Found problems: 698
The height drawn on the hypotenuse of a right triangle divides the hypotenuse into two sections with a length ratio of $9: 1$ and two triangles of the starting triangle with a difference of areas of $48$ cm$^2$. Find the original triangle sidelengths.
A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$
Points $A$ and $B$ are given on a circle. With the help of a compass and a ruler, construct on this circle the points $C,$ $D$, $E$ that lie on one side of the straight line $AB$ and for which the pentagon with vertices $A$, $B$, $C$, $D$, $E$ has the largest possible area
The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQRS$.
[img]https://1.bp.blogspot.com/--fMGH2lX6Go/XzcDqhgGKfI/AAAAAAAAMXo/x4NATcMDJ2MeUe-O0xBGKZ_B4l_QzROjACLcBGAsYHQ/s0/2000%2BMohr%2Bp1.png[/img]
A convex quadrilateral $ABCD$ is given. Let $O_a$ be the circumcenter of the triangle $DBC$, and define $O_b,O_c$ and $O_d$ similarly. The points $O_a, O_b, O_c, O_d$ are the vertices of a convex quadrilateral. Prove that its area is equal to half of the absolute value of the difference between the areas of $AO_bCO_d$ and $BO_cDO_a$.
[i]Proposed by V. Dubrovsky[/i]
Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFE$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.
The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon.
(a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.)
(b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter.
(c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths.
It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle.
(d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon
can be increased without changing the perimeter.
(e) Prove that the polygon of maximum area must be equiangular, or have all angles equal.
(f ) Prove that the polygon of maximum area is a regular polygon.
PS. You had better use hide for answers.
Point $P$ is $\sqrt3$ units away from plane $A$. Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$?
A triangle of area $1$ is cut out of paper. Prove that it can be bent along a straight segment so that the area of the resulting figure is less than $s_0$, where $s_0=\frac{\sqrt5-1}{2}$.
Note. The value $s_0$ specified in the condition can be reduced (the smallest value of$s_0$ is unknown to the authors of the problem). If you manage to do this (and justify it), write.
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 $\frac{S}{S'} \ge 3 +2\sqrt2$
Prove that there are in finitely many positive integers $n$ such that $\Delta = nr^2$, where $\Delta$ and $r$ are respectively the area and the inradius of a triangle with integer sides.
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 $F$.
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]
One of the sides of the triangle is divided by the ratio $p: q$, and the other by $m: n: k$. The obtained division points of the sides are connected to the opposite vertices of the triangle by straight lines. Find the ratio of the area of this triangle to the area of the quadrilateral formed by three such lines and one of the sides of the triangle.
Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$.
[i]Alternative version.[/i] Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with
[b]a)[/b] vertices on the sides of the polygon (or)
[b]b)[/b] vertices among the vertices of the polygon
such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon.
[i]Proposed by Ben Green and Edward Crane, United Kingdom[/i]
The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure?
[img]https://1.bp.blogspot.com/-oWnh8uhyTIo/XzP30gZueKI/AAAAAAAAMUY/GlC3puNU_6g6YRf6hPpbQW8IE8IqMP3ugCLcBGAsYHQ/s0/2018%2BMohr%2Bp2.png[/img]
In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment
In the plane, the segments $AB$ and $CD$ are given, while the lines $AB$ and $CD$ intersect. Prove that the set of all points $P$ in the plane such that triangles $ABP$ and $CDP$ have equal areas , form two lines intersecting at the intersection of the lines $AB$ and $CD$.
$AC' BA'C B'$ is a convex hexagon such that $AB' = AC'$, $BC' = BA'$, $CA' = CB'$ and $\angle A +\angle B + \angle C = \angle A' + \angle B' + \angle C'$. Prove that the area of the triangle $ABC$ is half the area of the hexagon.
(V Proizvolov)
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]
(a) Let $ABC$ be any triangle. Let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively.
Suppose that $BX \leq XC, CY \leq YA, AZ \leq ZB$. Show that the area of the triangle $XYZ$ $\geq 1\slash 4$ times the area of $ABC.$
(b) Let $ABC$ be any triangle, and let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively. Using (a) or by any other method, show: One of the three corner triangles $AZY, BXZ, CYX$ has an area $\leq$ area of the triangle $XYZ.$
The point $M$ varies on the segment $AB$ that measures $2$ m.
a) Find the equation and the graphical representation of the locus of the points of the plane whose coordinates, $x$, and $y$, are, respectively, the areas of the squares of sides $AM$ and $MB$ .
b) Find out what kind of curve it is. (Suggestion: make a $45^o$ axis rotation).
c) Find the area of the enclosure between the curve obtained and the coordinate axes.
A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area $A$ of a quadrilateral with consecutive sides $a,b,c,d$:
$A =\frac{a+c}{2}\frac{b+d}{2}$ (1) and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$ (2) where $p = (a+b+c+d)/2$.
However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.
Through an arbitrary point inside a triangle, lines parallel to the sides of the triangle are drawn, dividing the triangle into three triangles with areas $T_1,T_2,T_3$ and three parallelograms. If $T$ is the area of the original triangle, prove that
$$T=(\sqrt{T_1}+\sqrt{T_2}+\sqrt{T_3})^2$$
.
Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.