Olympiad problems

Denmark (Mohr) - geometry, 1993.4

In triangle $ABC$, points $D, E$, and $F$ intersect one-third of the respective sides. Show that the sum of the areas of the three gray triangles is equal to the area of middle triangle. [img]https://1.bp.blogspot.com/-KWrhwMHXfDk/XzcIkhWnK5I/AAAAAAAAMYk/Tj6-PnvTy9ksHgke8cDlAjsj2u421Xa9QCLcBGAsYHQ/s0/1993%2BMohr%2Bp4.png[/img]

May Olympiad L2 - geometry, 1998.2

Tags: geometry , areas , Equilateral Triangle

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 $\frac{ (MNP)}{(ABC)}$

1980 Polish MO Finals, 1

Tags: geometry , octagon , area , areas

Compute the area of an octagon inscribed in a circle, whose four sides have length $1$ and the other four sides have length $2$.

2015 Iran Geometry Olympiad, 4

Tags: geometry , areas , parallelogram , rectangle

In rectangle $ABCD$, the points $M,N,P, Q$ lie on $AB$, $BC$, $CD$, $DA$ respectively such that the area of triangles $AQM$, $BMN$, $CNP$, $DPQ$ are equal. Prove that the quadrilateral $MNPQ$ is parallelogram. by Mahdi Etesami Fard

2016 BMT Spring, 17

Tags: geometry , analytic geometry , areas

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 $ADBC$?

2017 Denmark MO - Mohr Contest, 3

Tags: geometry , areas , arc

The figure shows an arc $\ell$ on the unit circle and two regions $A$ and $B$. Prove that the area of $A$ plus the area of $B$ equals the length of $\ell$. [img]https://1.bp.blogspot.com/-SYoSrFowZ30/XzRz0ygiOVI/AAAAAAAAMUs/0FCduUoxKGwq0gSR-b3dtb3SvDjZ89x_ACLcBGAsYHQ/s0/2017%2BMohr%2Bp3.png[/img]

V Soros Olympiad 1998 - 99 (Russia), 9.9

Tags: geometry , geometric inequality , areas

What is the largest area of a right triangle, the vertices of which are located at distances $a$, $b$ and $c$ from a certain point (where $a$ is the distance to the vertex of the right angle)?

2012 Dutch BxMO/EGMO TST, 4

Tags: geometry , areas , equal segments

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.

2008 Switzerland - Final Round, 1

Tags: geometry , circumcircle , areas , equal areas

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 $ABC$.

2007 Swedish Mathematical Competition, 6

Tags: geometry , areas , equal areas

In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the vertices of the triangle.

Estonia Open Senior - geometry, 1998.2.5

Tags: diameter , areas , chord , geometry , Circumference

The plane has a semicircle with center $O$ and diameter $AB$. Chord $CD$ is parallel to the diameter $AB$ and $\angle AOC = \angle DOB = \frac{7}{16}$ (radians). Which of the two parts it divides into a semicircle is larger area?

2009 Swedish Mathematical Competition, 1

Tags: geometry , Squares , areas

Five square carpets have been bought for a square hall with a side of $6$ m , two with the side $2$ m, one with the side $2.1$ m and two with the side $2.5$ m. Is it possible to place the five carpets so that they do not overlap in any way each other? The edges of the carpets do not have to be parallel to the cradles in the hall.

1998 Estonia National Olympiad, 3

Tags: ratio , perpendicular , triangle area , areas , geometry , irrational

In a triangle $ABC$, the bisector of the largest angle $\angle A$ meets $BC$ at point $D$. Let $E$ and $F$ be the feet of perpendiculars from $D$ to $AC$ and $AB$, respectively. Let $R$ denote the ratio between the areas of triangles $DEB$ and $DFC$. (a) Prove that, for every real number $r > 0$, one can construct a triangle ABC for which $R$ is equal to $r$. (b) Prove that if $R$ is irrational, then at least one side length of $\vartriangle ABC$ is irrational. (c) Give an example of a triangle $ABC$ with exactly two sides of irrational length, but with rational $R$.

