Olympiad problems

Durer Math Competition CD 1st Round - geometry, 2013.D3

The area of the triangle $ABC$ shown in the figure is $1$ unit. Points $D$ and $E$ lie on sides $AC$ and $BC$ respectively, and also are its ''one third'' points closer to $C$. Let $F$ be that $AE$ and $G$ are the midpoints of segment $BD$. What is the area of the marked quadrilateral $ABGF$? [img]https://cdn.artofproblemsolving.com/attachments/4/e/305673f429c86bbc58a8d40272dd6c9a8f0ab2.png[/img]

Estonia Open Senior - geometry, 1998.2.1

Tags: circles , area , geometry

Circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively lie on a plane such that that the circle $C_2$ passes through $O_1$. The ratio of radius of circle $C_1$ to $O_1O_2$ is $\sqrt{2+\sqrt3}$. a) Prove that the circles $C_1$ and $C_2$ intersect at two distinct points. b) Let $A,B$ be these points of intersection. What proportion of the area of circle is $C_1$ is the area of the sector $AO_1B$ ?

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 $XYZ$.

2016 BMT Spring, 17

Tags: geometry , analytic geometry , area

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 $ADBC$?

2004 Bosnia and Herzegovina Team Selection Test, 2

Tags: number theory , geometry , area

Determine whether does exists a triangle with area $2004$ with his sides positive integers.

2024 Mozambique National Olympiad, P6

Tags: right triangle , parallelogram , area , geometry

Let $ABC$ be an isosceles right triangle with $\angle BCA=90^{\circ}, BC=AC=10$. Let $P$ be a point on $AB$ that is a distance $x$ from $A$, $Q$ be a point on $AC$ such that $PQ$ is parallel to $BC$. Let $R$ and $S$ be points on $BC$ such that $QR$ is parallel to $AB$ and $PS$ is parallel to $AC$. The union of the quadrilaterals $PBRQ$ and $PSCQ$ determine a shaded area $f(x)$. Evaluate $f(2)$

2016 Portugal MO, 3

Tags: geometry , equilateral , area , equal area

Let $[ABC]$ be an equilateral triangle on the side $1$. Determine the length of the smallest segment $[DE]$, where $D$ and $E$ are on the sides of the triangle, which divides $[ABC]$ into two figures with equal area.

1955 Moscow Mathematical Olympiad, 288

Tags: bisects segment , geometry , area , right triangle , angle

We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.

2019 Portugal MO, 1

Tags: geometry , square , area

In a square of side $10$ cm , the vertices are joined to the midpoints on the opposite sides, as shown in the figure. How much does the area of the colored region measure? [img]https://1.bp.blogspot.com/-bHrc1Nu0PQI/X4KaJysLAcI/AAAAAAAAMk0/LLGv1fotQO0Tk1AXqQymG_nNdpyWcbjyACLcBGAsYHQ/s109/2019%2BPortugal%2Bp1.png[/img]

1998 Austrian-Polish Competition, 8

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

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 $P$.)

2017 BMT Spring, 13

Tags: geometry , equilateral , square , area

$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles?

1998 Tuymaada Olympiad, 3

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

The segment of length $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where $r$ is the radius of the inscribed circle of the triangle.

1998 May Olympiad, 2

Tags: geometry , equilateral triangle , area

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

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?

Durer Math Competition CD 1st Round - geometry, 2012.D2

Tags: geometry , area

Durer drew a regular triangle and then poked at an interior point. He made perpendiculars from it sides and connected it to the vertices. In this way, $6$ small triangles were created, of which (moving clockwise) all the second one is painted gray, as shown in figure. Show that the sum of the gray areas is just half the area of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/e/7/a84ad28b3cd45bd0ce455cee2446222fd3eac2.png[/img]

1999 Estonia National Olympiad, 3

Tags: geometry , incircle , area

The incircle of the triangle $ABC$, with the center $I$ , touches the sides $AB, AC$ and $BC$ in the points $K, L$ and $M$ respectively. Points $P$ and $Q$ are taken on the sides $AC$ and $BC$ respectively, such that $|AP| = |CL|$ and $|BQ| = |CM|$. Prove that the difference of areas of the figures $APIQB$ and $CPIQ$ is equal to the area of the quadrangle $CLIM$

VII Soros Olympiad 2000 - 01, 8.6

Tags: geometry , area

Three cyclists started simultaneously on three parallel straight paths (at the time of the start, the athletes were on the same straight line). Cyclists travel at constant speeds. $1$ second after the start, the triangle formed by the cyclists had an area of $5$ m$^2$. What area will such a triangle have in $10$ seconds after the start?

1996 Estonia National Olympiad, 3

Tags: rotation , area , equilateral , geometry

An equilateral triangle of side$ 1$ is rotated around its center, yielding another equilareral triangle. Find the area of the intersection of these two triangles.

1993 Abels Math Contest (Norwegian MO), 1a

Tags: geometry , triangle area , area , midpoint

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 $a+c = b+d$.

1993 Chile National Olympiad, 2

Tags: geometry , rectangle , area

Given a rectangle, circumscribe a rectangle of maximum area.

2023 Novosibirsk Oral Olympiad in Geometry, 2

Tags: geometry , square , area

In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area? [img]https://cdn.artofproblemsolving.com/attachments/d/f/2be7bcda3fa04943687de9e043bd8baf40c98c.png[/img]

1998 Iran MO (3rd Round), 2

Tags: geometry , trapezoid , ratio , area , circumcircle

Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

1951 Moscow Mathematical Olympiad, 205

Tags: geometry , 3d geometry , tetrahedron , maximum , area , projection

Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.

2019 BMT Spring, 16

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

Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ and the sides of $\vartriangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.

2005 Sharygin Geometry Olympiad, 6

Tags: geometry , area of a triangle , area

Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ..., n$). What part of the area of the triangle is painted over?