Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?
In triangle $ABC$, $E$ is a point on $AC$ and $F$ is a point on $AB$. $BE$ and $CF$ intersect at $D$. If the areas of triangles $BDF$, $BCD$ and $CDE$ are 3, 7 and 7 respectively, what is the area of the quadrilateral $AEDF$?
A farmer has four straight fences, with respective lengths 1, 4, 7 and 8 metres. What is the maximum area of the quadrilateral the farmer can enclose?
A triangle's perimeter is 2016, and the ratio of its three altitudes is 3:5:7. Find the area of this triangle.
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?
What is the area of the shaded region of the given $8 \times 5$ rectangle?
Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is $\tfrac{a-\sqrt{2}}{b}$, where $a$ and $b$ are positive integers. What is $a+b$ ?
[asy] real x=.369; draw((0,0)--(0,1)--(1,1)--(1,0)--cycle); filldraw((0,0)--(0,x)--(x,x)--(x,0)--cycle, gray); filldraw((0,1)--(0,1-x)--(x,1-x)--(x,1)--cycle, gray); filldraw((1,1)--(1,1-x)--(1-x,1-x)--(1-x,1)--cycle, gray); filldraw((1,0)--(1,x)--(1-x,x)--(1-x,0)--cycle, gray); filldraw((.5,.5-x*sqrt(2)/2)--(.5+x*sqrt(2)/2,.5)--(.5,.5+x*sqrt(2)/2)--(.5-x*sqrt(2)/2,.5)--cycle, gray); [/asy]
Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$?
Given $\triangle{ABC}$, let $m_a, m_b,$ and $m_c$ be the lengths of three medians. Find its area $S_{\triangle{ABC}}$ with respect to $m_a, m_b,$ and $m_c$.
Let $ABC$ be a right triangle where $AB=4, BC=3$ and $AC=5$. Draw square $ACEF$ and $BCDG$ as shown. Find the area of $\triangle{CDE}$.
Squares $ABCD$ and $EFGH$ have a common center at $\overline{AB} || \overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest positive integer values for the area of $IJKL$.
In $\triangle{ABC}$, let $AB=c$, $AC=b$, and $\angle{BAC}=\alpha$. If $AD$ bisects $\angle{BAC}$ and intersects $BC$ at $D$, find the length of $AD$.
As shown, prove $$\frac{\sin(\alpha+\beta)}{PC}=\frac{\sin{\alpha}}{PB}+\frac{\sin{\beta}}{PA}$$
As show, three squares are arranged side-by-side such that their bases are collinear. The sides of two squares are known and marked. Find the area of shaded triangle.
Let $ABCD$ be a rectangle where $AB=4$ and $BC=6$. If $AE=CG=3$, $BF=DH=4$, and $S_{AEPH}=5$. Find the area of $PFCG$.
As shown in the diagram, both $ABCD$ and $BEFG$ are squares, where point $E$ is on $AB$. If $AD=2$, compute the area of $\triangle{AFC}$.
Let $P$ be a point inside $\triangle{ABC}$. If $AP$, $BP$, and $CP$ intersect the opposite sides at $D$, $E$, and $F$, respectively. Show that $$\frac{PD}{AD}+\frac{PE}{BE}+\frac{PF}{CF}=1$$
Let real numbers $x_1$ and $x_2$ satisfy $ \frac{\pi}{2} > x_1 > x_2 > 0$, show $$\frac{\tan x_1}{x_1} > \frac{\tan x_2}{x_2}$$
Three squares are drawn on the sides of $\triangle{ABC}$ (i.e. the square on $AB$ has $AB$ as one of its sides and lies outside $\triangle{ABC}$). Show that the lines drawn from the vertices $A, B, C$ to the centers of the opposite squares are concurrent.
Two 8-inch by 10-inch sheets of paper are placed flat on top of a 2-foot by 3-foot rectangular table. Nothing else is on the table, and the area of the table not covered by the sheets of paper is $708 in^2$. In square inches, what is the area of the overlap between the two sheets of paper?
As shown, $ABCD$ is a square with side length equaling 10 cm, $CE\perp BE$, and $CE=8$. Find the area of the shaded triangle.
As shown, a regular hexagon is inscribed in the bigger circle. If the area of the bigger circle is 2016 $cm^2$, find the total area of shaded regions.
