Found problems: 698
It is known that there is a straight line dividing the perimeter and area of a certain polygon circumscribed around a circle in the same ratio. Prove that this line passes through the center of the indicated circle.
In space there is a line $ k $ and a cube with a vertex $ M $ and edges $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, of length$ 1$. Prove that the length of the orthogonal projection of edge $ MA $ on the line $ k $ is equal to the area of the orthogonal projection of a square with sides $ MB $ and $ MC $ onto a plane perpendicular to the line $ k $.
[hide=original wording]W przestrzeni dana jest prosta $ k $ oraz sześcian o wierzchołku $ M $ i krawędziach $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, długości 1. Udowodnić, że długość rzutu prostokątnego krawędzi $ MA $ na prostą $ k $ jest równa polu rzutu prostokątnego kwadratu o bokach $ MB $ i $ MC $ na płaszczyznę prostopadłą do prostej $ k $.[/hide]
Two equal chess-boards ($8\times 8$) have the same centre, but one is rotated by $45$ degrees with respect to another. Find the total area of black fields intersection, if the fields have unit length sides.
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 $AD$ .
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?
Five points are given in the plane that each of $10$ triangles they define has area greater than $2$. Prove that there exists a triangle of area greater than $3$.
Is there a planar polygon whose vertices have integer coordinates and whose area is $1/2$, such that this polygon is
(a) a triangle with at least two sides longer than $1000$?
(b) a triangle whose sides are all longer than $1000$?
(c) a quadrangle?
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
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 $XYZ$
Seven triangles of area $7$ lie in a square of area $27$. Prove that among the $7$ triangles there are $2$ that intersect in a region of area not less than $1$.
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 $p + q + r$.
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The point $M$ is the midpoint of the median $BD$ of the triangle $ABC$, the area of which is $S$. The line $AM$ intersects the side $BC$ at the point $K$. Determine the area of the triangle $BKM$.
$ABC$ is a triangle. $X$, $Y$, $Z$ lie on $BC$, $CA$, $AB$ respectively. Show that area $XYZ$ cannot be smaller than each of area $AYZ$, area $BZX$, area $CXY$.
Through a point $P$ inside the triangle $ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal areas. If the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is:
$ \textbf{(A)}\ \frac12 \qquad\textbf{(B)}\ \frac14\qquad\textbf{(C)}\ 2-\sqrt2\qquad\textbf{(D)}\ \frac{2-\sqrt2}{2}\qquad\textbf{(E)}\ \frac{2+\sqrt2}{8} $
One side of a convex polygon is equal to $a$, the sum of exterior angles at the vertices not adjacent to this side are equal to $120^o$. Among such polygons, find the polygon of the largest area.
Two circles in the plane, both of radius $R$, intersect at a right angle. Compute the area of the intersection of the interiors of the two circles.
Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$.
Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.
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 $BCEF$ .
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Let $P = (7, 1)$ and let $O = (0, 0)$.
(a) If $S$ is a point on the line $y = x$ and $T$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $ST$, determine the minimum possible area of triangle $OST$.
(b) If $U$ is a point on the line $y = x$ and $V$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $UV$, determine the minimum possible perimeter of triangle $OUV$.
The pentagon $ABCDE$ has $AB = BC, CD = DE, \angle ABC = 120^o, \angle CDE = 60^o$ and $BD = 2$. Calculate the area of the pentagon.
$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.
Three sides of a tetrahedron are right-angled triangles having the right angle at their common vertex. The areas of these sides are $A, B$, and $C$. Find the total surface area of the tetrahedron.
Prove that the largest area of a triangle with sides $a, b, c$ satisfying the relation $a^2 +b^2 c^2 = 3m^2$, equals to $\frac{\sqrt3}{4}m^2$.
In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.
Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice’s view. The total area in the room Alice can see can be expressed in the form $\frac{m\pi}{n} +p\sqrt{q}$, where $m$ and $n$ are relatively prime positive integers and $p$ and $q$ are integers such that $q$ is square-free. Compute $m + n + p + q$. (Note that the pillar is not included in the total area of the room.)
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