Found problems: 414
1996 Estonia National Olympiad, 4
Let $K, L, M$, and $N$ be the midpoints of $CD,DA,AB$ and $BC$ of a square $ABCD$ respectively. Find the are of the triangles $AKB, BLC, CMD$ and $DNA$ if the square $ABCD$ has area $1$.
2009 Balkan MO Shortlist, G5
Let $ABCD$ be a convex quadrilateral and $S$ an arbitrary point in its interior. Let also $E$ be the symmetric point of $S$ with respect to the midpoint $K$ of the side $AB$ and let $Z$ be the symmetric point of $S$ with respect to the midpoint $L$ of the side $CD$. Prove that $(AECZ) = (EBZD) = (ABCD)$.
2023 Israel TST, P2
Let $SABCDE$ be a pyramid whose base $ABCDE$ is a regular pentagon and whose other faces are acute triangles. The altitudes from $S$ to the base sides dissect them into ten triangles, colored red and blue alternatingly. Prove that the sum of the squared areas of the red triangles is equal to the sum of the squared areas of the blue triangles.
Ukrainian TYM Qualifying - geometry, I.8
One of the sides of the triangle is divided by the ratio $p: q$, and the other by $m: n: k$. The obtained division points of the sides are connected to the opposite vertices of the triangle by straight lines. Find the ratio of the area of ​​this triangle to the area of ​​the quadrilateral formed by three such lines and one of the sides of the triangle.
VII Soros Olympiad 2000 - 01, 9.7
Sides $AB$ and $CD$ of quadrilateral $ABCD$ intersect at point $E$. On the diagonals$ AC$ and $BD$ points $M$ and $N$ are taken, respectively, so that $AM / AC = BN / BD = k$. Find the area of ​​a triangle $EMN$ if the area of ​​$ABCD$ is $S$.
1997 Tournament Of Towns, (526) 3
The vertical diameter of a circle is moved a centimetres to the right, and the horizontal diameter of this circle is moved $b$ centimetres up. These two lines divide the circle into four pieces. Consider the sum of the areas of the largest and the smallest pieces, and the sum of the areas of the other two pieces. Find the difference between these two sums.
(G Galperin, NB Vassiliev)
1962 All Russian Mathematical Olympiad, 013
Given points $A' ,B' ,C' ,D',$ on the extension of the $[AB], [BC], [CD], [DA]$ sides of the convex quadrangle $ABCD$, such, that the following pairs of vectors are equal: $$[BB']=[AB], [CC']=[BC], [DD']=[CD], [AA']=[DA].$$ Prove that the quadrangle $A'B'C'D'$ area is five times more than the quadrangle $ABCD$ area.
1970 Spain Mathematical Olympiad, 3
An arbitrary triangle $ABC$ is given and a point $P$ lies on the side $AB$. It is requested to draw through $P$ a line that divides the triangle into two figures of the same area.
2015 Oral Moscow Geometry Olympiad, 3
$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.
1942 Eotvos Mathematical Competition, 3
Let $A'$, $B'$ and $C'$ be points on the sides $BC$, $CA$ and $AB$, respectively, of an equilateral triangle $ABC$. If $AC' = 2C'B$, $BA' = 2A'C$ and $CB' = 2B'A$, prove that the lines $AA'$, $BB'$ and $CC'$ enclose a triangle whose area is $1/7$ that of $ABC$.
2020-21 IOQM India, 1
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$.)
II Soros Olympiad 1995 - 96 (Russia), 9.7
Through a point located on a side of a triangle of area $1$, two straight lines are drawn parallel to the two remaining sides. They divided the triangle into three parts. Let $s$ be the largest of the areas of these parts. Find the smallest possible value of $s$.
1984 All Soviet Union Mathematical Olympiad, 384
The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).
1953 Moscow Mathematical Olympiad, 242
Let $A$ be a vertex of a regular star-shaped pentagon, the angle at $A$ being less than $180^o$ and the broken line $AA_1BB_1CC_1DD_1EE_1$ being its contour. Lines $AB$ and $DE$ meet at $F$. Prove that polygon $ABB_1CC_1DED_1$ has the same area as the quadrilateral $AD_1EF$.
Note: A regular star pentagon is a figure formed along the diagonals of a regular pentagon.
2011 May Olympiad, 3
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]
2003 Junior Balkan Team Selection Tests - Moldova, 3
The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$
2003 Paraguay Mathematical Olympiad, 5
In a square $ABCD$, $E$ is the midpoint of side $BC$. Line $AE$ intersects line $DC$ at $F$ and diagonal $BD$ at $G$. If the area $(EFC) = 8$, determine the area $(GBE)$.
1966 IMO Longlists, 63
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
1981 Swedish Mathematical Competition, 5
$ABC$ is a triangle. $X$, $Y$, $Z$ lie on $BC$, $CA$, $AB$ respectively. Show that area $XYZ$ cannot be smaller than each of area $AYZ$, area $BZX$, area $CXY$.
2001 May Olympiad, 4
Ten coins of $1$ cm radius are placed around a circle as indicated in the figure.
Each coin is tangent to the circle and its two neighboring coins.
Prove that the sum of the areas of the ten coins is twice the area of the circle.
[img]https://cdn.artofproblemsolving.com/attachments/5/e/edf7a7d39d749748f4ae818853cb3f8b2b35b5.gif[/img]
III Soros Olympiad 1996 - 97 (Russia), 10.5
Two circles intersect at two points $A$ and $B$. The radii of these circles are equal to $R$ and $r$, respectively; the angle between the radii going to the points of intersection is equal to $a$. A chord $KM$ of length $b$ is taken in a circle of radius $r$. Straight lines $KA$, $KB$, $MA$ and $MB$ intersect the other circle for second time at four points. Find the area of the quadrilateral with vertices at these points.
1915 Eotvos Mathematical Competition, 3
Prove that a triangle inscribed in a parallelogram has at most half the area of the parallelogram.
Mathley 2014-15, 2
Let $ABC$ be a triangle with a circumcircle $(K)$. A circle touching the sides $AB,AC$ is internally tangent to $(K)$ at $K_a$; two other points $K_b,K_c$ are defined in the same manner. Prove that the area of triangle $K_aK_bK_c$ does not exceed that of triangle $ABC$.
Nguyen Minh Ha, Hanoi University of Education, Xuan Thuy, Cau Giay, Hanoi.
1986 Tournament Of Towns, (107) 1
Through vertices $A$ and $B$ of triangle $ABC$ are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral .
(G . Galperin , A . Savin, Moscow)
Estonia Open Junior - geometry, 1996.2.4
A pentagon (not necessarily convex) has all sides of length $1$ and its product of cosine of any four angles equal to zero. Find all possible values of the area of such a pentagon.