Olympiad problems

2014 BMT Spring, 2

Suppose $ \vartriangle ABC$ is similar to $\vartriangle DEF$, with $ A$, $ B$, and $C$ corresponding to $D, E$, and $F$ respectively. If $\overline{AB} = \overline{EF}$, $\overline{BC} = \overline{FD}$, and $\overline{CA} = \overline{DE} = 2$, determine the area of $ \vartriangle ABC$.

1989 All Soviet Union Mathematical Olympiad, 497

Tags: equal ratios , geometry , areas , geometric inequality

$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.

1955 Poland - Second Round, 5

Tags: rectangle , geometry , Geometric Inequalities , areas

Given a triangle $ ABC $. Find the rectangle of smallest area containing the triangle.

2009 Hanoi Open Mathematics Competitions, 9

Tags: geometry , area of a triangle , areas

Let be given $ \vartriangle ABC$ with area $ (\vartriangle ABC) = 60$ cm$^2$. Let $R,S $ lie in $BC$ such that $BR = RS = SC$ and $P,Q$ be midpoints of $AB$ and $AC$, respectively. Suppose that $PS$ intersects $QR$ at $T$. Evaluate area $(\vartriangle PQT)$.

2019 PUMaC Individual Finals A, B, A3

Tags: areas , geometry , inequalities

Let $ABCDEF$ be a convex hexagon with area $S$ such that $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$ holds, and whose all angles are obtuse and opposite sides are not the same length. Prove that the following inequality holds: $$A_{ABC} + A_{BCD} + A_{CDE} + A_{DEF} + A_{EFA} + A_{FAB} < S$$ , where $A_{XYZ}$ is the area of triangle $XYZ$

1999 Cono Sur Olympiad, 2

Tags: geometry , construction , areas , right triangle , rectangle

Let $ABC$ be a triangle right in $A$. Construct a point $P$ on the hypotenuse $BC$ such that if $Q$ is the foot of the perpendicular drawn from $P$ to side $AC$, then the area of the square of side $PQ$ is equal to the area of the rectangle of sides $PB$ and $PC$. Show construction steps.

2015 Portugal MO, 4

Tags: geometry , areas , parallelogram

Let $[ABCD]$ be a parallelogram and $P$ a point between $C$ and $D$. The line parallel to $AD$ that passes through $P$ intersects the diagonal $AC$ in $Q$. Knowing that the area of $[PBQ]$ is $2$ and the area of $[ABP]$ is $6$, determine the area of $[PBC]$. [img]https://cdn.artofproblemsolving.com/attachments/0/8/664a00020065b7ad6300a062613fca4650b8d0.png[/img]

May Olympiad L2 - geometry, 2003.4

Tags: geometry , combinatorics , combinatorial geometry , areas

Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$. Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.

2016 Flanders Math Olympiad, 3

Tags: geometry , areas , Geometric Inequalities

Three line segments divide a triangle into five triangles. The area of these triangles is called $u, v, x,$ yand $z$, as in the figure. (a) Prove that $uv = yz$. (b) Prove that the area of the great triangle is at most $ \frac{xz}{y}$ [img]https://cdn.artofproblemsolving.com/attachments/9/4/2041d62d014cf742876e01dd8c604c4d38a167.png[/img]

Estonia Open Junior - geometry, 2000.1.3

Tags: ratio , areas , Squares

Consider a shape obtained from two equal squares with the same center. Prove that the ratio of the area of this shape to the perimeter does not change when the squares are rotated around their center. [img]http://4.bp.blogspot.com/-1AI4FxsNSr4/XovZWkvAwiI/AAAAAAAALvY/-kIzOgXB5rk3iIqGbpoKRCW9rwJPcZ3uQCK4BGAYYCw/s400/estonia%2B2000%2Bo.j.1.3.png[/img]

Denmark (Mohr) - geometry, 1996.1

Tags: geometry , right triangle , areas

In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.

2004 Denmark MO - Mohr Contest, 1

Tags: ratio , geometry , rectangle , areas

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]

Kyiv City MO Seniors 2003+ geometry, 2010.10.3

Tags: geometry , homothety , square , areas

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 $D'DAA'$.

