Found problems: 97
2008 Harvard-MIT Mathematics Tournament, 6
Let $ ABC$ be a triangle with $ \angle A \equal{} 45^\circ$. Let $ P$ be a point on side $ BC$ with $ PB \equal{} 3$ and $ PC \equal{} 5$. Let $ O$ be the circumcenter of $ ABC$. Determine the length $ OP$.
2000 National Olympiad First Round, 5
$[BD]$ is a median of $\triangle ABC$. $m(\widehat{ABD})=90^\circ$, $|AB|=2$, and $|AC|=6$. $|BC|=?$
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 3\sqrt2
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 4\sqrt2
\qquad\textbf{(E)}\ 2\sqrt6
$
2014 Online Math Open Problems, 11
Let $X$ be a point inside convex quadrilateral $ABCD$ with $\angle AXB+\angle CXD=180^{\circ}$. If $AX=14$, $BX=11$, $CX=5$, $DX=10$, and $AB=CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$.
[i]Proposed by Michael Kural[/i]
2016 Middle European Mathematical Olympiad, 5
Let $ABC$ be an acute triangle for which $AB \neq AC$, and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$, and the line $BC$ in $E$. The circumcircle of $CDE$ meets the line $CA$ again in $P$. The lines $PE$ and $AB$ intersect in $Q$. Line passing through $O$ parallel to the line $PE$ intersects the $A$-altitude of $ABC$ in $F$.
Prove that $FP = FQ$.
2011 Purple Comet Problems, 24
The diagram below shows a regular hexagon with an inscribed square where two sides of the square are parallel to two sides of the hexagon. There are positive integers $m$, $n$, and $p$ such that the ratio of the area of the hexagon to the area of the square can be written as $\tfrac{m+\sqrt{n}}{p}$ where $m$ and $p$ are relatively prime. Find $m + n + p$.
[asy]
import graph; size(4cm);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle);
filldraw((1.13,2.5)--(-0.13,2.5)--(-0.13,1.23)--(1.13,1.23)--cycle,grey);
draw((0,1)--(1,1));
draw((1,1)--(1.5,1.87));
draw((1.5,1.87)--(1,2.73));
draw((1,2.73)--(0,2.73));
draw((0,2.73)--(-0.5,1.87));
draw((-0.5,1.87)--(0,1));
draw((1.13,2.5)--(-0.13,2.5));
draw((-0.13,2.5)--(-0.13,1.23));
draw((-0.13,1.23)--(1.13,1.23));
draw((1.13,1.23)--(1.13,2.5)); [/asy]
2012 Online Math Open Problems, 8
In triangle $ABC$ let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$.
[i]Ray Li.[/i]
[hide="Clarifications"][list=1][*]Triangle $ABC$ is acute.[/list][/hide]
1973 AMC 12/AHSME, 4
Two congruent $ 30^{\circ}$-$ 60^{\circ}$-$ 90^{\circ}$ are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is
$ \textbf{(A)}\ 6\sqrt3 \qquad
\textbf{(B)}\ 8\sqrt3 \qquad
\textbf{(C)}\ 9\sqrt3 \qquad
\textbf{(D)}\ 12\sqrt3 \qquad
\textbf{(E)}\ 24$
2010 AMC 12/AHSME, 22
Let $ ABCD$ be a cyclic quadrilateral. The side lengths of $ ABCD$ are distinct integers less than $ 15$ such that $ BC\cdot CD\equal{}AB\cdot DA$. What is the largest possible value of $ BD$?
$ \textbf{(A)}\ \sqrt{\frac{325}{2}} \qquad \textbf{(B)}\ \sqrt{185} \qquad \textbf{(C)}\ \sqrt{\frac{389}{2}} \qquad \textbf{(D)}\ \sqrt{\frac{425}{2}} \qquad \textbf{(E)}\ \sqrt{\frac{533}{2}}$
2003 AIME Problems, 11
Triangle $ABC$ is a right triangle with $AC=7,$ $BC=24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD=BD=15.$ Given that the area of triangle $CDM$ may be expressed as $\frac{m\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p.$
2013 AMC 12/AHSME, 12
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$?
