Found problems: 216
Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true?
[asy]/* AMC8 2002 #16 Problem */
draw((0,0)--(4,0)--(4,3)--cycle);
draw((4,3)--(-4,4)--(0,0));
draw((-0.15,0.1)--(0,0.25)--(.15,0.1));
draw((0,0)--(4,-4)--(4,0));
draw((4,0.2)--(3.8,0.2)--(3.8,-0.2)--(4,-0.2));
draw((4,0)--(7,3)--(4,3));
draw((4,2.8)--(4.2,2.8)--(4.2,3));
label(scale(0.8)*"$Z$", (0, 3), S);
label(scale(0.8)*"$Y$", (3,-2));
label(scale(0.8)*"$X$", (5.5, 2.5));
label(scale(0.8)*"$W$", (2.6,1));
label(scale(0.65)*"5", (2,2));
label(scale(0.65)*"4", (2.3,-0.4));
label(scale(0.65)*"3", (4.3,1.5));[/asy]
$ \textbf{(A)}\ X\plus{}Z\equal{}W\plus{}Y \qquad \textbf{(B)}\ W\plus{}X\equal{}Z \qquad\textbf{(C)}\ 3X\plus{}4Y\equal{}5Z \qquad $
$\textbf{(D)}\ X\plus{}W\equal{}\frac{1}{2}(Y\plus{}Z) \qquad\textbf{(E)}\ X\plus{}Y\equal{}Z$
On side $\overline{AB}$ of square $ABCD$, point $E$ is selected. Points $F$ and $G$ are located on sides $\overline{AB}$ and $\overline{AD}$, respectively, such that $\overline{FG} \perp \overline{CE}$. Let $P$ be the intersection point of segments $\overline{FG}$ and $\overline{CE}$. Given that $[EPF] = 1$, $[EPGA] = 8$, and $[CPFB] = 15$, compute $[PGDC]$. (Here $[\mathcal P]$ denotes the area of the polygon $\mathcal P$.)
[i]Proposed by Aaron Lin[/i]
Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$. Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$, determine the number of possible lengths of segment $AD$.
[i]Ray Li[/i]
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ \sqrt2 \qquad
\textbf{(C)}\ \sqrt3 \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 2\sqrt2$
Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$
The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?
$\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$
In the figure, BC is a diameter of the circle, where $BC=\sqrt{901}, BD=1$, and $DA=16$. If $EC=x$, what is the value of x?
[asy]size(2inch);
pair O,A,B,C,D,E;
B=(0,0);
O=(2,0);
C=(4,0);
D=(.333,1.333);
A=(.75,2.67);
E=(1.8,2);
draw(Arc(O,2,0,360));
draw(B--C--A--B);
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",E,N);
label("Figure not drawn to scale",(2,-2.5),S);
[/asy]
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 $(\tfrac{AB}{BC})^2$.
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 $BC$.
A $1\times 2$ rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?
$\textbf{(A)}\ \frac\pi2 \qquad \textbf{(B)}\ \frac{2\pi}3 \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}3 \qquad \textbf{(E)}\ \frac{5\pi}3$
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 $5 : 3$.
A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ And $HB$, with $HB = 16$. The area of $\triangle ABC$ is:
[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(11pt));
pair A = origin, H = (5,0), B = (13,0), C = (5,6.5);
draw(C--A--B--C--H^^rightanglemark(C,H,B,16));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,N);
label("$H$",H,S);
label("$15$",C/2,NW);
label("$16$",(H+B)/2,S);
[/asy]
$\textbf{(A) }120\qquad
\textbf{(B) }144\qquad
\textbf{(C) }150\qquad
\textbf{(D) }216\qquad
\textbf{(E) }144\sqrt5$
Let $ABC$ be an acute triangle. Denote by $B_0$ and $C_0$ the feet of the altitudes from vertices $B$ and $C$, respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_0$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_0$. Show that the line $AX$ is perpendicular to $BC$.
Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$
Let $d$ be one of the common tangent lines of externally tangent circles $k_1$ and $k_2$. $d$ touches $k_1$ at $A$. Let $[AB]$ be a diameter of $k_1$. The tangent from $B$ to $k_2$ touches $k_2$ at $C$. If $|AB|=8$ and the diameter of $k_2$ is $7$, then what is $|BC|$?
$ \textbf{(A)}\ 7
\qquad\textbf{(B)}\ 6\sqrt 2
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ 5\sqrt 3
$
Circles with centers $A ,~ B,$ and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals
$\textbf{(A) }3r-2\qquad\textbf{(B) }r^2\qquad\textbf{(C) }r+\sqrt{3(r-1)}\qquad$
$\textbf{(D) }1+\sqrt{3(r^2-1)}\qquad\textbf{(E) }\text{none of these}$
[asy]
//Holy crap, CSE5 is freaking amazing!
import cse5;
pathpen=black;
pointpen=black;
dotfactor=3;
size(200);
pair A=(1,2),B=(2,0),C=(0,0);
D(CR(A,1.5));
D(CR(B,1.5));
D(CR(C,1.5));
D(MP("$A$",A));
D(MP("$B$",B));
D(MP("$C$",C));
pair[] BB,CC;
CC=IPs(CR(A,1.5),CR(B,1.5));
BB=IPs(CR(A,1.5),CR(C,1.5));
D(BB[0]--CC[1]);
MP("$B'$",BB[0],NW);MP("$C'$",CC[1],NE);
//Credit to TheMaskedMagician for the diagram
[/asy]
Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point D on the south edge of the field. Cao arrives at point D at the same time that Ana and Bob arrive at D for the first time. The ratio of the field's length to the field's width to the distance from point D to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with p and q relatively prime. Find $p + q + r$.
Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$.
[asy]
size(200);
defaultpen(fontsize(10));
real x=22.61986495;
pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x);
draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7));
dot(A^^B^^C^^D^^E^^F);
pair point=(13,13);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));[/asy]
$ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Then $\sin \theta=$
[asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((0,0)--(2,1));
draw((0,0)--(1,2));
label("A", (0,0), SW);
label("B", (0,2), NW);
label("C", (2,2), NE);
label("D", (2,0), SE);
label("M", (1,2), N);
label("N", (2,1), E);
label("$\theta$", (.5,.5), SW);
[/asy]
$ \textbf{(A)}\ \frac{\sqrt{5}}{5} \qquad\textbf{(B)}\ \frac{3}{5} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{5} \qquad\textbf{(D)}\ \frac{4}{5} \qquad\textbf{(E)}\ \text{none of these} $
Let $ ABCD$ be a rhombus with $ AC\equal{}16$ and $ BD\equal{}30$. Let $ N$ be a point on $ \overline{AB}$, and let $ P$ and $ Q$ be the feet of the perpendiculars from $ N$ to $ \overline{AC}$ and $ \overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $ PQ$?
[asy]unitsize(2.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
pair D=(0,0), C=dir(0), A=dir(aSin(240/289)), B=shift(A)*C;
pair Np=waypoint(B--A,0.6), P=foot(Np,A,C), Q=foot(Np,B,D);
draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
draw(Np--Q);
draw(Np--P);
label("$D$",D,SW);
label("$C$",C,SE);
label("$B$",B,NE);
label("$A$",A,NW);
label("$N$",Np,N);
label("$P$",P,SW);
label("$Q$",Q,SSE);
draw(rightanglemark(Np,P,C,2));
draw(rightanglemark(Np,Q,D,2));[/asy]$ \textbf{(A)}\ 6.5 \qquad
\textbf{(B)}\ 6.75 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 7.25 \qquad
\textbf{(E)}\ 7.5$
Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation
\[ (a_n \minus{} a_{n \minus{} 1})(a_n \minus{} a_{n \minus{} 2}) \plus{} (b_n \minus{} b_{n \minus{} 1})(b_n \minus{} b_{n \minus{} 2}) \equal{} 0
\]
for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k \equal{} a_{k \plus{} 2008}$.