Found problems: 126
2014 AMC 12/AHSME, 24
Let $ABCDE$ be a pentagon inscribed in a circle such that $AB=CD=3$, $BC=DE=10$, and $AE=14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A) }129\qquad
\textbf{(B) }247\qquad
\textbf{(C) }353\qquad
\textbf{(D) }391\qquad
\textbf{(E) }421\qquad$
2007 Princeton University Math Competition, 6
Triangle $ABC$ has $AC = 3$, $BC = 5$, $AB = 7$. A circle is drawn internally tangent to the circumcircle of $ABC$ at $C$, and tangent to $AB$. Let $D$ be its point of tangency with $AB$. Find $BD - DA$.
[asy]
/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(6cm);
real labelscalefactor = 2.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -4.5, xmax = 7.01, ymin = -3, ymax = 8.02; /* image dimensions */
/* draw figures */
draw(circle((1.37,2.54), 5.17));
draw((-2.62,-0.76)--(-3.53,4.2));
draw((-3.53,4.2)--(5.6,-0.44));
draw((5.6,-0.44)--(-2.62,-0.76));
draw(circle((-0.9,0.48), 2.12));
/* dots and labels */
dot((-2.62,-0.76),dotstyle);
label("$C$", (-2.46,-0.51), SW * labelscalefactor);
dot((-3.53,4.2),dotstyle);
label("$A$", (-3.36,4.46), NW * labelscalefactor);
dot((5.6,-0.44),dotstyle);
label("$B$", (5.77,-0.17), SE * labelscalefactor);
dot((0.08,2.37),dotstyle);
label("$D$", (0.24,2.61), SW * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$7$",(-3.36,4.46)--(5.77,-0.17), NE * labelscalefactor);
label("$3$",(-3.36,4.46)--(-2.46,-0.51),SW * labelscalefactor);
label("$5$",(-2.46,-0.51)--(5.77,-0.17), SE * labelscalefactor);
/* end of picture */
[/asy]
2015 AMC 10, 8
The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
[asy]
import cse5;pathpen=black;pointpen=black;
size(2cm);
D((0,-2)--MP("y",(0,7),N));
D((-3,0)--MP("x",(5,0),E));
D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5));
[/asy][asy]
import cse5;pathpen=black;pointpen=black;
unitsize(0.2cm);
D((0,-2)--MP("y",(0,7),N));
D(MP("\textbf{(A) }",(-3,0),W)--MP("x",(5,0),E));
D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5));
//
D((18,-2)--MP("y",(18,7),N));
D(MP("\textbf{(B) }",(13,0),W)--MP("x",(21,0),E));
D((17,0)--(17,2)--(16,2)--(16,3)--(17,3)--(17,4)--(15,4)--(15,5)--(18,5));
//
D((36,-2)--MP("y",(36,7),N));
D(MP("\textbf{(C) }",(29,0),W)--MP("x",(38,0),E));
D((31,0)--(31,1)--(33,1)--(33,2)--(34,2)--(34,1)--(35,1)--(35,3)--(36,3));
//
D((0,-17)--MP("y",(0,-8),N));
D(MP("\textbf{(D) }",(-3,-15),W)--MP("x",(5,-15),E));
D((3,-15)--(3,-14)--(1,-14)--(1,-13)--(2,-13)--(2,-12)--(1,-12)--(1,-10)--(0,-10));
//
D((15,-17)--MP("y",(15,-8),N));
D(MP("\textbf{(E) }",(13,-15),W)--MP("x",(22,-15),E));
D((15,-14)--(17,-14)--(17,-13)--(18,-13)--(18,-14)--(19,-14)--(19,-12)--(20,-12)--(20,-15));
[/asy]
2014 China Western Mathematical Olympiad, 2
Let $ AB$ be the diameter of semicircle $O$ ,
$C, D $ be points on the arc $AB$,
$P, Q$ be respectively the circumcenter of $\triangle OAC $ and $\triangle OBD $ .
