Found problems: 9
2014 South East Mathematical Olympiad, 1
Let $ABC$ be a triangle with $AB<AC$ and let $M $ be the midpoint of $BC$. $MI$ ($I$ incenter) intersects $AB$ at $D$ and $CI$ intersects the circumcircle of $ABC$ at $E$. Prove that $\frac{ED }{ EI} = \frac{IB }{IC}$
[img]https://cdn.artofproblemsolving.com/attachments/0/5/4639d82d183247b875128a842a013ed7415fba.jpg[/img]
[hide=.][url=http://artofproblemsolving.com/community/c6h602657p10667541]source[/url], translated by Antreas Hatzipolakis in fb, corrected by me in order to be compatible with it's figure[/hide]
2003 JHMMC 8, 3
On an exam with $80$ problems, Roger solved $68$ of them. What percentage of the problems did he solve?
2013 APMO, 1
Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.
2013 Brazil Team Selection Test, 1
Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.
2013 USAJMO, 5
Quadrilateral $XABY$ is inscribed in the semicircle $\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \[\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.\]
2017 AMC 10, 11
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
$\textbf{(A) } 10\%\qquad \textbf{(B) } 12\%\qquad \textbf{(C) } 20\%\qquad \textbf{(D) } 25\%\qquad \textbf{(E) } 33\frac{1}{3}\%$
2012 Sharygin Geometry Olympiad, 2
We say that a point inside a triangle is good if the lengths of the cevians passing through this point are inversely proportional to the respective side lengths. Find all the triangles for which the number of good points is maximal.
(A.Zaslavsky, B.Frenkin)
2023 ISI Entrance UGB, 3
In $\triangle ABC$, consider points $D$ and $E$ on $AC$ and $AB$, respectively, and assume that they do not coincide with any of the vertices $A$, $B$, $C$. If the segments $BD$ and $CE$ intersect at $F$, consider areas $w$, $x$, $y$, $z$ of the quadrilateral $AEFD$ and the triangles $BEF$, $BFC$, $CDF$, respectively.
[list=a]
[*] Prove that $y^2 > xz$.
[*] Determine $w$ in terms of $x$, $y$, $z$.
[/list]
[asy]
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(12); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -2.8465032978885407, xmax = 9.445649196374966, ymin = -1.7618066305534972, ymax = 4.389732795464592; /* image dimensions */
draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282)--cycle, linewidth(0.5));
/* draw figures */
draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117), linewidth(0.5));
draw((-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282), linewidth(0.5));
draw((5.672613975760373,-0.636902634996282)--(3.8295013012181283,2.816337276198864), linewidth(0.5));
draw((-0.7368327629589799,-0.5920813291311117)--(4.569287648059735,1.430279997142299), linewidth(0.5));
draw((5.672613975760373,-0.636902634996282)--(1.8844000180622977,1.3644681598392678), linewidth(0.5));
label("$y$",(2.74779188172294,0.23771684184669772),SE*labelscalefactor);
label("$w$",(3.2941097703568736,1.8657441499758196),SE*labelscalefactor);
label("$x$",(1.6660824622277512,1.0025618859342047),SE*labelscalefactor);
label("$z$",(4.288408327670633,0.8168138037986672),SE*labelscalefactor);
/* dots and labels */
dot((3.8295013012181283,2.816337276198864),dotstyle);
label("$A$", (3.8732067323088435,2.925600853925651), NE * labelscalefactor);
dot((-0.7368327629589799,-0.5920813291311117),dotstyle);
label("$B$", (-1.1,-0.7565817154670613), NE * labelscalefactor);
dot((5.672613975760373,-0.636902634996282),dotstyle);
label("$C$", (5.763466626982254,-0.7784344310124186), NE * labelscalefactor);
dot((4.569287648059735,1.430279997142299),dotstyle);
label("$D$", (4.692683565259744,1.5051743434774234), NE * labelscalefactor);
dot((1.8844000180622977,1.3644681598392678),dotstyle);
label("$E$", (1.775346039954538,1.4942479857047448), NE * labelscalefactor);
dot((2.937230516274804,0.8082418657164665),linewidth(4.pt) + dotstyle);
label("$F$", (2.889834532767763,0.954), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]
2017 AMC 12/AHSME, 10
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
$\textbf{(A) } 10\%\qquad \textbf{(B) } 12\%\qquad \textbf{(C) } 20\%\qquad \textbf{(D) } 25\%\qquad \textbf{(E) } 33\frac{1}{3}\%$