Found problems: 216
2006 Stanford Mathematics Tournament, 9
$\triangle ABC$ 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 $(\tfrac{AB}{BC})^2$.
2009 AMC 10, 22
A cubical cake with edge length $ 2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $ M$ is the midpoint of a top edge. The piece whose top is triangle $ B$ contains $ c$ cubic inches of cake and $ s$ square inches of icing. What is $ c\plus{}s$?
[asy]unitsize(1cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle);
draw((1,1)--(-1,0));
pair P=foot((1,-1),(1,1),(-1,0));
draw((1,-1)--P);
draw(rightanglemark((-1,0),P,(1,-1),4));
label("$M$",(-1,0),W);
label("$C$",(-0.1,-0.3));
label("$A$",(-0.4,0.7));
label("$B$",(0.7,0.4));[/asy]$ \textbf{(A)}\ \frac{24}{5} \qquad
\textbf{(B)}\ \frac{32}{5} \qquad
\textbf{(C)}\ 8\plus{}\sqrt5 \qquad
\textbf{(D)}\ 5\plus{}\frac{16\sqrt5}{5} \qquad
\textbf{(E)}\ 10\plus{}5\sqrt5$
2005 France Team Selection Test, 2
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle).
Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.
2006 AMC 10, 16
A circle of radius 1 is tangent to a circle of radius 2. The sides of $ \triangle ABC$ are tangent to the circles as shown, and the sides $ \overline{AB}$ and $ \overline{AC}$ are congruent. What is the area of $ \triangle ABC$?
[asy]defaultpen(black+linewidth(0.7));
size(7cm);
real t=2^0.5;
D((0,0)--(4*t,0)--(2*t,8)--cycle, black);
D(CR((2*t,2),2), black);
D(CR((2*t,5),1), black);
dot(origin^^(4t,0)^^(2t,8));
label("B", (0,0), SW);
label("C", (4*t,0), SE);
label("A", (2*t,8), N);
D((2*t,2)--(2*t,4), black); D((2*t,5)--(2*t,6), black);
MP('2', (2*t,3), W); MP('1',(2*t, 5.5), W);[/asy]
$ \textbf{(A) } \frac {35}2 \qquad \textbf{(B) } 15\sqrt {2} \qquad \textbf{(C) } \frac {64}3 \qquad \textbf{(D) } 16\sqrt {2} \qquad \textbf{(E) } 24$
2007 AMC 8, 14
The base of isosceles $\triangle{ABC}$ is $24$ and its area is $60$. What is the length of one of the congruent sides?
$\textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 13 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 18$
2008 National Olympiad First Round, 17
Let the vertices $A$ and $C$ of a right triangle $ABC$ be on the arc with center $B$ and radius $20$. A semicircle with diameter $[AB]$ is drawn to the inner region of the arc. The tangent from $C$ to the semicircle touches the semicircle at $D$ other than $B$. Let $CD$ intersect the arc at $F$. What is $|FD|$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ \frac 52
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
2003 AMC 12-AHSME, 22
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$
1993 India Regional Mathematical Olympiad, 1
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 $BC$.
2000 AMC 12/AHSME, 24
If circular arcs $ AC$ and $ BC$ have centers at $ B$ and $ A$, respectively, then there exists a circle tangent to both $ \stackrel{\frown}{AC}$ and $ \stackrel{\frown}{BC}$, and to $ \overline{AB}$. If the length of $ \stackrel{\frown}{BC}$ is $ 12$, then the circumference of the circle is
[asy]unitsize(4cm);
defaultpen(fontsize(8pt)+linewidth(.8pt));
dotfactor=3;
pair O=(0,.375);
pair A=(-.5,0);
pair B=(.5,0);
pair C=shift(-.5,0)*dir(60);
draw(Arc(A,1,0,60));
draw(Arc(B,1,120,180));
draw(A--B);
draw(Circle(O,.375));
dot(A);
dot(B);
dot(C);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);[/asy]$ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 28$
2008 Harvard-MIT Mathematics Tournament, 1
Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$, and also that $ AX \equal{} \frac {\sqrt {2}}{2}$. Determine the value of $ CX^2$.
2015 AMC 12/AHSME, 24
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$?
$ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $
2006 AMC 10, 8
A square of area $40$ is inscribed in a semicircle as shown. What is the area of the semicircle?
