Found problems: 414
Kyiv City MO 1984-93 - geometry, 1986.10.5
Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.
Swiss NMO - geometry, 2008.1
Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.
2009 Thailand Mathematical Olympiad, 4
In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.
2005 Junior Tuymaada Olympiad, 2
Points $ X $ and $ Y $ are the midpoints of the sides $ AB $ and $ AC $ of the triangle $ ABC $, $ I $ is the center of its inscribed circle, $ K $ is the point of tangency of the inscribed circles with side $ BC $. The external angle bisector at the vertex $ B $ intersects the line $ XY $ at the point $ P $, and the external angle bisector at the vertex of $ C $ intersects $ XY $ at $ Q $. Prove that the area of the quadrilateral $ PKQI $ is equal to half the area of the triangle $ ABC $.
1974 All Soviet Union Mathematical Olympiad, 202
Given a convex polygon. You can put no triangle with area $1$ inside it. Prove that you can put the polygon inside a triangle with the area $4$.
1952 Polish MO Finals, 2
On the sides $ BC $, $ CA $, $ AB $ of the triangle $ ABC $, the points $ M $, $ N $, $ P $ are taken, respectively, in such a way that $$\frac{BM}{MC} = \frac{CN}{NA} = \frac{AP}{PB} = k, $$
where $ k $ means a given number greater than $ 1 $, then the segments $ AM $, $ BN $, $ CP $ were drawn . Given the area $ S $ of the triangle $ ABC $, calculate the area of the triangle bounded by the lines $ AM $, $ BN $ and $ CP $.
2007 Singapore Junior Math Olympiad, 1
Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$. Suppose the area of $\vartriangle DOC$ is $2S/9$. Find the value of $a/b$.
Estonia Open Senior - geometry, 1998.2.1
Circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively lie on a plane such that that the circle $C_2$ passes through $O_1$. The ratio of radius of circle $C_1$ to $O_1O_2$ is $\sqrt{2+\sqrt3}$.
a) Prove that the circles $C_1$ and $C_2$ intersect at two distinct points.
b) Let $A,B$ be these points of intersection. What proportion of the area of circle is $C_1$ is the area of the sector $AO_1B$ ?
1996 Tournament Of Towns, (520) 3
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)
2016 India PRMO, 5
Consider a triangle $ABC$ with $AB = 13, BC = 14, CA = 15$. A line perpendicular to $BC$ divides the interior of $\vartriangle BC$ into two regions of equal area. Suppose that the aforesaid perpendicular cuts $BC$ at $D$, and cuts $\vartriangle ABC$ again at $E$. If $L$ is the length of the line segment $DE$, find $L^2$.
2006 Denmark MO - Mohr Contest, 1
The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star.
[img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]
2010 Saudi Arabia Pre-TST, 4.2
Let $a$ be a real number.
1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$.
2) Prove that the area of this triangle does not depend on $a$.
2022 Chile Junior Math Olympiad, 2
In a trapezoid $ABCD$ whose parallel sides $AB$ and $CD$ are in ratio $\frac{AB}{CD}=\frac32$, the points $ N$ and $M$ are marked on the sides $BC$ and $AB$ respectively, in such a way that $BN = 3NC$ and $AM = 2MB$ and segments $AN$ and $DM$ are drawn that intersect at point $P$, find the ratio between the areas of triangle $APM$ and trapezoid $ABCD$.
[img]https://cdn.artofproblemsolving.com/attachments/7/8/21d59ca995d638dfcb76f9508e439fd93a5468.png[/img]
2003 Estonia National Olympiad, 1
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]
2022 Durer Math Competition Finals, 11
In rectangle $ABCD$, diagonal $AC$ is met by the angle bisector from $B$ at $B'$ and the angle bisector from $D$ at $D'$. Diagonal $BD$ is met by the angle bisector from $A$ at $A'$ and the angle bisector from $C$ at $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ the area of rectangle $ABCD$. What is the ratio of the longer side and shorter side of rectangle $ABCD$?
2004 Thailand Mathematical Olympiad, 4
Let $ABCD$ be a convex quadrilateral. Prove that area $(ABCD) \le \frac{AB^2 + BC^2 + CD^2 + DA^2}{4}$
2022 Yasinsky Geometry Olympiad, 1
From the triangle $ABC$, are gicen only the incenter $I$, the touchpoint $K$ of the inscribed circle with the side $AB$, as well as the center $I_a$ of the exscribed circle, that touches the side $BC$ . Construct a triangle equal in size to triangle $ABC$.
(Gryhoriy Filippovskyi)
1981 IMO Shortlist, 18
Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.
2006 Sharygin Geometry Olympiad, 21
On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken.
Prove that for the areas of the corresponding triangles, the inequality holds:
$$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$
and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.
1976 Chisinau City MO, 133
A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.
2019 Durer Math Competition Finals, 7
We choose a point on each side of a parallelogram $ABCD$, let these four points be $P, Q, R$ and $S$. Then we divide the parallelogram into several regions using line segments as shown in the figure. The areas of the grey regions are given, except for one (see the figure). Find the area of the region marked with a question mark.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/dbd009042dabdb2eafc8fc74960e9011038dae.png[/img]
2009 IMAC Arhimede, 2
In the triangle $ABC$, the circle with the center at the point $O$ touches the pages $AB, BC$ and $CA$ in the points $C_1, A_1$ and $B_1$, respectively. Lines $AO, BO$ and $CO$ cut the inscribed circle at points $A_2, B_2$ and $C_2,$ respectively. Prove that it is the area of the triangle $A_2B_2C_2$ is double from the surface of the hexagon $B_1A_2C_1B_2A_1C_2$.
(Moldova)
2011 Abels Math Contest (Norwegian MO), 2b
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]
2001 Paraguay Mathematical Olympiad, 4
In a parallelogram $ABCD$ of surface area $60$ cm$^2$ , a line is drawn by $D$ that intersects $BC$ at $P$ and the extension of $AB$ at $Q$. If the area of the quadrilateral $ABPD$ is $46$ cm$^2$ , find the area of triangle $CPQ$.
2001 Regional Competition For Advanced Students, 3
In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$. What is the area of the triangle $BCE$ ?