Found problems: 124
2018 India National Olympiad, 1
Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$ ; let $G$ be the centroid of $\Delta{ABC}$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta{ABC}$.
1989 AMC 12/AHSME, 19
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$, $4$, and $5$. What is the area of the triangle?
$\textbf{(A)}\ 6 \qquad
\textbf{(B)}\ \frac{18}{\pi^2} \qquad
\textbf{(C)}\ \frac{9}{\pi^2}\left(\sqrt{3}-1\right) \qquad
\textbf{(D)}\ \frac{9}{\pi^2}\left(\sqrt{3}+1\right) \qquad
\textbf{(E)}\ \frac{9}{\pi^2}\left(\sqrt{3}+3\right)$
2013 Sharygin Geometry Olympiad, 6
The altitudes $AA_1, BB_1, CC_1$ of an acute triangle $ABC$ concur at $H$. The perpendicular lines from $H$ to $B_1C_1, A_1C_1$ meet rays $CA, CB$ at $P, Q$ respectively. Prove that the line from $C$ perpendicular to $A_1B_1$ passes through the midpoint of $PQ$.
2015 AMC 10, 19
The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\angle{ACB}$ intersect $AB$ at $D$ and $E$. What is the area of $\triangle{CDE}$?
$\textbf{(A) }\frac{5\sqrt{2}}{3}\qquad\textbf{(B) }\frac{50\sqrt{3}-75}{4}\qquad\textbf{(C) }\frac{15\sqrt{3}}{8}\qquad\textbf{(D) }\frac{50-25\sqrt{3}}{2}\qquad\textbf{(E) }\frac{25}{6}$
1966 AMC 12/AHSME, 6
$AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, the length of chord $AC$, expressed in inches, is:
$\text{(A)} \ 3 \qquad \text{(B)} \ \frac{5\sqrt{2}}{2} \qquad \text{(C)} \frac{5\sqrt3}{2} \ \qquad \text{(D)} \ 3\sqrt3 \qquad \text{(E)} \ \text{none of these}$
2004 China Team Selection Test, 2
Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.
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]
2010 Contests, 1
$D, \: E , \: F$ are points on the sides $AB, \: BC, \: CA,$ respectively, of a triangle $ABC$ such that $AD=AF, \: BD=BE,$ and $DE=DF.$ Let $I$ be the incenter of the triangle $ABC,$ and let $K$ be the point of intersection of the line $BI$ and the tangent line through $A$ to the circumcircle of the triangle $ABI.$ Show that $AK=EK$ if $AK=AD.$
1979 AMC 12/AHSME, 24
Sides $AB,~ BC,$ and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4,~ 5,$ and $20$, respectively. If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B =\frac{3}{5} $, then side $AD$ has length
$\textbf{(A) }24\qquad\textbf{(B) }24.5\qquad\textbf{(C) }24.6\qquad\textbf{(D) }24.8\qquad\textbf{(E) }25$
[size=70]*A polygon is called “simple” if it is not self intersecting.[/size]
1953 AMC 12/AHSME, 34
If one side of a triangle is $ 12$ inches and the opposite angle is $ 30$ degrees, then the diameter of the circumscribed circle is:
$ \textbf{(A)}\ 18\text{ inches} \qquad\textbf{(B)}\ 30\text{ inches} \qquad\textbf{(C)}\ 24\text{ inches} \qquad\textbf{(D)}\ 20\text{ inches}\\
\textbf{(E)}\ \text{none of these}$
2014 Dutch IMO TST, 2
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.
2006 Switzerland Team Selection Test, 1
In the triangle $A,B,C$, let $D$ be the middle of $BC$ and $E$ the projection of $C$ on $AD$. Suppose $\angle ACE = \angle ABC$. Show that the triangle $ABC$ is isosceles or rectangle.
