Found problems: 698
Ukrainian From Tasks to Tasks - geometry, 2016.8
Let $ABCD$ be a convex quadrilateral. It is known that $S_{ABD} = 7$, $S_{BCD}= 5$ and $S_{ABC}= 3$. Inside the quadrilateral mark the point $X$ so that $ABCX$ is a parallelogram. Find $S_{ADX}$ and $S_{BDX}$.
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 $.
1999 Abels Math Contest (Norwegian MO), 3
An isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$ is inscribed in a circle with center $O$. Point $D$ lies on the shorter arc $AC$ so that $\angle DOC = 30^o$, and point $G$ lies on the shorter arc $AB$ so that $DG = AC$ and $AG < BG$. The line $BG$ intersects $AC$ and $AB$ at $E$ and $F$, respectively.
(a) Prove that triangle $AFG$ is equilateral.
(b) Find the ratio between the areas of triangles $AFE$ and $ABC$.
1993 IMO Shortlist, 6
For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane,
\[ m(ABC) \leq m(ABX) + m(AXC) + m(XBC). \]
1989 IMO Longlists, 62
Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$
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 $s$.
2003 Junior Balkan Team Selection Tests - Romania, 4
Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.
2016 Sharygin Geometry Olympiad, 4
The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid.
[i]Proposed by Nikolai Beluhov, Bulgaria[/i]
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 $.
2006 IMO Shortlist, 10
Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at least twice the area of $P$.
Denmark (Mohr) - geometry, 2020.2
A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes.
The grey stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral.
[img]https://1.bp.blogspot.com/-ia13b4RsNs0/XzP0cepAcEI/AAAAAAAAMT8/0UuCogTRyj4yMJPhfSK3OQihRqfUT7uSgCLcBGAsYHQ/s0/2020%2Bmohr%2Bp2.png[/img]
2009 Hanoi Open Mathematics Competitions, 9
Let be given $ \vartriangle ABC$ with area $ (\vartriangle ABC) = 60$ cm$^2$. Let $R,S $ lie in $BC$ such that $BR = RS = SC$ and $P,Q$ be midpoints of $AB$ and $AC$, respectively. Suppose that $PS$ intersects $QR$ at $T$. Evaluate area $(\vartriangle PQT)$.
1985 Austrian-Polish Competition, 3
In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.
2016 Puerto Rico Team Selection Test, 5
$ABCD$ is a quadrilateral, $E, F, G, H$ are the midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Find the point $P$ such that area $(PHAE) =$ area $(PEBF) =$ area $(PFCG) =$ area $(PGDH).$
1998 May Olympiad, 2
Let $ABC$ be an equilateral triangle. $N$ is a point on the side $AC$ such that $\vec{AC} = 7\vec{AN}$, $M$ is a point on the side $AB$ such that $MN$ is parallel to $BC$ and $P$ is a point on the side $BC$ such that $MP$ is parallel to $AC$. Find the ratio of areas $\frac{ (MNP)}{(ABC)}$
2010 Contests, 1b
The edges of the square in the figure have length $1$.
Find the area of the marked region in terms of $a$, where $0 \le a \le 1$.
[img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]
2003 Dutch Mathematical Olympiad, 1
A Pythagorean triangle is a right triangle whose three sides are integers.
The best known example is the triangle with rectangular sides $3$ and $4$ and hypotenuse $5$.
Determine all Pythagorean triangles whose area is twice the perimeter.
2020-21 IOQM India, 4
Let $ABCD$ be a rectangle in which $AB + BC + CD = 20$ and $AE = 9$ where $E$ is the midpoint of the side $BC$. Find the area of the rectangle.
2018 Korea Winter Program Practice Test, 2
Let $\Delta ABC$ be a triangle and $P$ be a point in its interior. Prove that \[ \frac{[BPC]}{PA^2}+\frac{[CPA]}{PB^2}+\frac{[APB]}{PC^2} \ge \frac{[ABC]}{R^2} \]
where $R$ is the radius of the circumcircle of $\Delta ABC$, and $[XYZ]$ is the area of $\Delta XYZ$.
1983 Austrian-Polish Competition, 3
A bounded planar region of area $S$ is covered by a finite family $F$ of closed discs. Prove that $F$ contains a subfamily consisting of pairwise disjoint discs, of joint area not less than $S/9$.
2020 Paraguay Mathematical Olympiad, 3
In triangle $ABC$, side $AC$ is $8$ cm. Two segments are drawn parallel to $AC$ that have their ends on $AB$ and $BC$ and that divide the triangle into three parts of equal area. What is the length of the parallel segment closest to $AC$?
1962 All Russian Mathematical Olympiad, 023
What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?
2008 Postal Coaching, 5
Prove that there are in finitely many positive integers $n$ such that $\Delta = nr^2$, where $\Delta$ and $r$ are respectively the area and the inradius of a triangle with integer sides.
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 $CXY$.
1995 Chile National Olympiad, 2
In a circle of radius $1$, six arcs of radius $1$ are drawn, which cut the circle as in the figure. Determine the black area.
[img]https://cdn.artofproblemsolving.com/attachments/8/9/0323935be8406ea0c452b3c8417a8148c977e3.jpg[/img]