Olympiad problems

1957 Moscow Mathematical Olympiad, 367

Two rectangles on a plane intersect at eight points. Consider every other intersection point, they are connected with line segments, these segments form a quadrilateral. Prove that the area of this quadrilateral does not vary under translations of one of the rectangles.

1989 Tournament Of Towns, (228) 2

Tags: hexagon , cyclic , geometry , area

The hexagon $ABCDEF$ is inscribed in a circle, $AB = BC = a, CD = DE = b$, and $EF = FA = c$. Prove that the area of triangle $BDF$ equals half the area of the hexagon. (I.P. Nagel, Yevpatoria).

Mathley 2014-15, 2

Tags: mixtilinear , geometric inequality , area , geometry

Let $ABC$ be a triangle with a circumcircle $(K)$. A circle touching the sides $AB,AC$ is internally tangent to $(K)$ at $K_a$; two other points $K_b,K_c$ are defined in the same manner. Prove that the area of triangle $K_aK_bK_c$ does not exceed that of triangle $ABC$. Nguyen Minh Ha, Hanoi University of Education, Xuan Thuy, Cau Giay, Hanoi.

Denmark (Mohr) - geometry, 2004.1

Tags: ratio , geometry , rectangle , area

The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$. Which fraction of the area of the big rectangle is that of the small one? [img]https://1.bp.blogspot.com/-aeqefhbBh5E/XzcBjhgg7sI/AAAAAAAAMXM/B0qSgWDBuqc3ysd-mOitP1LarOtBdJJ3gCLcBGAsYHQ/s0/2004%2BMohr%2Bp1.png[/img]

2010 Balkan MO Shortlist, G3

Tags: area of a triangle , geometry , incenter , incircle , area , concurrency

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 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 $A_n <\frac23$

1974 Czech and Slovak Olympiad III A, 5

Tags: geometry , geometric inequalities , geometrical inequalities , inequalities , cyclic , hexagon , area

Let $ABCDEF$ be a cyclic hexagon such that \[AB=BC,\quad CD=DE,\quad EF=FA.\] Show that \[[ACE]\le[BDF]\] and determine when the equality holds. ($[XYZ]$ denotes the area of the triangle $XYZ.$)

Ukrainian TYM Qualifying - geometry, I.8

Tags: ratio , area , geometry

One of the sides of the triangle is divided by the ratio $p: q$, and the other by $m: n: k$. The obtained division points of the sides are connected to the opposite vertices of the triangle by straight lines. Find the ratio of the area of this triangle to the area of the quadrilateral formed by three such lines and one of the sides of the triangle.

2004 IMO Shortlist, 6

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

Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$. [i]Alternative version.[/i] Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with [b]a)[/b] vertices on the sides of the polygon (or) [b]b)[/b] vertices among the vertices of the polygon such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon. [i]Proposed by Ben Green and Edward Crane, United Kingdom[/i]

1993 Tournament Of Towns, (384) 2

Tags: geometry , area

The square $ PQRS$ is placed inside the square $ABCD$ in such a way that the segments $AP$, $BQ$, $CR$ and $DS$ intersect neither each other nor the square $PQRS$. Prove that the sum of areas of quadrilaterals $ABQP$ and $CDSR$ is equal to the sum of the areas of quadrilaterals $BCRQ$ and $DAPS$. (Folklore)

1993 Abels Math Contest (Norwegian MO), 1a

Tags: midpoint , geometry , triangle area , area

Let $ABCD$ be a convex quadrilateral and $A',B'C',D'$ be the midpoints of $AB,BC,CD,DA$, respectively. Let $a,b,c,d$ denote the areas of quadrilaterals into which lines $A'C'$ and $B'D'$ divide the quadrilateral $ABCD$ (where a corresponds to vertex $A$ etc.). Prove that $a+c = b+d$.

2013 Hanoi Open Mathematics Competitions, 6

Tags: geometry , geometric inequality , area of a triangle , area

Let $ABC$ be a triangle with area $1$ (cm$^2$). Points $D,E$ and $F$ lie on the sides $AB, BC$ and CA, respectively. Prove that $min\{$area of $\vartriangle ADF,$ area of $\vartriangle BED,$ area of $\vartriangle CEF\} \le \frac14$ (cm$^2$).

2013 Thailand Mathematical Olympiad, 12

Tags: geometry , incircle , area

Let $\omega$ be the incircle of $\vartriangle ABC$, $\omega$ is tangent to sides $BC$ and $AC$ at $D$ and $E$ respectively. The line perpendicular to $BC$ at $D$ intersects $\omega$ again at $P$. Lines $AP$ and $BC$ intersect at $M$. Let $N$ be a point on segment $AC$ so that $AE = CN$. Line $BN$ intersects $\omega$ at $Q$ (closer to $B$) and intersect $AM$ at $R$. Show that the area of $\vartriangle ABR$ is equal to the area of $PQMN$.

1970 IMO, 3

Tags: limit , inequality , calculus , integration , area

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

2012 Dutch BxMO/EGMO TST, 4

Tags: geometry , equal segments , area

Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$. Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.

2003 Belarusian National Olympiad, 8

Tags: area , convex , pentagon , geometry theorems

Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$. (I.Voronovich)

1999 Greece JBMO TST, 5

Tags: geometry , locus , locus problems , area of triangle , area , symmetry

$\Phi$ is the union of all triangles that are symmetric of the triangle $ABC$ wrt a point $O$, as point $O$ moves along the triangle's sides. If the area of the triangle is $E$, find the area of $\Phi$.

Kyiv City MO 1984-93 - geometry, 1986.10.5

Tags: geometry , max , geometric inequality , area

Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.

1951 Moscow Mathematical Olympiad, 194

Tags: geometry , maximum , convex polygon , area

One side of a convex polygon is equal to $a$, the sum of exterior angles at the vertices not adjacent to this side are equal to $120^o$. Among such polygons, find the polygon of the largest area.

Swiss NMO - geometry, 2008.1

Tags: equal area , geometry , circumcircle , area

Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.

1965 Polish MO Finals, 5

Tags: inscribed , geometry , area

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 $.

1966 IMO Shortlist, 52

Tags: geometry , combinatorics , area , combinatorial geometry , dissection

A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$

1970 IMO Longlists, 52

Tags: limit , inequality , calculus , integration , area

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

2023 New Zealand MO, 2

Tags: parallelogram , area of a triangle , geometry , area

Let $ABCD$ be a parallelogram, and let $P$ be a point on the side $AB$. Let the line through $P$ parallel to $BC$ intersect the diagonal $AC$ at point $Q$. Prove that $$|DAQ|^2 = |PAQ| \times |BCD| ,$$ where $|XY Z|$ denotes the area of triangle $XY Z$.

1994 ITAMO, 5

Tags: area , minimum , maximum , cube , plane , geometry

Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.