Olympiad problems

1985 Traian Lălescu, 1.1

Consider the function $ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max (x-3,2) . $ Find the perimeter and the area of the figure delimited by the lines $ x=-3,x=1, $ the $ Ox $ axis, and the graph of $ f. $

2022 Novosibirsk Oral Olympiad in Geometry, 5

Tags: geometry , area , isosceles

Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]

2020 Ukrainian Geometry Olympiad - December, 2

Tags: geometry , ratio , area

Let $ABCD$ be a cyclic quadrilateral such that $AC =56, BD = 65, BC>DA$ and $AB: BC =CD: DA$. Find the ratio of areas $S (ABC): S (ADC)$.

2016 BMT Spring, 5

Tags: geometry , area of a triangle , area

Let $ABC$ be a right triangle with $AB = BC = 2$. Let $ACD$ be a right triangle with angle $\angle DAC = 30$ degrees and $\angle DCA = 60$ degrees. Given that $ABC$ and $ACD$ do not overlap, what is the area of triangle $BCD$?

2015 Bangladesh Mathematical Olympiad, 5

Tags: tetrahedron , volume , area , geometry , 3d geometry , algebra

A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.

2018 Denmark MO - Mohr Contest, 2

Tags: geometry , circles , area

The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure? [img]https://1.bp.blogspot.com/-oWnh8uhyTIo/XzP30gZueKI/AAAAAAAAMUY/GlC3puNU_6g6YRf6hPpbQW8IE8IqMP3ugCLcBGAsYHQ/s0/2018%2BMohr%2Bp2.png[/img]

2007 Denmark MO - Mohr Contest, 1

Tags: decagon , geometry , area

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]

1984 All Soviet Union Mathematical Olympiad, 384

Tags: polygon , perimeter , disc , area , circles , geometry

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).

2005 Sharygin Geometry Olympiad, 2

Tags: geometry , perimeter , area , polygon

Cut a cross made up of five identical squares into three polygons, equal in area and perimeter.

1989 IMO Shortlist, 3

Tags: calculus , algebra , rectangle , equation , area

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

2005 Cuba MO, 8

Tags: geometry , integer , geometric inequality , area

Find the smallest real number $A$, such that there are two different triangles, with integer sidelengths and so that the area of each be $A$.

2000 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: geometry , geometric inequality , area

In a triangle $ABC$, points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $$\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$$ Note: $(XYZ)$ is the area of triangle $XYZ$.

1968 German National Olympiad, 2

Tags: 3d geometry , area , geometric inequality , geometry

Which of all planes, the one and the same body diagonal of a cube with the edge length $a$, cuts out a cut figure with the smallest area from the cube? Calculate the area of such a cut figure. [hide=original wording]Welche von allen Ebenen, die eine und dieselbe Korperdiagonale eines Wurfels mit der Kantenlange a enthalten, schneiden aus den W¨urfel eine Schnittfigur kleinsten Flacheninhaltes heraus? Berechnen Sie den Fl¨acheninhalt solch einer Schnittfigur![/hide]

2020 BMT Fall, 13

Tags: geometry , hexagon , area

Sheila is making a regular-hexagon-shaped sign with side length $ 1$. Let $ABCDEF$ be the regular hexagon, and let $R, S,T$ and U be the midpoints of $FA$, $BC$, $CD$ and $EF$, respectively. Sheila splits the hexagon into four regions of equal width: trapezoids $ABSR$, $RSCF$ , $FCTU$, and $UTDE$. She then paints the middle two regions gold. The fraction of the total hexagon that is gold can be written in the form $m/n$ , where m and n are relatively prime positive integers. Compute $m + n$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9lLzIwOTVmZmViZjU3OTMzZmRlMzFmMjM1ZWRmM2RkODMyMTA0ZjNlLnBuZw==&rn=MjAyMCBCTVQgSW5kaXZpZHVhbCAxMy5wbmc=[/img]

2023 AMC 12/AHSME, 11

Tags: geometry , trapezoid , area

What is the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other? $ \textbf{(A) }\frac 54 \qquad \textbf{(B) } \frac 87 \qquad \textbf{(C)} \frac{5\sqrt2}4 \qquad \textbf{(D) } \frac 32 \qquad \textbf{(E) } \frac{3\sqrt3}4 $

2009 Hanoi Open Mathematics Competitions, 9

Tags: geometry , area of a triangle , area

Give an acute-angled triangle $ABC$ with area $S$, let points $A',B',C'$ be located as follows: $A'$ is the point where altitude from $A$ on $BC$ meets the outwards facing semicirle drawn on $BX$ as diameter.Points $B',C'$ are located similarly. Evaluate the sum $T=($area $\vartriangle BCA')^2+($area $\vartriangle CAB')^2+($area $\vartriangle ABC')^2$.

2013 IFYM, Sozopol, 8

Tags: geometry , parallelogram , geometric inequality , area , polygon

Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\] where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively. [i]Proposed by Witold Szczechla, Poland[/i]

2020 AMC 10, 20

Tags: quadrilateral , geometry , area

Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$? $\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370$

Denmark (Mohr) - geometry, 2000.1

Tags: square , area , midpoint

The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQRS$. [img]https://1.bp.blogspot.com/--fMGH2lX6Go/XzcDqhgGKfI/AAAAAAAAMXo/x4NATcMDJ2MeUe-O0xBGKZ_B4l_QzROjACLcBGAsYHQ/s0/2000%2BMohr%2Bp1.png[/img]

2016 BMT Spring, 17

Tags: analytic geometry , area , geometry

Consider triangle $ABC$ in $xy$-plane where $ A$ is at the origin, $ B$ lies on the positive $x$-axis, $C$ is on the upper right quadrant, and $\angle A = 30^o$, $\angle B = 60^o$ ,$\angle C = 90^o$. Let the length $BC = 1$. Draw the angle bisector of angle $\angle C$, and let this intersect the $y$-axis at $D$. What is the area of quadrilateral $ADBC$?

1983 All Soviet Union Mathematical Olympiad, 365

Tags: geometry , rectangle , geometric inequality , inequalities , area

One side of the rectangle is $1$ cm. It is known that the rectangle can be divided by two orthogonal lines onto four rectangles, and each of the smaller rectangles has the area not less than $1$ square centimetre, and one of them is not less than $2$ square centimetres. What is the least possible length of another side of big rectangle?