May Olympiad L1 - geometry, 2011.3

Tags: geometry , rectangle , area of a triangle , areas

In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ . [img]https://1.bp.blogspot.com/-4w6e729AF9o/XNY9hqHaBaI/AAAAAAAAKL0/eCaNnWmgc7Yj9uV4z29JAvTcWCe21NIMgCK4BGAYYCw/s400/may%2B2011%2Bl1.png[/img]

1973 All Soviet Union Mathematical Olympiad, 185

Tags: geometry , areas , geometric inequality

Given a triangle with $a,b,c$ sides and with the area $1$ ($a \ge b \ge c$). Prove that $b^2 \ge 2$.

Estonia Open Junior - geometry, 2018.2.5

Tags: geometry , geometric inequality , areas , Centroid

Medians $AD, BE$, and $CF$ of triangle $ABC$ intersect at point $M$. Is it possible that the circles with radii $MD, ME$, and $MF$ a) all have areas smaller than the area of triangle $ABC$, b) all have areas greater than the area of triangle $ABC$, c) all have areas equal to the area of triangle $ABC$?

1999 Mexico National Olympiad, 3

Tags: geometry , areas , concurrency

A point $P$ is given inside a triangle $ABC$. Let $D,E,F$ be the midpoints of $AP,BP,CP$, and let $L,M,N$ be the intersection points of $ BF$ and $CE, AF$ and $CD, AE$ and $BD$, respectively. (a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$. (b) Prove that $DL,EM$, and $FN$ are concurrent.

2021 Bulgaria EGMO TST, 1

Tags: geometry , areas , Menelaus , lengths

On the side $AB$ of a triangle $ABC$ is chosen a point $P$. Let $Q$ be the midpoint of $BC$ and let $CP$ and $AQ$ intersect at $R$. If $AB + AP = CP$, prove that $CR = AB$.

Ukrainian TYM Qualifying - geometry, 2015.18

Tags: areas , circle , geometry , equal areas , Ukrainian TYM

Is it possible to divide a circle by three chords, different from diameters, into several equal parts?

2019 Adygea Teachers' Geometry Olympiad, 1

Tags: geometry , areas , midpoint

Inside the quadrangle, a point is taken and connected with the midpoint of all sides. Areas of the three out of four formed quadrangles are $S_1, S_2, S_3$. Find the area of the fourth quadrangle.

1998 May Olympiad, 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 $\frac{ (MNP)}{(ABC)}$

2018 Korea Winter Program Practice Test, 2

Tags: geometric inequality , areas , geometry , circumcircle , inequalities , Korea

Let $\Delta ABC$ be a triangle and $P$ be a point in its interior. Prove that \[ \frac{[BPC]}{PA^2}+\frac{[CPA]}{PB^2}+\frac{[APB]}{PC^2} \ge \frac{[ABC]}{R^2} \] where $R$ is the radius of the circumcircle of $\Delta ABC$, and $[XYZ]$ is the area of $\Delta XYZ$.

2002 All-Russian Olympiad Regional Round, 9.6

Tags: geometry , trapezoid , areas , symmetry

Let $A'$ be a point on one of the sides of the trapezoid $ABCD$ such that line $AA'$ divides the area of the trapezoid in half. Points $B'$, $C'$, $D'$ are defined similarly. Prove that the intersection points of the diagonals of quadrilaterals $ABCD$ and $A'B'C'D'$ are symmetrical wrt the midpoint of midline of trapezoid $ABCD$.

Novosibirsk Oral Geo Oly VIII, 2023.1

Tags: geometry , areas , square

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]

2017 BMT Spring, 16

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

Let $ABC$ be a triangle with $AB = 3$, $BC = 5$, $AC = 7$, and let $ P$ be a point in its interior. If $G_A$, $G_B$, $G_C$ are the centroids of $\vartriangle PBC$, $\vartriangle PAC$, $\vartriangle PAB$, respectively, find the maximum possible area of $\vartriangle G_AG_BG_C$.