of the triangle $ABC$.
2020 Nordic, 3
Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$.
1953 Poland - Second Round, 3
A triangular piece of sheet metal weighs $900$ g. Prove that by cutting this sheet metal along a straight line passing through the center of gravity of the triangle, it is impossible to cut off a piece weighing less than $400$ g.
Durer Math Competition CD 1st Round - geometry, 2013.D3
The area of the triangle $ABC$ shown in the figure is $1$ unit. Points $D$ and $E$ lie on sides $AC$ and $BC$ respectively, and also are its ''one third'' points closer to $C$. Let $F$ be that $AE$ and $G$ are the midpoints of segment $BD$. What is the area of the marked quadrilateral $ABGF$?
[img]https://cdn.artofproblemsolving.com/attachments/4/e/305673f429c86bbc58a8d40272dd6c9a8f0ab2.png[/img]
2016 May Olympiad, 4
In a triangle $ABC$, let $D$ and $E$ be points of the sides $ BC$ and $AC$ respectively. Segments $AD$ and $BE$ intersect at $O$. Suppose that the line connecting midpoints of the triangle and parallel to $AB$, bisects the segment $DE$. Prove that the triangle $ABO$ and the quadrilateral $ODCE$ have equal areas.
1988 Tournament Of Towns, (165) 2
We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .
2023 Novosibirsk Oral Olympiad in Geometry, 2
In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area?
[img]https://cdn.artofproblemsolving.com/attachments/d/f/2be7bcda3fa04943687de9e043bd8baf40c98c.png[/img]
1983 All Soviet Union Mathematical Olympiad, 365
One side of the rectangle is $1$ cm. It is known that the rectangle can be divided by two orthogonal lines onto four rectangles, and each of the smaller rectangles has the area not less than $1$ square centimetre, and one of them is not less than $2$ square centimetres. What is the least possible length of another side of big rectangle?
2024 Canadian Mathematical Olympiad Qualification, 3
Let $\vartriangle ABC$ be an acute triangle with $AB < AC$. Let $H$ be its orthocentre and $M$ be the midpoint of arc $BAC$ on the circumcircle. It is given that $B$, $H$, $M$ are collinear, the length of the altitude from $M$ to $AB$ is $1$, and the length of the altitude from $M$ to $BC$ is $6$. Determine all possible areas for $\vartriangle ABC$ .
2021 Iranian Geometry Olympiad, 2
Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$ is parallel to some side of $ABCD$.
[i]Proposed by Josef Tkadlec - Czech Republic[/i]
1980 Spain Mathematical Olympiad, 7
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.
IV Soros Olympiad 1997 - 98 (Russia), 11.5
The sides of the parallelogram serve as the diagonals of the four squares. The vertices of the squares lying in the part of the plane external to the parallelogram (the sides of the squares emerging from these vertices do not have common points with the parallelogram) serve as the vertices of a quadrilateral of area $a$, the four vertices opposite to them form a quadrilateral of area $b$. Find the area of the parallelogram.
1953 Moscow Mathematical Olympiad, 249
Let $a, b, c, d$ be the lengths of consecutive sides of a quadrilateral, and $S$ its area. Prove that $S \le \frac{ (a + b)(c + d)}{4}$
VMEO I 2004, 3
In the plane, given an angle $Axy$.
a) Given a triangle $MNP$ of area $T$, describe how to construct a triangle of given area $T$ and altitude $h$. Using this, describe how to construct parallelogram A$BCD$ with two sides lying on $Ax$ and $Ay$, the area $T$ and the distance between the two opposite sides equal to d given.
b) From an arbitrary point $I$ on the line $CD$, construct a line that intersects the lines $A$B, $BC$, $AD$ at $E$, $G$ and $F$ respectively so that the area of triangle $AEF$ is equal to the area of parallelogram $ABCD$.
c) Apply the above two sentences: Given any point $O$ in the plane. From $O$, construct a line that intersects two rays $Ax$ and $Ay$ at $E$ and $F$ respectively so that the area of triangle $AEF$ is equal to the area of any given triangle.
2006 JBMO ShortLists, 15
Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.