Found problems: 216
1984 IMO Shortlist, 9
Let $a, b, c$ be positive numbers with $\sqrt a +\sqrt b +\sqrt c = \frac{\sqrt 3}{2}$. Prove that the system of equations
\[\sqrt{y-a}+\sqrt{z-a}=1,\] \[\sqrt{z-b}+\sqrt{x-b}=1,\] \[\sqrt{x-c}+\sqrt{y-c}=1\]
has exactly one solution $(x, y, z)$ in real numbers.
2012 Turkey Team Selection Test, 2
In a plane, the six different points $A, B, C, A', B', C'$ are given such that triangles $ABC$ and $A'B'C'$ are congruent, i.e. $AB=A'B', BC=B'C', CA=C'A'.$ Let $G$ be the centroid of $ABC$ and $A_1$ be an intersection point of the circle with diameter $AA'$ and the circle with center $A'$ and passing through $G.$ Define $B_1$ and $C_1$ similarly. Prove that
\[ AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2 \]
2004 China Second Round Olympiad, 1
In an acute triangle $ABC$, point $H$ is the intersection point of altitude $CE$ to $AB$ and altitude $BD$ to $AC$. A circle with $DE$ as its diameter intersects $AB$ and $AC$ at $F$ and $G$, respectively. $FG$ and $AH$ intersect at point $K$. If $BC=25$, $BD=20$, and $BE=7$, find the length of $AK$.
2014 AMC 10, 9
The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$. How long is the third altitude of the triangle?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $
1980 AMC 12/AHSME, 7
Sides $AB,BC,CD$ and $DA$ of convex polygon $ABCD$ have lengths 3,4,12, and 13, respectively, and $\measuredangle CBA$ is a right angle. The area of the quadrilateral is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
real r=degrees((12,5)), s=degrees((3,4));
pair D=origin, A=(13,0), C=D+12*dir(r), B=A+3*dir(180-(90-r+s));
draw(A--B--C--D--cycle);
markscalefactor=0.05;
draw(rightanglemark(A,B,C));
pair point=incenter(A,C,D);
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("$3$", A--B, dir(A--B)*dir(-90));
label("$4$", B--C, dir(B--C)*dir(-90));
label("$12$", C--D, dir(C--D)*dir(-90));
label("$13$", D--A, dir(D--A)*dir(-90));[/asy]
$\text{(A)} \ 32 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 39 \qquad \text{(D)} \ 42 \qquad \text{(E)} \ 48$
2000 India National Olympiad, 4
In a convex quadrilateral $PQRS$, $PQ =RS$, $(\sqrt{3} +1 )QR = SP$ and $\angle RSP - \angle SQP = 30^{\circ}$. Prove that $\angle PQR - \angle QRS = 90^{\circ}.$
2004 AMC 12/AHSME, 14
In $ \triangle ABC$ , $ AB \equal{} 13$, $ AC \equal{} 5$, and $ BC \equal{} 12$. Points $ M$ and $ N$ lie on $ \overline{AC}$ and $ \overline{BC}$, respectively, with $ CM \equal{} CN \equal{} 4$. Points $ J$ and $ K$ are on $ \overline{AB}$ so that $ \overline{MJ}$ and $ \overline{NK}$ are perpendicular to $ \overline{AB}$. What is the area of pentagon $ CMJKN$?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair C=(0,0), B=(12,0), A=(0,5), M=(0,4), Np=(4,0);
pair K=foot(Np,A,B), J=foot(M,A,B);
draw(A--B--C--cycle);
draw(M--J);
draw(Np--K);
label("$C$",C,SW);
label("$A$",A,NW);
label("$B$",B,SE);
label("$N$",Np,S);
label("$M$",M,W);
label("$J$",J,NE);
label("$K$",K,NE);[/asy]$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ \frac{81}{5} \qquad
\textbf{(C)}\ \frac{205}{12} \qquad
\textbf{(D)}\ \frac{240}{13} \qquad
\textbf{(E)}\ 20$
1973 AMC 12/AHSME, 20
A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is
$ \textbf{(A)}\ 4\plus{}\sqrt{185} \qquad
\textbf{(B)}\ 16 \qquad
\textbf{(C)}\ 17 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ \sqrt{32}\plus{}\sqrt{137}$
1952 AMC 12/AHSME, 29
In a circle of radius $ 5$ units, $ CD$ and $ AB$ are perpendicular diameters. A chord $ CH$ cutting $ AB$ at $ K$ is $ 8$ units long. The diameter $ AB$ is divided into two segments whose dimensions are:
$ \textbf{(A)}\ 1.25, 8.75 \qquad\textbf{(B)}\ 2.75,7.25 \qquad\textbf{(C)}\ 2,8 \qquad\textbf{(D)}\ 4,6$
$ \textbf{(E)}\ \text{none of these}$
2006 AMC 12/AHSME, 16
Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE \equal{} 5$. What is $ CD$?
