Found problems: 970
The points $(0,0),$ $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.
A fixed point $A$ inside a circle is given. Consider all chords $XY$ of the circle such that $\angle XAY$ is a right angle, and for all such chords construct the point $M$ symmetric to $A$ with respect to $XY$ . Find the locus of points $M$.
Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$.
[i]Proposed by Evan Chen[/i]
In triangle $ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.
${{ \textbf{(A)}\ 4\qquad\textbf{(B)}\ \sqrt{17} \qquad\textbf{(C)}\ 4.25\qquad\textbf{(D)}\ 2\sqrt{5} }\qquad\textbf{(E)}\ 4.5} $
$\triangle ABC$ has $AB=AC$. Points $M$ and $N$ are midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. The medians $\overline{MC}$ and $\overline{NB}$ intersect at a right angle. Find $(\tfrac{AB}{BC})^2$.
In triangle $ABC$, $D$ is the midpoint of $BC$, $E$ is the foot of the perpendicular from $A$ to $BC$, and $F$ is the foot of the perpendicular from $D$ to $AC$. Given that $BE=5$, $EC=9$, and the area of triangle $ABC$ is $84$, compute $|EF|$.
$HOW,BOW,$ and $DAH$ are equilateral triangles in a plane such that $WO=7$ and $AH=2$. Given that $D,A,B$ are collinear in that order, find the length of $BA$.
The diagram shows a circle with radius $24$ which contains two circles with radius $12$ tangent to each other and the larger circle. The smallest circle is tangent to the three other circles. What is the radius of the smallest circle?
[asy]
size(150);
defaultpen(linewidth(0.8));
draw(unitcircle^^circle((0,0.5),0.5)^^circle((0,-0.5),0.5)^^circle((2/3,0),1/3));[/asy]
Circles $C_1$, $C_2$, and $C_3$ are inside a rectangle $WXYZ$ such that $C_1$ is tangent to $\overline{WX}$, $\overline{ZW}$, and $\overline{YZ}$; $C_2$ is tangent to $\overline{WX}$ and $\overline{XY}$; and $C_3$ is tangent to $\overline{YZ}$, $C_1$, and $C_2$. If the radii of $C_1$, $C_2$, and $C_3$ are $1$, $\tfrac 12$, and $\tfrac 23$ respectively, compute the area of the triangle formed by the centers of $C_1$, $C_2$, and $C_3$.
[i]Proposed by Connor Gordon[/i]
Assume a triangle $ABC$ satisfies $|AB| = 1, |AC| = 2$ and $\angle ABC = \angle ACB + 90^\circ.$ What is the area of $ABC?$
\[
\mathrm a. ~ 6/7\qquad \mathrm b.~5/7\qquad \mathrm c. ~1/2 \qquad \mathrm d. ~4/5 \qquad \mathrm e. ~3/5
\]
Consider 2 concentric circle radii $ R$ and $ r$ ($ R > r$) with centre $ O.$ Fix $ P$ on the small circle and consider the variable chord $ PA$ of the small circle. Points $ B$ and $ C$ lie on the large circle; $ B,P,C$ are collinear and $ BC$ is perpendicular to $ AP.$
[b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal?
[b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
[asy]
real r=(3+sqrt(5))/2;
real s=sqrt(r);
real Brad=r;
real brad=1;
real Fht = 2*s;
import graph3;
import solids;
currentprojection=orthographic(1,0,.2);
currentlight=(10,10,5);
revolution sph=sphere((0,0,Fht/2),Fht/2);
//draw(surface(sph),green+white+opacity(0.5));
//triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));}
triple f(pair t) {
triple v0 = Brad*(cos(t.x),sin(t.x),0);
triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht);
return (v0 + t.y*(v1-v0));
}
triple g(pair t) {
return (t.y*cos(t.x),t.y*sin(t.x),0);
}
surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2);
surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2);
surface base = surface(g,(0,0),(2pi,Brad),80,2);
draw(sback,rgb(0,1,0));
draw(sfront,rgb(.3,1,.3));
draw(base,rgb(.4,1,.4));
draw(surface(sph),rgb(.3,1,.3));
[/asy]
$ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $
Given a square $ABCD$ of side $a$, we consider the circle $\omega$, tangent to side $BC$ and to the two semicircles of diameters $AB$ and $CD$. Calculate the radius of circle $\omega$,
What is the area of a square in square feet, if each of its diagonals is $4$ feet long?
