Found problems: 321
2018 Estonia Team Selection Test, 5
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
1957 Moscow Mathematical Olympiad, 362
(a) A circle is inscribed in a triangle. The tangent points are the vertices of a second triangle in which another circle is inscribed. Its tangency points are the vertices of a third triangle. The angles of this triangle are identical to those of the first triangle. Find these angles.
(b) A circle is inscribed in a scalene triangle. The tangent points are vertices of another triangle, in which a circle is inscribed whose tangent points are vertices of a third triangle, in which a third circle is inscribed, etc. Prove that the resulting sequence does not contain a pair of similar triangles.
1982 IMO Shortlist, 17
The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.
1990 Mexico National Olympiad, 6
$ABC$ is a triangle with $\angle C = 90^o$. $E$ is a point on $AC$, and $F$ is the midpoint of $EC$. $CH$ is an altitude. $I$ is the circumcenter of $AHE$, and $G$ is the midpoint of $BC$. Show that $ABC$ and $IGF$ are similar.
2007 Stanford Mathematics Tournament, 13
A rope of length 10 [i]m[/i] is tied tautly from the top of a flagpole to the ground 6 [i]m[/i] away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 [i]cm/min[/i]. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)
1996 China National Olympiad, 1
Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.
2025 Kosovo National Mathematical Olympiad`, P3
Let $g_a$, $g_b$ and $g_c$ be the medians of a triangle $\triangle ABC$ erected from the vertices $A$, $B$ and $C$, respectively.
Similarly, let $g_x$, $g_y$ and $g_z$ be the medians of an another triangle $\triangle XYZ$. Show that if
$$g_a : g_b : g_c = g_x : g_y : g_z, $$
then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.
2021 Sharygin Geometry Olympiad, 10-11.7
Let $I$ be the incenter of a right-angled triangle $ABC$, and $M$ be the midpoint of hypothenuse $AB$. The tangent to the circumcircle of $ABC$ at $C$ meets the line passing through $I$ and parallel to $AB$ at point $P$. Let $H$ be the orthocenter of triangle $PAB$. Prove that lines $CH$ and $PM$ meet at the incircle of triangle $ABC$.
1990 APMO, 1
Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?
2009 International Zhautykov Olympiad, 2
Given a quadrilateral $ ABCD$ with $ \angle B\equal{}\angle D\equal{}90^{\circ}$. Point $ M$ is chosen on segment $ AB$ so taht $ AD\equal{}AM$. Rays $ DM$ and $ CB$ intersect at point $ N$. Points $ H$ and $ K$ are feet of perpendiculars from points $ D$ and $ C$ to lines $ AC$ and $ AN$, respectively.
Prove that $ \angle MHN\equal{}\angle MCK$.
2009 Baltic Way, 14
For which $n\ge 2$ is it possible to find $n$ pairwise non-similar triangles $A_1, A_2,\ldots , A_n$ such that each of them can be divided into $n$ pairwise non-similar triangles, each of them similar to one of $A_1,A_2 ,\ldots ,A_n$?
Cono Sur Shortlist - geometry, 1993.8
In a triangle $ABC$, let $D$, $E$ and $F$ be the touchpoints of the inscribed circle and the sides $AB$, $BC$ and $CA$. Show that the triangles $DEF$ and $ABC$ are similar if and only if $ABC$ is equilateral.
2010 National Olympiad First Round, 7
A frog is at the center of a circular shaped island with radius $r$. The frog jumps $1/2$ meters at first. After the first jump, it turns right or left at exactly $90^\circ$, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least $r$ that yields the frog can never fall into the water?
$ \textbf{(A)}\ \frac{\sqrt 5}{3}
\qquad\textbf{(B)}\ \frac{\sqrt {13}}{5}
\qquad\textbf{(C)}\ \frac{\sqrt {19}}{6}
\qquad\textbf{(D)}\ \frac{1}{\sqrt 2}
\qquad\textbf{(E)}\ \frac34
$
1984 AMC 12/AHSME, 27
In $\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB \perp AC, AF \perp BC$, and $BD = DC = FC = 1$. Find $AC$.
