Olympiad problems

2015 JBMO Shortlist, C4

Let $n\ge 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of side length $1$. Find the number of parallelograms which have vertices among the vertices of the $n^2$ squares of side length $1$, with both sides smaller or equal to $2$, and which have tha area equal to $2$. (Greece)

Durer Math Competition CD 1st Round - geometry, 2011.C4

Tags: geometry , area

Given a grid rectangle of size $2010 \times 1340$. A grid point is called [i]fair [/i] if the $2$ axis-parallel lines passing through it from the upper left and lower right corners of the large rectangle cut out a rectangle of equal area (such a point is shown in the figure). How many fair grid points lie inside the rectangle? [img]https://cdn.artofproblemsolving.com/attachments/1/b/21d4fb47c94b774994ac1c3aae7690bb98c7ae.png[/img]

Kyiv City MO Seniors 2003+ geometry, 2010.10.3

Tags: homothety , geometry , square , area

A point $O$ is chosen inside the square $ABCD$. The square $A'B'C'D'$ is the image of the square $ABCD$ under the homothety with center at point $O$ and coefficient $k> 1$ (points $A', B', C', D' $ are images of points $A, B, C, D$ respectively). Prove that the sum of the areas of the quadrilaterals $A'ABB'$ and $C'CDD'$ is equal to the sum of the areas quadrilaterals $B'BCC'$ and $D'DAA'$.

2015 Oral Moscow Geometry Olympiad, 3

Tags: trapezoid , area , geometry

$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.

1905 Eotvos Mathematical Competition, 2

Tags: geometry , area

Divide the unit square into $9$ equal squares by means of two pairs of lines parallel to the sides (see figure). Now remove the central square. Treat the remaining $8$ squares the same way, and repeat the process $n$ times. (a) How many squares of side length $1/3^n$ remain? (b) What is the sum of the areas of the removed squares as $n$ becomes infinite? [center][img]https://cdn.artofproblemsolving.com/attachments/7/d/3e6e68559919583c24d4457f946bc4cef3922f.png[/img][/center]

2010 QEDMO 7th, 4

Tags: geometry , square , area

Let $ABCD$ and $A'B'C'D'$ be two squares, both are oriented clockwise. In addition, it is assumed that all points are arranged as shown in the figure.Then it has to be shown that the sum of the areas of the quadrilaterals $ABB'A'$ and $CDD'C'$ equal to the sum of the areas of the quadrilaterals $BCC'B'$ and $DAA'D'$. [img]https://cdn.artofproblemsolving.com/attachments/0/2/6f7f793ded22fe05a7b0a912ef6c4e132f963e.png[/img]

2019 Costa Rica - Final Round, LR3

Tags: geometry , square , area

Consider the following sequence of squares (side $1$), in each step the central square is divided into equal parts and colored as shown in the figure: [img]https://cdn.artofproblemsolving.com/attachments/9/0/6874ab5aecadf2112fbe4a196ab3091ab8b31a.png[/img] Square 1 Square 2 Square 3 Let $A_n$ with $n \in N$, $n> 1$ be the shaded area of square $n$, show that $A_n <\frac23$

2018 Malaysia National Olympiad, A1

Tags: geometry , area

Quadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its sidelengths are integers, $AB = 6$, $BC = 7$, and $\angle B = \angle D = 90^o$. Find the area of$ ABCD$.

2005 Estonia National Olympiad, 1

Tags: right triangle , area , geometry

The height drawn on the hypotenuse of a right triangle divides the hypotenuse into two sections with a length ratio of $9: 1$ and two triangles of the starting triangle with a difference of areas of $48$ cm$^2$. Find the original triangle sidelengths.

1965 Polish MO Finals, 5

Tags: area , geometry , inscribed

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 $.