$ \textbf{(A)}\ 36\qquad\textbf{(B)}\ 38\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 44$
1982 AMC 12/AHSME, 18
In the adjoining figure of a rectangular solid, $\angle DHG=45^\circ$ and $\angle FHB=60^\circ$. Find the cosine of $\angle BHD$.
[asy]
size(200);
import three;defaultpen(linewidth(0.7)+fontsize(10));
currentprojection=orthographic(1/3+1/10,1-1/10,1/3);
real r=sqrt(3);
triple A=(0,0,r), B=(0,r,r), C=(1,r,r), D=(1,0,r), E=O, F=(0,r,0), G=(1,0,0), H=(1,r,0);
draw(D--G--H--D--A--B--C--D--B--F--H--B^^C--H);
draw(A--E^^G--E^^F--E, linetype("4 4"));
label("$A$", A, N);
label("$B$", B, dir(0));
label("$C$", C, N);
label("$D$", D, W);
label("$E$", E, NW);
label("$F$", F, S);
label("$G$", G, W);
label("$H$", H, S);
triple H45=(1,r-0.15,0.1), H60=(1-0.05, r, 0.07);
label("$45^\circ$", H45, dir(125), fontsize(8));
label("$60^\circ$", H60, dir(25), fontsize(8));[/asy]
$\textbf {(A) } \frac{\sqrt{3}}{6} \qquad \textbf {(B) } \frac{\sqrt{2}}{6} \qquad \textbf {(C) } \frac{\sqrt{6}}{3} \qquad \textbf {(D) } \frac{\sqrt{6}}{4} \qquad \textbf {(E) } \frac{\sqrt{6}-\sqrt{2}}{4}$
2007 Harvard-MIT Mathematics Tournament, 7
Convex quadrilateral $ABCD$ has sides $AB=BC=7$, $CD=5$, and $AD=3$. Given additionally that $m\angle ABC=60^\circ$, find $BD$.
2014 Math Prize For Girls Problems, 8
A triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.
JBMO Geometry Collection, 2002
The triangle $ABC$ has $CA = CB$. $P$ is a point on the circumcircle between $A$ and $B$ (and on the opposite side of the line $AB$ to $C$). $D$ is the foot of the perpendicular from $C$ to $PB$. Show that $PA + PB = 2 \cdot PD$.
2003 AIME Problems, 10
Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ$. Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ$. Find the number of degrees in $\angle CMB$.
2002 Junior Balkan MO, 1
The triangle $ABC$ has $CA = CB$. $P$ is a point on the circumcircle between $A$ and $B$ (and on the opposite side of the line $AB$ to $C$). $D$ is the foot of the perpendicular from $C$ to $PB$. Show that $PA + PB = 2 \cdot PD$.
1998 AMC 12/AHSME, 26
In quadrilateral $ ABCD$, it is given that $ \angle A \equal{} 120^\circ$, angles $ B$ and $ D$ are right angles, $ AB \equal{} 13$, and $ AD \equal{} 46$. Then $ AC \equal{}$
$ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 62 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 65 \qquad \textbf{(E)}\ 72$
2020 Candian MO, 2#
Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.
1958 AMC 12/AHSME, 36
The sides of a triangle are $ 30$, $ 70$, and $ 80$ units. If an altitude is dropped upon the side of length $ 80$, the larger segment cut off on this side is:
$ \textbf{(A)}\ 62\qquad
\textbf{(B)}\ 63\qquad
\textbf{(C)}\ 64\qquad
\textbf{(D)}\ 65\qquad
\textbf{(E)}\ 66$
1963 AMC 12/AHSME, 34
In triangle ABC, side $a = \sqrt{3}$, side $b = \sqrt{3}$, and side $c > 3$. Let $x$ be the largest number such that the magnitude, in degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals:
$\textbf{(A)}\ 150 \qquad
\textbf{(B)}\ 120\qquad
\textbf{(C)}\ 105 \qquad
\textbf{(D)}\ 90 \qquad
\textbf{(E)}\ 60$
2011 Indonesia MO, 3
Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.
2005 AIME Problems, 7
In quadrilateral $ABCD$, $BC=8$, $CD=12$, $AD=10$, and $m\angle A= m\angle B = 60^\circ$. Given that $AB=p + \sqrt{q}$, where $p$ and $q$ are positive integers, find $p+q$.