Prove that:$CP\cdot CQ=DP \cdot DQ$.[asy]
import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black;
real h=sqrt(55/64);
pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B);
D(arc(O,1,0,180),darkgreen);
D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue);
D(O);
[/asy]
1985 AIME Problems, 6
As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$.
[asy]
size(200);
pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C);
draw(F--C--A--B--C^^A--D^^B--E);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("84", centroid(H, C, E), fontsize(9.5));
label("35", centroid(H, B, D), fontsize(9.5));
label("30", centroid(H, F, B), fontsize(9.5));
label("40", centroid(H, A, F), fontsize(9.5));[/asy]
1971 AMC 12/AHSME, 9
An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. If the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance between the centers of the pulleys in inches is
$\textbf{(A) }24\qquad\textbf{(B) }2\sqrt{119}\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad \textbf{(E) }4\sqrt{35}$
2011 Indonesia TST, 3
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
2000 All-Russian Olympiad, 3
Let $O$ be the center of the circumcircle $\omega$ of an acute-angle triangle $ABC$. A circle $\omega_1$ with center $K$ passes through $A$, $O$, $C$ and intersects $AB$ at $M$ and $BC$ at $N$. Point $L$ is symmetric to $K$ with respect to line $NM$. Prove that $BL \perp AC$.
2013 ELMO Shortlist, 5
Let $\omega_1$ and $\omega_2$ be two orthogonal circles, and let the center of $\omega_1$ be $O$. Diameter $AB$ of $\omega_1$ is selected so that $B$ lies strictly inside $\omega_2$. The two circles tangent to $\omega_2$, passing through $O$ and $A$, touch $\omega_2$ at $F$ and $G$. Prove that $FGOB$ is cyclic.
[i]Proposed by Eric Chen[/i]
1997 All-Russian Olympiad, 3
The incircle of triangle $ABC$ touches sides $AB$;$BC$;$CA$ at $M$;$N$;$K$, respectively.
The line through $A$ parallel to $NK$ meets $MN$ at $D$.
The line through $A$ parallel to $MN$ meets $NK$ at $E$.
Show that the line $DE$ bisects sides $AB$ and $AC$ of triangle $ABC$.
[i]M. Sonkin[/i]
1976 AMC 12/AHSME, 3
The sum of the distances from one vertex of a square with sides of length two to the midpoints of each of the sides of the square is
$\textbf{(A) }2\sqrt{5}\qquad\textbf{(B) }2+\sqrt{3}\qquad\textbf{(C) }2+2\sqrt{3}\qquad\textbf{(D) }2+\sqrt{5}\qquad \textbf{(E) }2+2\sqrt{5}$
2000 India National Olympiad, 1
The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. The line through $A$ parallel to $LK$ meets $MK$ at $P$, and the line through $A$ parallel to $MK$ meets $LK$ at $Q$. Show that the line $PQ$ bisects $AB$ and bisects $AC$.
2014 AMC 8, 9
In $\bigtriangleup ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$?