1996 Cono Sur Olympiad, 1

Tags: rectangle , square , area , geometry

In the following figure, the largest square is divided into two squares and three rectangles, as shown: The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$. If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square. [img]https://cdn.artofproblemsolving.com/attachments/f/a/0b424d9c293889b24d9f31b1531bed5081081f.png[/img]

Kyiv City MO 1984-93 - geometry, 1992.10.2

Tags: area , geometry

In the triangle $ABC$, the median $BD$ is drawn and through its midpoint and vertex $A$ the line $\ell$. Thus the triangle $ABC$ is divided into three triangles and one quadrilateral. Determine the areas of these figures if the area of triangle $ABC$ is equal to $S$.

2012 India PRMO, 20

Tags: area , funky circles , semicircle , geometry

$PS$ is a line segment of length $4$ and $O$ is the midpoint of $PS$. A semicircular arc is drawn with $PS$ as diameter. Let $X$ be the midpoint of this arc. $Q$ and $R$ are points on the arc $PXS$ such that $QR$ is parallel to $PS$ and the semicircular arc drawn with $QR$ as diameter is tangent to $PS$. What is the area of the region $QXROQ$ bounded by the two semicircular arcs?

1980 All Soviet Union Mathematical Olympiad, 289

Tags: area , geometry , circles , chord , maximum

Given a point $E$ on the diameter $AC$ of the certain circle. Draw a chord $BD$ to maximise the area of the quadrangle $ABCD$.

1985 All Soviet Union Mathematical Olympiad, 415

Tags: geometry , pentagon , area

All the points situated more close than $1$ cm to ALL the vertices of the regular pentagon with $1$ cm side, are deleted from that pentagon. Find the area of the remained figure.

2018 Polish Junior MO Second Round, 2

Tags: geometry , area , circumcircle , plane geometry , quadrilateral

Let $ABC$ be an acute traingle with $AC \neq BC$. Point $K$ is a foot of altitude through vertex $C$. Point $O$ is a circumcenter of $ABC$. Prove that areas of quadrilaterals $AKOC$ and $BKOC$ are equal.

2004 Thailand Mathematical Olympiad, 4

Tags: geometry , convex , area

Let $ABCD$ be a convex quadrilateral. Prove that area $(ABCD) \le \frac{AB^2 + BC^2 + CD^2 + DA^2}{4}$

Denmark (Mohr) - geometry, 2005.1

Tags: 3d geometry , pyramid , area , geometry

This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base. [img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]

2025 Kosovo National Mathematical Olympiad`, P3

Tags: geometry , area

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 $\triangle XYZ$.

1989 IMO Shortlist, 3

Tags: rectangle , calculus , algebra , 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?

Denmark (Mohr) - geometry, 2012.1

Tags: geometry , circles , area

Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$. [img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]

2008 Alexandru Myller, 3

Tags: geometry , area

For a convex pentagon, prove that $ \frac{\text{area} (ABC)}{\text{area} (ABCD)} +\frac{\text{area} (CDE)}{\text{area} (BCDE)} <1. $ [i]Dan Ismailescu[/i]

2008 Regional Olympiad of Mexico Northeast, 1

Tags: parallelogram , geometry , area

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.

2002 Federal Math Competition of S&M, Problem 2

Tags: geometry , inequalities , geometric inequality , area

Points $A_0,A_1,\ldots,A_{2k}$, in this order, divide a circumference into $2k+1$ equal arcs. Point $A_0$ is connected by chords to all the other points. These $2k$ chords divide the interior of the circle into $2k+1$ parts. These parts are alternately painted red and blue so that there are $k+1$ red and $k$ blue parts. Show that the blue area is larger than the red area.

2019 Ecuador NMO (OMEC), 6

Tags: geometry , rational , area

Let $n\ge 3$ be a positive integer. Danielle draws a math flower on the plane Cartesian as follows: first draw a unit circle centered on the origin, then draw a polygon of $n$ vertices with both rational coordinates on the circumference so that it has two diametrically opposite vertices, on each side draw a circumference that has the diameter of that side, and finally paints the area inside the $n$ small circles but outside the unit circle. If it is known that the painted area is rational, find all possible polygons drawn by Danielle.