This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 698

May Olympiad L1 - geometry, 2023.3
Tags: area , geometry
On a straight line $\ell$ there are four points, $A$, $B$, $C$ and $D$ in that order, such that $AB=BC=CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed . Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn , which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.

2019 Adygea Teachers' Geometry Olympiad, 1
Tags: geometry , midpoint , area
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.

Estonia Open Senior - geometry, 1999.2.3
Tags: incircle , circumcircle , geometric inequality , area of a triangle , geometry , right triangle , area
Two right triangles are given, of which the incircle of the first triangle is the circumcircle of the second triangle. Let the areas of the triangles be $S$ and $S'$ respectively. Prove that $\frac{S}{S'} \ge 3 +2\sqrt2$

Novosibirsk Oral Geo Oly VIII, 2022.5
Tags: geometry , area , isosceles
Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]

1974 Czech and Slovak Olympiad III A, 6
Tags: geometric transformation , rotation , locus , square , area , geometry
Let a unit square $\mathcal D$ be given in the plane. For any point $X$ in the plane denote $\mathcal D_X$ the image of $\mathcal D$ in rotation with respect to origin $X$ by $+90^\circ.$ Find the locus of all $X$ such that the area of union $\mathcal D\cup\mathcal D_X$ is at most 1.5.

2016 Peru Cono Sur TST, P2
Tags: geometry , geometric inequality , inequalities , polygon , area
Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$

Kvant 2020, M2590
Tags: geometry , area
In an acute triangle $ABC$ the point $O{}$ is the circumcenter, $H_1$ is the foot of the perpendicular from $A{}$ onto $BC$, and $M_H$ and $N_H$ are the projections of $H_1$ on $AC$ and $AB{}$, respectively. Prove that the polyline $M_HON_H$ divides the triangle $ABC$ in two figures of equal area. [i]Proposed by I. A. Kushner[/i]

2010 Saudi Arabia Pre-TST, 4.2
Tags: triangle inequality , geometry , area
Let $a$ be a real number. 1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$. 2) Prove that the area of this triangle does not depend on $a$.

1976 Bundeswettbewerb Mathematik, 2
Tags: geometry , area , quadratilateral
Each of the two opposite sides of a convex quadrilateral is divided into seven equal parts, and corresponding division points are connected by a segment, thus dividing the quadrilateral into seven smaller quadrilaterals. Prove that the area of at least one of the small quadrilaterals equals $1\slash 7$ slash of the area of the large quadrilateral.

May Olympiad L2 - geometry, 2016.5
Tags: game , combinatorics , geometry , winning strategy , area
Rosa and Sara play with a triangle $ABC$, right at $B$. Rosa begins by marking two interior points of the hypotenuse $AC$, then Sara marks an interior point of the hypotenuse $AC$ different from those of Rosa. Then, from these three points the perpendiculars to the sides $AB$ and $BC$ are drawn, forming the following figure. [img]https://cdn.artofproblemsolving.com/attachments/9/9/c964bbacc4a5960bee170865cc43902410e504.png[/img] Sara wins if the area of the shaded surface is equal to the area of the unshaded surface, in other case wins Rosa. Determine who of the two has a winning strategy.

Estonia Open Junior - geometry, 2018.2.5
Tags: geometry , geometric inequality , centroid , area
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 $ABC$?

Estonia Open Junior - geometry, 2016.2.4
Tags: quadratic , parabola , area , algebra , analytic geometry , conic
Let $d$ be a positive number. On the parabola, whose equation has the coefficient $1$ at the quadratic term, points $A, B$ and $C$ are chosen in such a way that the difference of the $x$-coordinates of points $A$ and $B$ is $d$ and the difference of the $x$-coordinates of points $B$ and $C$ is also $d$. Find the area of the triangle $ABC$.

2006 JBMO ShortLists, 15
Tags: geometry , minimum , prime number , integer , area
Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.

2022 Yasinsky Geometry Olympiad, 1
Tags: geometry , construction , area
From the triangle $ABC$, are gicen only the incenter $I$, the touchpoint $K$ of the inscribed circle with the side $AB$, as well as the center $I_a$ of the exscribed circle, that touches the side $BC$ . Construct a triangle equal in size to triangle $ABC$. (Gryhoriy Filippovskyi)

1979 All Soviet Union Mathematical Olympiad, 269
Tags: geometry , isosceles , inscribed , area
What is the least possible ratio of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?

Denmark (Mohr) - geometry, 2006.1
Tags: area , geometry
The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star. [img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]

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]

1985 Bulgaria National Olympiad, Problem 3
Tags: geometry , 3d geometry , pyramid , area
A pyramid $MABCD$ with the top-vertex $M$ is circumscribed about a sphere with center $O$ so that $O$ lies on the altitude of the pyramid. Each of the planes $ACM,BDM,ABO$ divides the lateral surface of the pyramid into two parts of equal areas. The areas of the sections of the planes $ACM$ and $ABO$ inside the pyramid are in ratio $(\sqrt2+2):4$. Determine the angle $\delta$ between the planes $ACM$ and $ABO$, and the dihedral angle of the pyramid at the edge $AB$.

Novosibirsk Oral Geo Oly VII, 2023.2
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]

2008 Oral Moscow Geometry Olympiad, 2
Tags: common tangent , geometry , area
The radii $r$ and $R$ of two non-intersecting circles are given. The common internal tangents of these circles are perpendicular. Find the area of the triangle bounded by these tangents, as well as the common external tangents.

1986 Poland - Second Round, 6
Tags: geometry , geometric inequalities , area
In the triangle $ ABC $, the point $ A' $ on the side $ BC $, the point $ B' $ on the side $ AC $, the point $ C' $ on the side $ AB $ are chosen so that the straight lines $ AA' $, $ CC' $ intersect at one point, i.e. equivalently $ |BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'| $. Prove that the area of triangle $ A'B'C' $ is not greater than $ 1/4 $ of the area of triangle $ ABC $.

1988 Tournament Of Towns, (165) 2
Tags: equal area , geometry , midpoint , area
We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .

2023-24 IOQM India, 5
Tags: geometry , area , area of a triangle , barycentric coordinates , barycentric
In a triangle $A B C$, let $E$ be the midpoint of $A C$ and $F$ be the midpoint of $A B$. The medians $B E$ and $C F$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $B E$ and $C F$ respectively. If the area of triangle $A B C$ is 480 , find the area of triangle $G Y Z$.

1997 Denmark MO - Mohr Contest, 2
Tags: square , area , geometry
Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

1915 Eotvos Mathematical Competition, 3
Tags: geometry , parallelogram , inscribed , geometric inequalities , area
Prove that a triangle inscribed in a parallelogram has at most half the area of the parallelogram.