A. $\sqrt{2}$
B. $\sqrt{3}$
C. $\sqrt[3] {2}$
D. $\sqrt[3] {3}$
E. $\sqrt[4] {3}$
2005 Bulgaria National Olympiad, 4
Let $ABC$ be a triangle with $AC\neq BC$, and let $A^{\prime }B^{\prime }C$ be a triangle obtained from $ABC$ after some rotation centered at $C$. Let $M,E,F$ be the midpoints of the segments $BA^{\prime },AC$ and $CB^{\prime }$ respectively. If $EM=FM$, find $\widehat{EMF}$.
2006 Mediterranean Mathematics Olympiad, 3
The side lengths $a,b,c$ of a triangle $ABC$ are integers with $\gcd(a,b,c)=1$. The bisector of angle $BAC$ meets $BC$ at $D$.
(a) show that if triangles $DBA$ and $ABC$ are similar then $c$ is a square.
(b) If $c=n^2$ is a square $(n\ge 2)$, find a triangle $ABC$ satisfying (a).
2010 Romania National Olympiad, 4
On the exterior of a non-equilateral triangle $ABC$ consider the similar triangles $ABM,BCN$ and $CAP$, such that the triangle $MNP$ is equilateral. Find the angles of the triangles $ABM,BCN$ and $CAP$.
[i]Nicolae Bourbacut[/i]
2005 All-Russian Olympiad Regional Round, 9.4
9.4, 10.3 Let $I$ be an incenter of $ABC$ ($AB<BC$), $M$ is a midpoint of $AC$, $N$ is a midpoint of circumcircle's arc $ABC$. Prove that $\angle IMA=\angle INB$.
([i]A. Badzyan[/i])
1985 ITAMO, 4
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985.
[asy]
size(200);
pair A=(0,1), B=(1,1), C=(1,0), D=origin;
draw(A--B--C--D--A--(1,1/6));
draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1));
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("$1/n$", (11/12,1), N, fontsize(9));[/asy]
2001 IMO Shortlist, 8
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?
Estonia Open Junior - geometry, 2008.1.3
Let $M$ be the intersection of the medians $ABC$ of the triangle and the midpoint of the side $BC$. $A$ line parallel to side $BC$ and passing through point $M$ intersects sides $AB$ and $AC$ at points $X$ and $Y$ respectively. Let the point of intersection of the lines $XC$ and $MB$ be $Q$ and let $P$ intersection point of the lines $YB$ and $MC$ be $P$ . Prove that the triangles $DPQ$ and $ABC$ are similar.
2005 Manhattan Mathematical Olympiad, 4
Circle of radius $r$ is inscribed in a triangle. Tangent lines parallel to the sides of triangle cut three small triangles. Let $r_1,r_2,r_3$ be radii of circles inscribed in these triangles. Prove that \[ r_1 + r_2 + r_3 = r. \]
2018 Romania Team Selection Tests, 1
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
2013 Online Math Open Problems, 40
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Let $M$ be the midpoint of $BC$ and let $\Gamma$ be the circle passing through $A$ and tangent to line $BC$ at $M$. Let $\Gamma$ intersect lines $AB$ and $AC$ at points $D$ and $E$, respectively, and let $N$ be the midpoint of $DE$. Suppose line $MN$ intersects lines $AB$ and $AC$ at points $P$ and $O$, respectively. If the ratio $MN:NO:OP$ can be written in the form $a:b:c$ with $a,b,c$ positive integers satisfying $\gcd(a,b,c)=1$, find $a+b+c$.
[i]James Tao[/i]
1972 AMC 12/AHSME, 18
Let $ABCD$ be a trapezoid with the measure of base $AB$ twice that of base $DC$, and let $E$ be the point of intersection of the diagonals. If the measure of diagonal $AC$ is $11$, then that of segment $EC$ is equal to
$\textbf{(A) }3\textstyle\frac{2}{3}\qquad\textbf{(B) }3\frac{3}{4}\qquad\textbf{(C) }4\qquad\textbf{(D) }3\frac{1}{2}\qquad \textbf{(E) }3$