[asy]unitsize(2.5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;
pair A=(0,0), Ep=(5,0), B=(5+40/3,0);
pair M=midpoint(A--Ep);
pair C=intersectionpoints(Circle(M,2.5),Circle(A,3))[1];
pair D=B+8*dir(180+degrees(C));
dot(A);
dot(C);
dot(B);
dot(D);
draw(C--D);
draw(A--B);
draw(Circle(A,3));
draw(Circle(B,8));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,SE);
label("$E$",Ep,SSE);
label("$D$",D,NW);[/asy]$ \textbf{(A) } 13\qquad \textbf{(B) } \frac {44}{3}\qquad \textbf{(C) } \sqrt {221}\qquad \textbf{(D) } \sqrt {255}\qquad \textbf{(E) } \frac {55}{3}$
2014 AIME Problems, 15
In $ \triangle ABC $, $ AB = 3 $, $ BC = 4 $, and $ CA = 5 $. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\tfrac{DG}{EG} = \tfrac{3}{4}$, length $DE=\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
1961 AMC 12/AHSME, 38
Triangle $ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$ \textbf{(A)}\ s^2\le8r^2$
$\qquad\textbf{(B)}\ s^2=8r^2$
$\qquad\textbf{(C)}\ s^2 \ge 8r^2$
${\qquad\textbf{(D)}\ s^2\le4r^2 }$
${\qquad\textbf{(E)}\ x^2=4r^2 } $
2011 AIME Problems, 14
Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
2004 AMC 12/AHSME, 19
A truncated cone has horizontal bases with radii $ 18$ and $ 2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 4\sqrt5 \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 6\sqrt3$
2012 AMC 10, 17
Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
$ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $
2008 Harvard-MIT Mathematics Tournament, 27
Cyclic pentagon $ ABCDE$ has a right angle $ \angle{ABC} \equal{} 90^{\circ}$ and side lengths $ AB \equal{} 15$ and $ BC \equal{} 20$. Supposing that $ AB \equal{} DE \equal{} EA$, find $ CD$.
2001 AIME Problems, 6
Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. The ratio of the area of square $EFGH$ to the area of square $ABCD$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers and $m<n$. Find $10n+m$.
2011 AMC 10, 14
A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?
$ \textbf{(A)}\ 52 \qquad
\textbf{(B)}\ 58 \qquad
\textbf{(C)}\ 62 \qquad
\textbf{(D)}\ 68 \qquad
\textbf{(E)}\ 70 $
2010 AMC 10, 7
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$
2014 AIME Problems, 3
A rectangle has sides of length $a$ and $36$. A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of $24,$ the hexagon has the same area as the original rectangle. Find $a^2$.
[asy]
pair A,B,C,D,E,F,R,S,T,X,Y,Z;
dotfactor = 2;
unitsize(.1cm);
A = (0,0);
B = (0,18);
C = (0,36);
// don't look here
D = (12*2.236, 36);
E = (12*2.236, 18);
F = (12*2.236, 0);
draw(A--B--C--D--E--F--cycle);
dot(" ",A,NW);
dot(" ",B,NW);
dot(" ",C,NW);
dot(" ",D,NW);
dot(" ",E,NW);
dot(" ",F,NW);
//don't look here
R = (12*2.236 +22,0);
S = (12*2.236 + 22 - 13.4164,12);
T = (12*2.236 + 22,24);
X = (12*4.472+ 22,24);
Y = (12*4.472+ 22 + 13.4164,12);
Z = (12*4.472+ 22,0);
draw(R--S--T--X--Y--Z--cycle);
dot(" ",R,NW);
dot(" ",S,NW);
dot(" ",T,NW);
dot(" ",X,NW);
dot(" ",Y,NW);
dot(" ",Z,NW);
// sqrt180 = 13.4164
// sqrt5 = 2.236
[/asy]
2011 Middle European Mathematical Olympiad, 6
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 $BC$.
1995 AIME Problems, 4
Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
1986 India National Olympiad, 3
Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that
\[ \frac{1}{\sqrt{c}}\equal{}\frac{1}{\sqrt{a}}\plus{}\frac{1}{\sqrt{b}}\]
2013 AMC 12/AHSME, 18
Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
$ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$
2010 AMC 10, 19
A circle with center $ O$ has area $ 156\pi$. Triangle $ ABC$ is equilateral, $ \overline{BC}$ is a chord on the circle, $ OA \equal{} 4\sqrt3$, and point $ O$ is outside $ \triangle ABC$. What is the side length of $ \triangle ABC$?
$ \textbf{(A)}\ 2\sqrt3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 4\sqrt3 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18$