Found problems: 414
2015 BMT Spring, 4
Triangle $ABC$ has side lengths $AB = 3$, $BC = 4$, and $CD = 5$. Draw line $\ell_A$ such that $\ell_A$ is parallel to $BC$ and splits the triangle into two polygons of equal area. Define lines $\ell_B$ and $\ell_C$ analogously. The intersection points of $\ell_A$, $\ell_B$, and $\ell_C$ form a triangle. Determine its area.
2017 BMT Spring, 13
Two points are located $10$ units apart, and a circle is drawn with radius $ r$ centered at one of the points. A tangent line to the circle is drawn from the other point. What value of $ r$ maximizes the area of the triangle formed by the two points and the point of tangency?
1956 Moscow Mathematical Olympiad, 340
a) * In a rectangle of area $5$ sq. units, $9$ rectangles of area $1$ are arranged. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$
b) In a rectangle of area $5$ sq. units, lie $9$ arbitrary polygons each of area $1$. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$
Denmark (Mohr) - geometry, 2007.1
Triangle $ABC$ lies in a regular decagon as shown in the figure.
What is the ratio of the area of the triangle to the area of the entire decagon?
Write the answer as a fraction of integers.
[img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]
Denmark (Mohr) - geometry, 2010.1
Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
[img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]
2016 Peru Cono Sur TST, P2
Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$
Estonia Open Senior - geometry, 1995.1.3
We call a tetrahedron a "trirectangular " if it has a vertex (we call this is called a "right-angled" vertex) in which the planes of the three sides of the tetrahedron intersect at right angles.
Prove the "three-dimensional Pythagorean theorem":
The square of the area of the opposite face of the "right-angled" vertex of the ""trirectangular " tetrahedron is equal to the sum of the squares of the areas of three other sides of the tetrahedron .
2021 Malaysia IMONST 1, 16
Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?
1979 All Soviet Union Mathematical Olympiad, 277
Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.
2004 May Olympiad, 4
In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$. If area of $ABCD$ equals two times the area of $PQRS$, and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$.
2019 Yasinsky Geometry Olympiad, p5
On the sides of the right triangle, outside are constructed regular nonagons, which are constructed on one of the catheti and on the hypotenuse, with areas equal to $1602$ $cm^2$ and $2019$ $cm^2$, respectively. What is the area of the nonagon that is constructed on the other cathetus of this triangle?
(Vladislav Kirilyuk)
2021 OMpD, 1
Let $ABCDEF$ be a regular hexagon with sides $1m$ and $O$ as its center. Suppose that $OPQRST$ is a regular hexagon, so that segments $OP$ and $AB$ intersect at $X$ and segments $OT$ and $CD$ intersect at $Y$, as shown in the figure below. Determine the area of the pentagon $OXBCY$.
1965 Polish MO Finals, 5
Points $ A_1 $, $ B_1 $, $ C_1 $ divide respectively the sides $ BC $, $ CA $, $ AB $ of the triangle $ ABC $ in the ratios $ k_1 $, $ k_2 $, $ k_3 $. Calculate the ratio of the areas of triangles $ A_1B_1C_1 $ and $ ABC $.
1986 Poland - Second Round, 6
In the triangle $ ABC $, the point $ A' $ on the side $ BC $, the point $ B' $ on the side $ AC $, the point $ C' $ on the side $ AB $ are chosen so that the straight lines $ AA' $, $ CC' $ intersect at one point, i.e. equivalently $ |BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'| $. Prove that the area of triangle $ A'B'C' $ is not greater than $ 1/4 $ of the area of triangle $ ABC $.
2010 Malaysia National Olympiad, 5
A circle and a square overlap such that the overlapping area is $50\%$ of the area of the circle, and is $25\%$ of the area of the square, as shown in the figure. Find the ratio of the area of the square outside the circle to the area of the whole figure.
[img]https://cdn.artofproblemsolving.com/attachments/e/2/c209a95f457dbf3c46f66f82c0a45cc4b5c1c8.png[/img]
2005 Oral Moscow Geometry Olympiad, 1
The hexagon has five $90^o$ angles and one $270^o$ angle (see picture). Use a straight-line ruler to divide it into two equal-sized polygons.
[img]https://cdn.artofproblemsolving.com/attachments/d/8/cdd4df68644bb8e04adbe1b265039b82a5382b.png[/img]
Estonia Open Senior - geometry, 1999.2.5
Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.
2017 May Olympiad, 3
Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFE$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.
1987 All Soviet Union Mathematical Olympiad, 458
The convex $n$-gon ($n\ge 5$) is cut along all its diagonals. Prove that there are at least a pair of parts with the different areas.
1997 Mexico National Olympiad, 5
Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.
2013 LMT, Hexagon Area
Let $ABC$ be a triangle and $O$ be its circumcircle. Let $A', B', C'$ be the midpoints of minor arcs $AB$, $BC$ and $CA$ respectively. Let $I$ be the center of incircle of $ABC$. If $AB = 13$, $BC = 14$ and $AC = 15$, what is the area of the hexagon $AA'BB'CC'$?
Suppose $m \angle BAC = \alpha$ , $m \angle CBA = \beta$, and $m \angle ACB = \gamma$.
[b]p10.[/b] Let the incircle of $ABC$ be tangent to $AB, BC$, and $AC$ at $J, K, L$, respectively. Compute the angles of triangles $JKL$ and $A'B'C'$ in terms of $\alpha$, $\beta$, and $\gamma$, and conclude that these two triangles are similar.
[b]p11.[/b] Show that triangle $AA'C'$ is congruent to triangle $IA'C'$. Show that $AA'BB'CC'$ has twice the area of $A'B'C'$.
[b]p12.[/b] Let $r = JL/A'C'$ and the area of triangle $JKL$ be $S$. Using the previous parts, determine the area of hexagon $AA'BB'CC'$ in terms of $ r$ and $S$.
[b]p13.[/b] Given that the circumradius of triangle $ABC$ is $65/8$ and that $S = 1344/65$, compute $ r$ and the exact value of the area of hexagon $AA'BB'CC'$.
PS. You had better use hide for answers.
1985 Tournament Of Towns, (099) 3
A teacher gives each student in the class the following task in their exercise book .
"Take two concentric circles of radius $1$ and $10$ . To the smaller circle produce three tangents whose intersections $A, B$ and $C$ lie in the larger circle . Measure the area $S$ of triangle $ABC$, and areas $S_1 , S_2$ and $S_3$ , the three sector-like regions with vertices at $A, B$ and $C$ (as in the diagram). Find the value of $S_1 +S_2 +S_3 -S$."
Prove that each student would obtain the same result .
[img]https://1.bp.blogspot.com/-K3kHWWWgxgU/XWHRQ8WqqPI/AAAAAAAAKjE/0iO4-Yz6p9AcM2mklprX_M18xTyg9O81gCK4BGAYYCw/s200/TOT%2B1985%2BAutumn%2BJ3.png[/img]
( A . K . Tolpygo, Kiev)
2003 Paraguay Mathematical Olympiad, 4
Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/2013de890e438f5bf88af446692b495917b1ff.png[/img]
1986 Tournament Of Towns, (126) 1
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
1991 Austrian-Polish Competition, 6
Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$.