[asy]
defaultpen(linewidth(0.8));
real r=sqrt(50), s=sqrt(10);
draw(Arc(origin, r, 0, 180));
draw((r,0)--(-r,0), dashed);
draw((s,0)--(s,2*s)--(-s,2*s)--(-s,0));[/asy]
$ \textbf{(A) }20\pi\qquad\textbf{(B) }25\pi\qquad\textbf{(C) }30\pi\qquad\textbf{(D) }40\pi\qquad\textbf{(E) }50\pi $
1958 AMC 12/AHSME, 31
The altitude drawn to the base of an isosceles triangle is $ 8$, and the perimeter $ 32$. The area of the triangle is:
$ \textbf{(A)}\ 56\qquad
\textbf{(B)}\ 48\qquad
\textbf{(C)}\ 40\qquad
\textbf{(D)}\ 32\qquad
\textbf{(E)}\ 24$
2004 AIME Problems, 4
A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.
1949-56 Chisinau City MO, 21
The sides of the triangle $ABC$ satisfy the relation $c^2 = a^2 + b^2$. Show that angle $C$ is right.
1986 AIME Problems, 15
Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.
2010 Princeton University Math Competition, 5
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$ say $C_1$, and then the midpoint of $A_1D$ which is $A_2$. He continues in this fashion, indefinitely. The total length of his path if $AB=5$ and $BC=12$ is of the form $a + b\sqrt{c}$. Find $\displaystyle\frac{abc}{4}$.
2004 AMC 12/AHSME, 18
Square $ ABCD$ has side length $ 2$. A semicircle with diameter $ \overline{AB}$ is constructed inside the square, and the tangent to the semicricle from $ C$ intersects side $ \overline{AD}$ at $ E$. What is the length of $ \overline{CE}$?
[asy]
defaultpen(linewidth(0.8));
pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=tangent(C, (0.5,0), 0.5, 1), F=C+2*dir(C--X), E=intersectionpoint(C--F, A--D);
draw(C--D--A--B--C--E);
draw(Arc((0.5,0), 0.5, 0, 180));
pair point=(0.5,0.5);
label("$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));[/asy]
$ \textbf{(A)}\ \frac {2 \plus{} \sqrt5}{2} \qquad \textbf{(B)}\ \sqrt 5 \qquad \textbf{(C)}\ \sqrt 6 \qquad \textbf{(D)}\ \frac52 \qquad \textbf{(E)}\ 5 \minus{} \sqrt5$
1961 AMC 12/AHSME, 36
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} $
2000 National Olympiad First Round, 13
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
$
2002 Bulgaria National Olympiad, 2
Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$
[i]Proposed by Alexander Ivanov[/i]
2008 ITest, 62
Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.
2003 AMC 8, 6
Given the areas of the three squares in the figure, what is the area of the interior triangle?
[asy]
real r=22.61986495;
pair A=origin, B=(12,0), C=(12,5);
draw(A--B--C--cycle);
markscalefactor=0.1;
draw(rightanglemark(C, B, A));
draw(scale(12)*shift(0,-1)*unitsquare);
draw(scale(5)*shift(12/5,0)*unitsquare);
draw(scale(13)*rotate(r)*unitsquare);
pair P=shift(0,-1)*(13/sqrt(2) * dir(r+45)), Q=(14.5,1.2), R=(6, -7);
label("169", P, N);
label("25", Q, N);
label("144", R, N);
[/asy]
$ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 60\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 1800$
2015 AMC 12/AHSME, 16
A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?
$\textbf{(A) }18\qquad\textbf{(B) }162\qquad\textbf{(C) }36\sqrt{21}\qquad\textbf{(D) }18\sqrt{138}\qquad\textbf{(E) }54\sqrt{21}$
1991 AMC 12/AHSME, 22
Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle. If $PA = AB = 4$, then the area of the smaller circle is
[asy]
size(250);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair O=origin, Q=(0,-3sqrt(2)), P=(0,-6sqrt(2)), A=(-4/3,3.77-6sqrt(2)), B=(-8/3,7.54-6sqrt(2)), C=(4/3,3.77-6sqrt(2)), D=(8/3,7.54-6sqrt(2));
draw(Arc(O,2sqrt(2),0,360));
draw(Arc(Q,sqrt(2),0,360));
dot(A);
dot(B);
dot(C);
dot(D);
dot(P);
draw(B--A--P--C--D);
label("$A$",A,dir(A));
label("$B$",B,dir(B));
label("$A'$",C,dir(C));
label("$B'$",D,dir(D));
label("$P$",P,S);[/asy]
$ \textbf{(A)}\ 1.44\pi\qquad\textbf{(B)}\ 2\pi\qquad\textbf{(C)}\ 2.56\pi\qquad\textbf{(D)}\ \sqrt{8}\pi\qquad\textbf{(E)}\ 4\pi $