Found problems: 414
2017 BMT Spring, 2
Barack is an equilateral triangle and Michelle is a square. If Barack and Michelle each have perimeter $ 12$, find the area of the polygon with larger area.
2019 BMT Spring, 7
Let $\vartriangle ABC$ be an equilateral triangle with side length $M$ such that points $E_1$ and $E_2$ lie on side $AB$, $F_1$ and $F_2$ lie on side $BC$, and $G1$ and $G2$ lie on side $AC$, such that $$m = \overline{AE_1} = \overline{BE_2} = \overline{BF_1} = \overline{CF_2} = \overline{CG_1} = \overline{AG_2}$$ and the area of polygon $E_1E_2F_1F_2G_1G_2$ equals the combined areas of $\vartriangle AE_1G_2$, $\vartriangle BF_1E_2$, and $\vartriangle CG_1F_2$. Find the ratio $\frac{m}{M}$.
[img]https://cdn.artofproblemsolving.com/attachments/a/0/88b36c6550c42d913cdddd4486a3dde251327b.png[/img]
2020 BMT Fall, 11
Equilateral triangle $ABC$ has side length $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the “lune” that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p}-\frac{q\pi}{r}$, where $p, q$, and $ r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/f349a807583a83f93ba413bebf07e013265551.png[/img]
Estonia Open Senior - geometry, 2010.2.1
The diagonals of trapezoid $ABCD$ with bases $AB$ and $CD$ meet at $P$. Prove the inequality $S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA}$, where $S_{XYZ}$ denotes the area of triangle $XYZ$.
2021 Denmark MO - Mohr Contest, 4
Given triangle $ABC$ with $|AC| > |BC|$. The point $M$ lies on the angle bisector of angle $C$, and $BM$ is perpendicular to the angle bisector. Prove that the area of triangle AMC is half of the area of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/4/2/1b541b76ec4a9c052b8866acbfea9a0ce04b56.png[/img]
Durer Math Competition CD Finals - geometry, 2018.C+1
Prove that you can select two adjacent sides of any quadrilateral and supplement them in order to create a parallelogram, the resulting parallelogram contains the original quadrilateral .
2007 Sharygin Geometry Olympiad, 5
A non-convex $n$-gon is cut into three parts by a straight line, and two parts are put together so that the resulting polygon is equal to the third part. Can $n$ be equal to:
a) five?
b) four?
2008 Mathcenter Contest, 7
$ABC$ is a triangle with an area of $1$ square meter. Given the point $D$ on $BC$, point $E$ on $CA$, point $F$ on $AB$, such that quadrilateral $AFDE$ is cyclic. Prove that the area of $DEF \le \frac{EF^2}{4 AD^2}$.
[i](holmes)[/i]
1946 Moscow Mathematical Olympiad, 122
On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn. Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB$, $\vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB$, $\vartriangle S_0CD$, $\vartriangle S0EF$. Consider separately the case $$\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}.$$
1994 Argentina National Olympiad, 4
A rectangle is divided into $9$ small rectangles if by parallel lines to its sides, as shown in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/e/d/1fd545862a3c7950249ec54a631c74e59fb9ed.png[/img]
The four numbers written indicate the areas of the four corresponding rectangles.
Prove that the total area of the rectangle is greater than or equal to $90$.
2023 Adygea Teachers' Geometry Olympiad, 1-2
Three cevians divided the triangle into six triangles, the areas of which are marked in the figure.
1) Prove that $S_1 \cdot S_2 \cdot S_3 =Q_1 \cdot Q_2 \cdot Q_3$.
2) Determine whether it is true that if $S_1 = S_2 = S_3$, then $Q_1 = Q_2 = Q_3$.
[img]https://cdn.artofproblemsolving.com/attachments/c/d/3e847223b24f783551373e612283e10e477e62.png[/img]
2003 May Olympiad, 4
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 $4$.