Let $\triangle ABC$ be an equilateral triangle of side length 1. Let $D,E,F$ be points on $BC,AC,AB$ respectively, such that $\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}$. Let $X,Y,Z$ be on lines $BC,CA,AB$ respectively, such that $XY\perp DE, YZ\perp EF, ZX\perp FD$. Find all possible values of $\frac{1}{[DEF]} + \frac{1}{[XYZ]}$.
Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB = 8$ and $CD = 6$, find the distance between the midpoints of $AD$ and $BC$.
Compute the area of the trapezoid $ABCD$ with right angles $BAD$ and $ADC$ and side lengths of $AB=3$, $BC=5$, and $CD=7$.
Let $ABCD$ be a parallelogram. Point $E$ lies on segment $CD$ such that \[2\angle AEB=\angle ADB+\angle ACB,\] and point $F$ lies on segment $BC$ such that \[2\angle DFA=\angle DCA+\angle DBA.\] Let $K$ be the circumcenter of triangle $ABD$. Prove that $KE=KF$.
[i]Merlijn Staps[/i]
A $1\times 2$ rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?
$\textbf{(A)}\ \frac\pi2 \qquad \textbf{(B)}\ \frac{2\pi}3 \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}3 \qquad \textbf{(E)}\ \frac{5\pi}3$
Let $\omega$ be a circle with center $O$ and radius $8$, and let $A$ be a point such that $AO = 17$. Let $P$ and $Q$ be points on $\omega$ such that line segments $\overline{AP}$ and $\overline{AQ}$ are tangent to $\omega$ . Let $B$ and $C$ be points chosen on $\overline{AP}$ and $\overline{AQ}$, respectively, such that $\overline{BC}$ is also tangent to $\omega$ . Compute the perimeter of triangle $\vartriangle ABC$.
In parallelogram $ABCD$, $\overline{DE}$ is the altitude to the base $\overline{AB}$ and $\overline{DF}$ is the altitude to the base $\overline{BC}$. ['''Note:''' ''Both pictures represent the same parallelogram.''] If $DC=12$, $EB=4$, and $DE=6$, then $DF=$
[asy]
unitsize(12);
pair A,B,C,D,P,Q,W,X,Y,Z;
A = (0,0); B = (12,0); C = (20,6); D = (8,6);
W = (18,0); X = (30,0); Y = (38,6); Z = (26,6);
draw(A--B--C--D--cycle);
draw(W--X--Y--Z--cycle);
P = (8,0); Q = (758/25,6/25);
dot(A); dot(B); dot(C); dot(D); dot(W); dot(X); dot(Y); dot(Z); dot(P); dot(Q);
draw(A--B--C--D--cycle);
draw(W--X--Y--Z--cycle);
draw(D--P);
draw(Z--Q);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",P,S);
label("$A$",W,SW);
label("$B$",X,S);
label("$C$",Y,NE);
label("$D$",Z,NW);
label("$F$",Q,E);
[/asy]
$\text{(A)}\ 6.4 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 7.2 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$
A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point of $M$ of $CD$. Prove that the crease will divide $BC$ in the ration $5 : 3$.
Rhombus $ABCD$ has $\angle BAD<90^{\circ}$. There is a point $P$ on the incircle of the rhombus such that the distances from $P$ to lines $DA$, $AB$, and $BC$ are $9$, $5$, and $16$, respectively. Find the perimeter of $ABCD$.
The incircle of scalene triangle $\triangle{ABC}$ is tangent to $\overline{BC}, \overline{AC},$ and $\overline{AB}$ at points $D, E,$ and $F,$ respectively. The line $EF$ intersects line $BC$ at $P$ and line $AD$ at $Q.$ The circumcircle of $\triangle{AEF}$ intersects line $AP$ again at point $R \neq A.$ If $QE=3, QF=4, $ and $QR=8,$ find the area of triangle $\triangle{AEF}.$
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
[asy]
size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); dot("$A$",A,dir(270)); dot("$B$",B,E); dot("$C$",C,N); dot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270));
[/asy]
Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$
$\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113$