1992 AMC 12/AHSME, 25
In triangle $ABC$, $\angle ABC = 120^{\circ}$, $AB = 3$ and $BC = 4$. If perpendiculars constructed to $\overline{AB}$ at $A$ and to $\overline{BC}$ at $C$ meet at $D$, then $CD = $
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ \frac{8}{\sqrt{3}}\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ \frac{11}{2}\qquad\textbf{(E)}\ \frac{10}{\sqrt{3}} $
1987 AMC 12/AHSME, 30
In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$ and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. (Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC$.) The ratio $\frac{AD}{AB}$ is
[asy]
size((220));
draw((0,0)--(20,0)--(7,6)--cycle);
draw((6,6)--(10,-1));
label("A", (0,0), W);
label("B", (20,0), E);
label("C", (7,6), NE);
label("D", (9.5,-1), W);
label("E", (5.9, 6.1), SW);
label("$45^{\circ}$", (2.5,.5));
label("$60^{\circ}$", (7.8,.5));
label("$30^{\circ}$", (16.5,.5));
[/asy]
$ \textbf{(A)}\ \frac{1}{\sqrt{2}} \qquad\textbf{(B)}\ \frac{2}{2+\sqrt{2}} \qquad\textbf{(C)}\ \frac{1}{\sqrt{3}} \qquad\textbf{(D)}\ \frac{1}{\sqrt[3]{6}} \qquad\textbf{(E)}\ \frac{1}{\sqrt[4]{12}} $
2006 Macedonia National Olympiad, 4
Let $M$ be a point on the smaller arc $A_1A_n$ of the circumcircle of a regular $n$-gon $A_1A_2\ldots A_n$ .
$(a)$ If $n$ is even, prove that $\sum_{i=1}^n(-1)^iMA_i^2=0$.
$(b)$ If $n$ is odd, prove that $\sum_{i=1}^n(-1)^iMA_i=0$.
2007 Moldova Team Selection Test, 2
If $I$ is the incenter of a triangle $ABC$ and $R$ is the radius of its circumcircle then \[AI+BI+CI\leq 3R\]
2014 Baltic Way, 13
Let $ABCD$ be a square inscribed in a circle $\omega$ and let $P$ be a point on the shorter arc $AB$ of $\omega$. Let $CP\cap BD = R$ and $DP \cap AC = S.$
Show that triangles $ARB$ and $DSR$ have equal areas.
1990 Kurschak Competition, 2
The incenter of $\triangle A_1A_2A_3$ is $I$, and the center of the $A_i$-excircle is $J_i$ ($i=1,2,3$). Let $B_i$ be the intersection point of side $A_{i+1}A_{i+2}$ and the bisector of $\angle A_{i+1}IA_{i+2}$ ($A_{i+3}:=A_i$ $\forall i$). Prove that the three lines $B_iJ_i$ are concurrent.
2011 AMC 12/AHSME, 10
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$?
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 60 \qquad
\textbf{(E)}\ 75 $
2012 National Olympiad First Round, 29
Let $D$ and $E$ be points on $[BC]$ and $[AC]$ of acute $\triangle ABC$, respectively. $AD$ and $BE$ meet at $F$. If $|AF|=|CD|=2|BF|=2|CE|$, and $Area(\triangle ABF) = Area(\triangle DEC)$, then $Area(\triangle AFC)/Area(\triangle BFC) = ?$
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt2 \qquad \textbf{(E)}\ 1$
2001 India National Olympiad, 5
$ABC$ is a triangle. $M$ is the midpoint of $BC$. $\angle MAB = \angle C$, and $\angle MAC = 15^{\circ}$. Show that $\angle AMC$ is obtuse. If $O$ is the circumcenter of $ADC$, show that $AOD$ is equilateral.
2013 Math Prize For Girls Problems, 7
In the figure below, $\triangle ABC$ is an equilateral triangle.
[asy]
import graph;
unitsize(60);
axes("$x$", "$y$", (0, 0), (1.5, 1.5), EndArrow);
real w = sqrt(3) - 1;
pair A = (1, 1);
pair B = (0, w);
pair C = (w, 0);
draw(A -- B -- C -- cycle);
dot(Label("$A(1, 1)$", A, NE), A);
dot(Label("$B$", B, W), B);
dot(Label("$C$", C, S), C);
[/asy]
Point $A$ has coordinates $(1, 1)$, point $B$ is on the positive $y$-axis, and point $C$ is on the positive $x$-axis. What is the area of $\triangle ABC$?
2014 China Team Selection Test, 4
Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$.
(Edited)
2000 Brazil Team Selection Test, Problem 1
Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$.
[color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]
2011 Switzerland - Final Round, 2
Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that \[\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}\] Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 2)[/i]