Found problems: 414
2013 Hanoi Open Mathematics Competitions, 7
Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.
1964 German National Olympiad, 3
Given a (not necessarily regular) tetrahedron, all of its sides are equal in area. Prove that the following points then coincide:
a) the center of the inscribed sphere, i.e. all four side surfaces internally touching sphere,
b) the center of the surrounding sphere, i.e. the sphere passing through the four vertixes.
2004 Switzerland Team Selection Test, 10
In an acute-angled triangle $ABC$ the altitudes $AU,BV,CW$ intersect at $H$.
Points $X,Y,Z$, different from $H$, are taken on segments $AU,BV$, and $CW$, respectively.
(a) Prove that if $X,Y,Z$ and $H$ lie on a circle, then the sum of the areas of triangles $ABZ, AYC, XBC$ equals the area of $ABC$.
(b) Prove the converse of (a).
2020 Estonia Team Selection Test, 2
The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$.
Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.
1996 Denmark MO - Mohr Contest, 1
In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.
2019 BAMO, B
In the figure below, parallelograms $ABCD$ and $BFEC$ have areas $1234$ cm$^2$ and $2804$ cm$^2$, respectively.
Points $M$ and $N$ are chosen on sides $AD$ and $FE$, respectively, so that segment $MN$ passes through $B$.
Find the area of $\vartriangle MNC$.
[img]https://cdn.artofproblemsolving.com/attachments/b/6/8b57b632191bdb3a27ab7c59e2376dab23950b.png[/img]
2001 Cuba MO, 9
In triangle $ABC$, right at $C$, let $F$ be the intersection point of the altitude $CD$ with the angle bisector $AE$ and $G$ be the intersection point of $ED$ with $BF$. Prove that the area of the quadrilateral $CEGF$ is equal to the area of the triangle $BDG$ .
2013 Denmark MO - Mohr Contest, 2
The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle.
[img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]
Cono Sur Shortlist - geometry, 1993.9
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.
1996 All-Russian Olympiad Regional Round, 10.6
Given triangle $A_0B_0C_0$. On the segment $A_0B_0$ points $A_1$, $A_2$, $...$, $A_n$, and on the segment $B_0C_0$ - points $C_1$, $C_2$, $...$, $Cn$ so that all segments $A_iC_{i+1}$ ($i = 0$, $1$, $...$,$n-1$) are parallel to each other and all segments $ C_iA_{i+1}$ ($i = 0$, $1$, $...$,$n-1$) are too. Segments $C_0A_1$, $A_1C_2$, $A_2C_1$ and $C_1A_0$ bound a certain parallelogram, segments $C_1A_2$, $A_2C_3$, $A_3C_2$ and $C_2A_1$ too, etc. Prove that the sum of the areas of all $n -1$ resulting parallelograms less than half the area of triangle $A_0B_0C_0$.
2008 Regional Olympiad of Mexico Northeast, 1
Let $ABCD$ be a parallelogram, $E$ a point on the line $AB$, beyond $B, F$ a point on the line $AD$, beyond $D$, and $K$ the point of intersection of the lines $ED$ and $BF$. Prove that quadrilaterals $ABKD$ and $CEKF$ have the same area.
1999 Junior Balkan Team Selection Tests - Moldova, 4
Let $ABC$ be an equilateral triangle of area $1998$ cm$^2$. Points $K, L, M$ divide the segments $[AB], [BC] ,[CA]$, respectively, in the ratio $3:4$ . Line $AL$ intersects the lines $CK$ and $BM$ respectively at the points $P$ and $Q$, and the line $BM$ intersects the line $CK$ at point $R$. Find the area of the triangle $PQR$.
VI Soros Olympiad 1999 - 2000 (Russia), 10.5
It is known that there is a straight line dividing the perimeter and area of a certain polygon circumscribed around a circle in the same ratio. Prove that this line passes through the center of the indicated circle.
2010 Balkan MO Shortlist, G3
The incircle of a triangle $A_0B_0C_0$ touches the sides $B_0C_0,C_0A_0,A_0B_0$ at the points $A,B,C$ respectively, and the incircle of the triangle $ABC$ with incenter $ I$ touches the sides $BC,CA, AB$ at the points $A_1, B_1,C_1$, respectively. Let $\sigma(ABC)$ and $\sigma(A_1B_1C)$ be the areas of the triangles $ABC$ and $A_1B_1C$ respectively. Show that if $\sigma(ABC) = 2 \sigma(A_1B_1C)$ , then the lines $AA_0, BB_0,IC_1$ pass through a common point .
2019 BMT Spring, 5
Point $P$ is $\sqrt3$ units away from plane $A$. Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$?
Estonia Open Junior - geometry, 2003.2.4
Consider the points $A_1$ and $A_2$ on the side $AB$ of the square $ABCD$ taken in such a way that $|AB| = 3 |AA_1| $ and $|AB| = 4 |A_2B|$, similarly consider points $B_1$ and $B_2, C_1$ and $C_2, D_1$ and $D_2$ respectively on the sides $BC$, $CD$ and $DA$. The intersection point of straight lines $D_2A_1$ and $A_2B_1$ is $E$, the intersection point of straight lines $A_2B_1$ and $B_2C_1$ is $F$, the intersection point of straight lines $B_2C_1$ and $C_2D_1$ is $G$ and the intersection point of straight lines $C_2D_1$ and $D_2A_1$ is $H$. Find the area of the square $EFGH$, knowing that the area of $ABCD$ is $1$.
Durer Math Competition CD 1st Round - geometry, 2013.C1
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]
2023 German National Olympiad, 2
In a triangle, the edges are extended past both vertices by the length of the edge opposite to the respective vertex.
Show that the area of the resulting hexagon is at least $13$ times the area of the original triangle.
2010 Balkan MO Shortlist, G2
Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle
1974 Poland - Second Round, 4
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} $.
2014 BMT Spring, 8
Line segment $AB$ has length $4$ and midpoint $M$. Let circle $C_1$ have diameter $AB$, and let circle $C_2$ have diameter $AM$. Suppose a tangent of circle $C_2$ goes through point $ B$ to intersect circle $C_1$ at $N$. Determine the area of triangle $AMN$.
2025 Kosovo National Mathematical Olympiad`, P3
On the side $AB$ of the parallelogram $ABCD$ we take the points $X$ and $Y$ such that the points $A$, $X$, $Y$ and $B$ appear in this order. The lines $DX$ and $CY$ intersect at the point $Z$. Suppose that the area of the triangle $\triangle XYZ$ is equal to the sum of the areas of the triangles $\triangle AXD$ and $\triangle CYB$. Prove that the area of the quadrilateral $XYCD$ is equal to $3$ times the area of the triangle $\triangle XYZ$.
V Soros Olympiad 1998 - 99 (Russia), 9.5
In triangle $ABC$, $\angle BAC= 60^o$. Point $P$ is taken inside the triangle so that angles $\angle APB=\angle BPC= \angle CP A=120^o$. It is known that $AP = a$. Find the area of triangle $BPC$.
2022 Bulgarian Spring Math Competition, Problem 12.1
$ABCD$ is circumscribed in a circle $k$, such that $[ACB]=s$, $[ACD]=t$, $s<t$. Determine the smallest value of $\frac{4s^2+t^2}{5st}$ and when this minimum is achieved.
2015 BMT Spring, 17
A circle intersects square $ABCD$ at points $A, E$, and $F$, where $E$ lies on $AB$ and $F$ lies on $AD$, such that $AE + AF = 2(BE + DF)$. If the square and the circle each have area $ 1$, determine the area of the union of the circle and square.