1979 All Soviet Union Mathematical Olympiad, 270

Tags: coordinates , areas , combinatorial geometry

A grasshopper is hopping in the angle $x\ge 0, y\ge 0$ of the coordinate plane (that means that it cannot land in the point with negative coordinate). If it is in the point $(x,y)$, it can either jump to the point $(x+1,y-1)$, or to the point $(x-5,y+7)$. Draw a set of such an initial points $(x,y)$, that having started from there, a grasshopper cannot reach any point farther than $1000$ from the point $(0,0)$. Find its area.

Estonia Open Senior - geometry, 2011.2.3

Tags: ratio , areas , geometry , rational

Let $ABC$ be a triangle with integral side lengths. The angle bisector drawn from $B$ and the altitude drawn from $C$ meet at point $P$ inside the triangle. Prove that the ratio of areas of triangles $APB$ and $APC$ is a rational number.

1996 Greece Junior Math Olympiad, 2

Tags: geometry , areas , midpoints

In a triangle $ABC$ let $D,E,Z,H,G$ be the midpoints of $BC,AD,BD,ED,EZ$ respectively. Let $I$ be the intersection of $BE,AC$ and let $K$ be the intersection of $HG,AC$. Prove that: a) $AK=3CK$ b) $HK=3HG$ c) $BE=3EI$ d) $(EGH)=\frac{1}{32}(ABC)$ Notation $(...)$ stands for area of $....$

1995 Mexico National Olympiad, 5

Tags: geometry , pentagon , areas

$ABCDE$ is a convex pentagon such that the triangles $ABC, BCD, CDE, DEA$ and $EAB$ have equal areas. Show that $(1/4)$ area $(ABCDE) <$ area $(ABC) < (1/3)$ area $(ABCDE)$.

1998 Austrian-Polish Competition, 8

Tags: polygon , areas , combinatorial geometry , combinatorics , grid

In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called [i]admissible[/i], where $n > 2$ is a given natural number. The [i]value [/i] of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$.)

Novosibirsk Oral Geo Oly IX, 2023.1

Tags: geometry , areas

In the triangle $ABC$ on the sides $AB$ and $AC$, points $D$ and E are chosen, respectively. Can the segments $CD$ and $BE$ divide $ABC$ into four parts of the same area? [img]https://cdn.artofproblemsolving.com/attachments/1/c/3bbadab162b22530f1b254e744ecd068dea65e.png[/img]

Denmark (Mohr) - geometry, 2013.2

Tags: geometry , circumcircle , semicircle , areas , rectangle

The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle. [img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]

2020 BMT Fall, 8

Tags: geometry , areas , square

Let $ABCD$ be a unit square and let $E$ and $F$ be points inside $ABCD$ such that the line containing $\overline{EF}$ is parallel to $\overline{AB}$. Point $E$ is closer to $\overline{AD}$ than point $F$ is to $\overline{AD}$. The line containing $\overline{EF}$ also bisects the square into two rectangles of equal area. Suppose $[AEF B] = [DEFC] = 2[AED] = 2[BFC]$. The length of segment $\overline{EF}$ can be expressed as $m/n$ , where m and $n$ are relatively prime positive integers. Compute $m + n$.

1981 All Soviet Union Mathematical Olympiad, 304

Tags: areas , geometry , Chessboard

Two equal chess-boards ($8\times 8$) have the same centre, but one is rotated by $45$ degrees with respect to another. Find the total area of black fields intersection, if the fields have unit length sides.

1993 Denmark MO - Mohr Contest, 4

Tags: areas , geometry , trisector

In triangle $ABC$, points $D, E$, and $F$ intersect one-third of the respective sides. Show that the sum of the areas of the three gray triangles is equal to the area of middle triangle. [img]https://1.bp.blogspot.com/-KWrhwMHXfDk/XzcIkhWnK5I/AAAAAAAAMYk/Tj6-PnvTy9ksHgke8cDlAjsj2u421Xa9QCLcBGAsYHQ/s0/1993%2BMohr%2Bp4.png[/img]

1985 All Soviet Union Mathematical Olympiad, 395

Tags: geometry , hexagon , areas

Two perpendiculars are drawn from the midpoints of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area.

2010 Saudi Arabia BMO TST, 2

Tags: geometry , areas , geometric inequality , inscribed , square

Let $ABC$ be an acute triangle and let $MNPQ$ be a square inscribed in the triangle such that $M ,N \in BC$, $P \in AC$, $Q \in AB$. Prove that $area \, [MNPQ] \le \frac12 area\, [ABC]$.