Found problems: 1001
2018 Stars of Mathematics, 1
Let $ABC$ be a triangle, and let $\ell$ be the line through $A$ and perpendicular to the line $BC$. The reflection of $\ell$ in the line $AB$ crosses the line through $B$ and perpendicular to $AB$ at $P$. The reflection of $\ell$ in the line $AC$ crosses the line through $C$ and perpendicular to $AC$ at $Q$. Show that the line $PQ$ passes through the orthocenter of the triangle $ABC$.
Flavian Georgescu
2010 JBMO Shortlist, 1
$\textbf{Problem G1}$
Consider a triangle $ABC$ with $\angle ACB=90^{\circ}$. Let $F$ be the foot of the altitude from
$C$. Circle $\omega$ touches the line segment $FB$ at point $P$, the altitude $CF$ at point $Q$ and the
circumcircle of $ABC$ at point $R$. Prove that points $A, Q, R$ are collinear and $AP = AC$.
2013 Romanian Master of Mathematics, 4
Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?
2017 Baltic Way, 11
Let $H$ and $I$ be the orthocenter and incenter, respectively, of an acute-angled triangle $ABC$. The circumcircle of the triangle $BCI$ intersects the segment $AB$ at the point $P$ different from $B$. Let $K$ be the projection of $H$ onto $AI$ and $Q$ the reflection of $P$ in $K$. Show that $B$, $H$ and $Q$ are collinear.
[i]Proposed by Mads Christensen, Denmark[/i]
2013 China Team Selection Test, 1
The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.
2018 Iran MO (1st Round), 21
The point $P$ is chosen inside or on the equilateral triangle $ABC$ of side length $1$. The reflection of $P$ with respect to $AB$ is $K$, the reflection of $K$ about $BC$ is $M$, and the reflection of $M$ with respect to $AC$ is $N$. What is the maximum length of $NP$?
$\textbf{(A)}\ 2\sqrt 3\qquad\textbf{(B)}\ \sqrt 3\qquad\textbf{(C)}\ \frac{\sqrt 3}{2} \qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 1$
2013 Romanian Masters In Mathematics, 1
Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?
2011 IMO Shortlist, 8
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$.
[i]Proposed by Japan[/i]
2010 APMO, 4
Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.
2013 Harvard-MIT Mathematics Tournament, 16
The walls of a room are in the shape of a triangle $ABC$ with $\angle ABC = 90^\circ$, $\angle BAC = 60^\circ$, and $AB=6$. Chong stands at the midpoint of $BC$ and rolls a ball toward $AB$. Suppose that the ball bounces off $AB$, then $AC$, then returns exactly to Chong. Find the length of the path of the ball.
1995 India National Olympiad, 1
In an acute angled triangle $ABC$, $\angle A = 30^{\circ}$, $H$ is the orthocenter, and $M$ is the midpoint of $BC$. On the line $HM$, take a point $T$ such that $HM = MT$. Show that $AT = 2 BC$.
2012 IMO Shortlist, G6
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. The points $D,E$ and $F$ on the sides $BC,CA$ and $AB$ respectively are such that $BD+BF=CA$ and $CD+CE=AB$. The circumcircles of the triangles $BFD$ and $CDE$ intersect at $P \neq D$. Prove that $OP=OI$.
2014 China Girls Math Olympiad, 6
In acute triangle $ABC$, $AB > AC$.
$D$ and $E$ are the midpoints of $AB$, $AC$ respectively.
The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$.
The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$.
Prove that $AP = AQ$.
1984 Balkan MO, 2
Let $ABCD$ be a cyclic quadrilateral and let $H_{A}, H_{B}, H_{C}, H_{D}$ be the orthocenters of the triangles $BCD$, $CDA$, $DAB$ and $ABC$ respectively. Show that the quadrilaterals $ABCD$ and $H_{A}H_{B}H_{C}H_{D}$ are congruent.
2007 Kyiv Mathematical Festival, 2
The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$
2022 Sharygin Geometry Olympiad, 8.1
Let $ABCD$ be a convex quadrilateral with $\angle{BAD} = 2\angle{BCD}$ and $AB = AD$. Let $P$ be a point such that $ABCP$ is a parallelogram. Prove that $CP = DP$.
1998 USAMTS Problems, 3
The integers from $1$ to $9$ can be arranged into a $3\times3$ array (as shown on the right) so that the sum of the numbers in every row, column, and diagonal is a multiple of $9$.
(a.) Prove that the number in the center of the array must be a multiple of $3$.
(b.) Give an example of such an array with $6$ in the center.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));size(100);
int i,j;
for(i=0; i<4; i=i+1) {
draw((0,2i)--(6,2i));
draw((2i,0)--(2i,6));
}
string[] letters={"G", "H", "I", "D", "E", "F", "A", "B", "C"};
for(i=0; i<3; i=i+1) {
for(j=0; j<3; j=j+1) {
label(letters[3i+j], (2j+1, 2i+1));
}}[/asy]
2013 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle which it not right-angled. De fine a sequence of triangles $A_iB_iC_i$, with $i \ge 0$, as follows: $A_0B_0C_0$ is the triangle $ABC$ and, for $i \ge 0$, $A_{i+1},B_{i+1},C_{i+1}$ are the reflections of the orthocentre of triangle $A_iB_iC_i$ in the sides $B_iC_i$,$C_iA_i$,$A_iB_i$, respectively.
Assume that $\angle A_m = \angle A_n$ for some distinct natural numbers $m,n$. Prove that $\angle A = 60^{\circ}$.
Estonia Open Senior - geometry, 2018.1.5
The midpoints of the sides $BC, CA$, and $AB$ of triangle $ABC$ are $D, E$, and $F$, respectively. The reflections of centroid $M$ of $ABC$ around points $D, E$, and $F$ are $X, Y$, and $Z$, respectively. Segments $XZ$ and $YZ$ intersect the side $AB$ in points $K$ and $L$, respectively. Prove that $AL = BK$.
2014 Switzerland - Final Round, 1
The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.
1969 IMO, 4
$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.
2008 All-Russian Olympiad, 6
In a scalene triangle $ ABC$ the altitudes $ AA_{1}$ and $ CC_{1}$ intersect at $ H, O$ is the circumcenter, and $ B_{0}$ the midpoint of side $ AC$. The line $ BO$ intersects side $ AC$ at $ P$, while the lines $ BH$ and $ A_{1}C_{1}$ meet at $ Q$. Prove that the lines $ HB_{0}$ and $ PQ$ are parallel.
2007 Germany Team Selection Test, 3
Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.
2014 Iran MO (3rd Round), 2
$\triangle{ABC}$ is isosceles$(AB=AC)$. Points $P$ and $Q$ exist inside the triangle such that $Q$ lies inside $\widehat{PAC}$ and $\widehat{PAQ} = \frac{\widehat{BAC}}{2}$. We also have $BP=PQ=CQ$.Let $X$ and $Y$ be the intersection points of $(AP,BQ)$ and $(AQ,CP)$ respectively. Prove that quadrilateral $PQYX$ is cyclic. [i](20 Points)[/i]
2008 Balkan MO, 1
Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.