[asy]
size(300);
defaultpen(linewidth(0.8));
pair A=(-1,0),C=(1,0),B=dir(40),D=origin;
draw(A--B--C--A);
draw(D--B);
dot("$A$", A, SW);
dot("$B$", B, NE);
dot("$C$", C, SE);
dot("$D$", D, S);
label("$70^\circ$",C,2*dir(180-35));
[/asy]
$\textbf{(A) }100\qquad\textbf{(B) }120\qquad\textbf{(C) }135\qquad\textbf{(D) }140\qquad \textbf{(E) }150$
1956 AMC 12/AHSME, 21
If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is:
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 2\text{ or }3 \qquad\textbf{(C)}\ 2\text{ or }4 \qquad\textbf{(D)}\ 3\text{ or }4 \qquad\textbf{(E)}\ 2,3,\text{ or }4$
1978 AMC 12/AHSME, 26
[asy]
import cse5;
size(180);
real a=4, b=3;
pathpen=black;
pair A=(a,0), B=(0,b), C=(0,0);
D(MP("A",A)--MP("B",B,N)--MP("C",C,SW)--cycle);
pair X=IP(B--A,(0,0)--(b,a));
D(CP((X+C)/2,C));
D(MP("R",IP(CP((X+C)/2,C),B--C),NW)--MP("Q",IP(CP((X+C)/2,C),A--C+(0.1,0))));
//Credit to chezbgone2 for the diagram[/asy]
In $\triangle ABC$, $AB = 10~ AC = 8$ and $BC = 6$. Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$, respectively. The length of segment $QR$ is
$\textbf{(A) }4.75\qquad\textbf{(B) }4.8\qquad\textbf{(C) }5\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }3\sqrt{3}$
1963 AMC 12/AHSME, 25
Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of triangle $CEF$ is $200$ square inches. Then the number of inches in $BE$ is:
[asy]
size(6cm);
pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7);
draw(A--B--C--D--cycle);
draw(F--C--E--B);
label("$A$", A, SW);
label("$B$", B, S);
label("$C$", C, N);
label("$D$", D, NW);
label("$E$", E, SE);
label("$F$", F, W);
//Credit to MSTang for the asymptote
[/asy]
$\textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 20$
2021 AMC 10 Fall, 7
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$
[asy]
size(6cm);
pair A = (0,10);
label("$A$", A, N);
pair B = (0,0);
label("$B$", B, S);
pair C = (10,0);
label("$C$", C, S);
pair D = (10,10);
label("$D$", D, SW);
pair EE = (15,11.8);
label("$E$", EE, N);
pair F = (3,10);
label("$F$", F, N);
filldraw(D--arc(D,2.5,270,380)--cycle,lightgray);
dot(A^^B^^C^^D^^EE^^F);
draw(A--B--C--D--cycle);
draw(D--EE--F--cycle);
label("$110^\circ$", (15,9), SW);
[/asy]
$\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$
2007 Princeton University Math Competition, 4
Find $\frac{area(CDF)}{area(CEF)}$ in the figure.
[asy]
/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(5.75cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -2, xmax = 21, ymin = -2, ymax = 16; /* image dimensions */
/* draw figures */
draw((0,0)--(20,0));
draw((13.48,14.62)--(7,0));
draw((0,0)--(15.93,9.12));
draw((13.48,14.62)--(20,0));
draw((13.48,14.62)--(0,0));
label("6",(15.16,12.72),SE*labelscalefactor);
label("10",(18.56,5.1),SE*labelscalefactor);
label("7",(3.26,-0.6),SE*labelscalefactor);
label("13",(13.18,-0.71),SE*labelscalefactor);
label("20",(5.07,8.33),SE*labelscalefactor);
/* dots and labels */
dot((0,0),dotstyle);
label("$B$", (-1.23,-1.48), NE * labelscalefactor);
dot((20,0),dotstyle);
label("$C$", (19.71,-1.59), NE * labelscalefactor);
dot((7,0),dotstyle);
label("$D$", (6.77,-1.64), NE * labelscalefactor);
dot((13.48,14.62),dotstyle);
label("$A$", (12.36,14.91), NE * labelscalefactor);
dot((15.93,9.12),dotstyle);
label("$E$", (16.42,9.21), NE * labelscalefactor);
dot((9.38,5.37),dotstyle);
label("$F$", (9.68,4.5), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
2008 AIME Problems, 12
On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let $ M$ be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when $ M$ is divided by 10.
