Found problems: 970
There is a rectangular field that measures $20\text{m}$ by $15\text{m}$. Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\text{m}$ sides such that he is $6\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly?
Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$?
Let $T = TNYWR$, and let $T = 10X + Y$ for an integer $X$ and a digit $Y$. Suppose that $a$ and $b$ are real numbers satisfying $a+\frac1b=Y$ and $\frac{b}a=X$. Compute $(ab)^4+\frac1{(ab)^4}$.
A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length
$ \textbf{(A)}\ 3\sqrt3 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 6\sqrt3 \qquad \textbf{(D)}\ 12\sqrt3 \qquad
\textbf{(E)}\ \text{ none of these}$
Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Lewis Chen
[/i]
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles.
A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid.
[asy]
size(150);
defaultpen(linewidth(0.8));
draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1));
draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4"));
draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4"));
label("$5$",(0,5/2),W);
label("$8$",(4,0),S);
[/asy]
The incircle of a right-angled triangle $ABC$ ($\angle C = 90^\circ$) touches $BC$ at point $K$. Prove that the chord of the incircle cut by line $AK$ is twice as large as the distance from $C$ to that line.
The sides of a triangle are of lengths $13$, $14$, and $15$. The altitudes of the triangle meet at point $H$. If $AD$ is the altitude to the side length $14$, what is the ratio $HD:HA$?
$\textbf{(A) } 3 : 11\qquad
\textbf{(B) } 5 : 11\qquad
\textbf{(C) } 1 : 2\qquad
\textbf{(D) }2 : 3\qquad
\textbf{(E) }25 : 33$
Square $ABCD$ has side length $5$ and arc $BD$ with center $A$. $E$ is the midpoint of $AB$ and $CE$ intersects arc $BD$ at $F$. $G$ is placed onto $BC$ such that $FG$ is perpendicular to $BC$. What is the length of $FG$?
In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.
[i]Proposed by Davood Vakili, Iran[/i]
Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$.
Rectangle $ABCD$ has $AB = 24$ and $BC = 7$. Let $d$ be the distance between the centers of the incircles of $\vartriangle ABC$ and $\vartriangle CDA$. Find $d^2$.
Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Let $E$ be the foot from $B$ to $AC$ and $F$ be the foot from $C$ to $AB.$ Denote $M$ the midpoint of $BC.$ The circumcircles of $\triangle BMF$ and $\triangle CME$ meet at another point $G.$ Compute the length of $GC.$
[i]Team #4[/i]
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
$ \textbf{(A) } 9\qquad \textbf{(B) } 12\frac{1}{2}\qquad \textbf{(C) } 15\qquad \textbf{(D) } 15\frac{1}{2}\qquad \textbf{(E) } 17$
[asy]
draw((0,0)--(0,5)--(5,5)--(5,0)--cycle);
filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray);
filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray);
filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray);
filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray);
[/asy]
A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.
[i]Proposed by Yannick Yao[/i]
Determine, with proof, whether or not there exists a [i]non-isosceles[/i] trapezoid $ABCD$ such that the lengths $AC$ and $BD$ both lie in the set $\{ DA+AB, AB+BC, BC+CD, CD+DA, AB+CD, BC+DA \}$.
Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.
Let triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $ABC$. The circle centered at $A$ of radius $AI$ intersects the circumcircle of $ABC$ at $H$ and $J$. Let $L$ be a point that lies on both the incircle of $ABC$ and line $HJ$. If the minimal possible value of $AL$ is $\sqrt{n}$, where $n \in \mathbb{Z}$, find $n$.
Let points $A=(0,0,0)$, $B=(1,0,0)$, $C=(0,2,0)$, and $D=(0,0,3)$. Points $E,F,G$, and $H$ are midpoints of line segments $\overline{BD},\overline{AB},\overline{AC}$, and $\overline{DC}$ respectively. What is the area of $EFGH$?
$ \textbf{(A)}\ \sqrt2
\qquad\textbf{(B)}\ \frac{2\sqrt5}{3}
\qquad\textbf{(C)}\ \frac{3\sqrt5}{4}
\qquad\textbf{(D)}\ \sqrt3
\qquad\textbf{(E)}\ \frac{2\sqrt7}{3}
$
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?
Given three squares as in the figure (where the vertex of B is touching square A --- the diagram had an error), where the largest square has area 1, and the area $ A$ is known. What is the area $ B$ of the smallest square?
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number4.jpg[/img]
A. $ A/8$
B. $ \frac {A^2}{2}$
C. $ \frac {A^4}{4}$
D. $ A(1 \minus{} A)$
E. $ \frac {(1 \minus{} A)^2}{4}$
Let $LOV ER$ be a convex pentagon such that $LOV E$ is a rectangle. Given that $OV = 20$ and $LO =V E = RE = RL = 23$, compute the radius of the circle passing through $R$, $O$, and $V$ .
A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder where $m+n$ is divided by 1000.
In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.
[i]Proposed by Davood Vakili, Iran[/i]