2014 Saudi Arabia Pre-TST, 1.2
Let $D$ be the midpoint of side $BC$ of triangle $ABC$ and $E$ the midpoint of median $AD$. Line $BE$ intersects side $CA$ at $F$. Prove that the area of quadrilateral $CDEF$ is $\frac{5}{12}$ the area of triangle $ABC$.
2006 Spain Mathematical Olympiad, 3
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?
2020 Princeton University Math Competition, 7
Let $X, Y$ , and $Z$ be concentric circles with radii $1$, $13$, and $22$, respectively. Draw points $A, B$, and $C$ on $X$, $Y$ , and $Z$, respectively, such that the area of triangle $ABC$ is as large as possible. If the area of the triangle is $\Delta$, find $\Delta^2$.
2013 Thailand Mathematical Olympiad, 12
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 $PQMN$.
1986 All Soviet Union Mathematical Olympiad, 438
A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$.
Is it possible to assert that it is more than $3.5$?
1999 Austrian-Polish Competition, 4
Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.
2022 Durer Math Competition Finals, 5
Annie drew a rectangle and partitioned it into $n$ rows and $k$ columns with horizontal and vertical lines. Annie knows the area of the resulting $n \cdot k$ little rectangles while Benny does not. Annie reveals the area of some of these small rectangles to Benny. Given $n$ and $k$ at least how many of the small rectangle’s areas did Annie have to reveal, if from the given information Benny can determine the areas of all the $n \cdot k$ little rectangles?
For example in the case $n = 3$ and $k = 4$ revealing the areas of the $10$ small rectangles if enough information to find the areas of the remaining two little rectangles.
[img]https://cdn.artofproblemsolving.com/attachments/b/1/c4b6e0ab6ba50068ced09d2a6fe51e24dd096a.png[/img]
May Olympiad L2 - geometry, 1996.1
Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .
1989 Chile National Olympiad, 3
In a right triangle with legs $a$, $b$ and hypotenuse $c$, draw semicircles with diameters on the sides of the triangle as indicated in the figure. The purple areas have values $X,Y$ . Calculate $X + Y$.
[img]https://cdn.artofproblemsolving.com/attachments/1/a/5086dc7172516b0a986ef1af192c15eba4d6fc.png[/img]
Durer Math Competition CD 1st Round - geometry, 2014.C4
$ABCDE$ is a convex pentagon with $AB = CD = EA = 1$, $\angle ABC = \angle DEA = 90^o$, and $BC + DE = 1$. What is the area of the pentagon?
1981 All Soviet Union Mathematical Olympiad, 308
Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $$y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a$$
LMT Team Rounds 2010-20, 2013 Hexagon
Let $ABC$ be a triangle and $O$ be its circumcircle. Let $A', B', C'$ be the midpoints of minor arcs $AB$, $BC$ and $CA$ respectively. Let $I$ be the center of incircle of $ABC$. If $AB = 13$, $BC = 14$ and $AC = 15$, what is the area of the hexagon $AA'BB'CC'$?
Suppose $m \angle BAC = \alpha$ , $m \angle CBA = \beta$, and $m \angle ACB = \gamma$.
[b]p10.[/b] Let the incircle of $ABC$ be tangent to $AB, BC$, and $AC$ at $J, K, L$, respectively. Compute the angles of triangles $JKL$ and $A'B'C'$ in terms of $\alpha$, $\beta$, and $\gamma$, and conclude that these two triangles are similar.
[b]p11.[/b] Show that triangle $AA'C'$ is congruent to triangle $IA'C'$. Show that $AA'BB'CC'$ has twice the area of $A'B'C'$.
[b]p12.[/b] Let $r = JL/A'C'$ and the area of triangle $JKL$ be $S$. Using the previous parts, determine the area of hexagon $AA'BB'CC'$ in terms of $ r$ and $S$.
[b]p13.[/b] Given that the circumradius of triangle $ABC$ is $65/8$ and that $S = 1344/65$, compute $ r$ and the exact value of the area of hexagon $AA'BB'CC'$.
PS. You had better use hide for answers.
2006 Tournament of Towns, 1
Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)