2011 Indonesia TST, 3
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
2012 China National Olympiad, 1
In the triangle $ABC$, $\angle A$ is biggest. On the circumcircle of $\triangle ABC$, let $D$ be the midpoint of $\widehat{ABC}$ and $E$ be the midpoint of $\widehat{ACB}$. The circle $c_1$ passes through $A,B$ and is tangent to $AC$ at $A$, the circle $c_2$ passes through $A,E$ and is tangent $AD$ at $A$. $c_1$ and $c_2$ intersect at $A$ and $P$. Prove that $AP$ bisects $\angle BAC$.
[hide="Diagram"][asy]
/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(14.4cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -5.23, xmax = 9.18, ymin = -2.97, ymax = 4.82; /* image dimensions */
/* draw figures */
draw(circle((-1.32,1.36), 2.98));
draw(circle((3.56,1.53), 3.18));
draw((0.92,3.31)--(-2.72,-1.27));
draw(circle((0.08,0.25), 3.18));
draw((-2.72,-1.27)--(3.13,-0.65));
draw((3.13,-0.65)--(0.92,3.31));
draw((0.92,3.31)--(2.71,-1.54));
draw((-2.41,-1.74)--(0.92,3.31));
draw((0.92,3.31)--(1.05,-0.43));
/* dots and labels */
dot((-1.32,1.36),dotstyle);
dot((0.92,3.31),dotstyle);
label("$A$", (0.81,3.72), NE * labelscalefactor);
label("$c_1$", (-2.81,3.53), NE * labelscalefactor);
dot((3.56,1.53),dotstyle);
label("$c_2$", (3.43,3.98), NE * labelscalefactor);
dot((1.05,-0.43),dotstyle);
label("$P$", (0.5,-0.43), NE * labelscalefactor);
dot((-2.72,-1.27),dotstyle);
label("$B$", (-3.02,-1.57), NE * labelscalefactor);
dot((2.71,-1.54),dotstyle);
label("$E$", (2.71,-1.86), NE * labelscalefactor);
dot((3.13,-0.65),dotstyle);
label("$C$", (3.39,-0.9), NE * labelscalefactor);
dot((-2.41,-1.74),dotstyle);
label("$D$", (-2.78,-2.07), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy][/hide]
2015 AMC 10, 9
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata?
[asy]
import cse5;pathpen=black;pointpen=black;
size(1.5inch);
D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W));
path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle;
draw(P,linewidth(2));
fill(P,gray);
[/asy]
$\textbf{(A) } \dfrac{4\pi}{5}
\qquad\textbf{(B) } \dfrac{9\pi}{8}
\qquad\textbf{(C) } \dfrac{4\pi}{3}
\qquad\textbf{(D) } \dfrac{7\pi}{5}
\qquad\textbf{(E) } \dfrac{3\pi}{2}
$
2009 Purple Comet Problems, 7
The figure $ABCD$ is bounded by a semicircle $CDA$ and a quarter circle $ABC$. Given that the distance from $A$ to $C$ is $18$, find the area of the figure.
[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=(-9,0),B=(0,9*sqrt(2)-9),C=(9,0),D=(0,9);
dot(A^^B^^C^^D);
draw(arc(origin,9,0,180)^^arc((0,-9),9*sqrt(2),45,135));
label("$A$",A,S);
label("$B$",B,N);
label("$C$",C,S);
label("$D$",D,N);
[/asy]
2007 USA Team Selection Test, 1
Circles $ \omega_1$ and $ \omega_2$ meet at $ P$ and $ Q$. Segments $ AC$ and $ BD$ are chords of $ \omega_1$ and $ \omega_2$ respectively, such that segment $ AB$ and ray $ CD$ meet at $ P$. Ray $ BD$ and segment $ AC$ meet at $ X$. Point $ Y$ lies on $ \omega_1$ such that $ PY \parallel BD$. Point $ Z$ lies on $ \omega_2$ such that $ PZ \parallel AC$. Prove that points $ Q,X,Y,Z$ are collinear.
1998 Harvard-MIT Mathematics Tournament, 3
Find the area of the region bounded by the graphs $y=x^2$, $y=x$, and $x=2$.