Found problems: 25757
2022 Taiwan TST Round 2, G
Let $I$, $O$, $H$, and $\Omega$ be the incenter, circumcenter, orthocenter, and the circumcircle of the triangle $ABC$, respectively. Assume that line $AI$ intersects with $\Omega$ again at point $M\neq A$, line $IH$ and $BC$ meets at point $D$, and line $MD$ intersects with $\Omega$ again at point $E\neq M$. Prove that line $OI$ is tangent to the circumcircle of triangle $IHE$.
[i]Proposed by Li4 and Leo Chang.[/i]
2014 Sharygin Geometry Olympiad, 3
Let $ABC$ be an isosceles triangle with base $AB$. Line $\ell$ touches its circumcircle at point $B$. Let $CD$ be a perpendicular from $C$ to $\ell$, and $AE$, $BF$ be the altitudes of $ABC$. Prove that $D$, $E$, and $F$ are collinear.
1983 IMO Shortlist, 13
Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?
2019 Germany Team Selection Test, 2
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2018 Dutch IMO TST, 4
In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.
2004 Korea - Final Round, 2
An acute triangle $ABC$ has circumradius $R$, inradius $r$. $A$ is the biggest angle among $A,B,C$. Let $M$ be the midpoint of $BC$, and $X$ be the intersection of two lines that touches circumcircle of $ABC$ and goes through $B,C$ respectively. Prove the following inequality : $ \frac{r}{R} \geq \frac{AM}{AX}$.
2010 Contests, 1
Two circles $\Gamma_1$ and $\Gamma_2$ meet at $A$ and $B$. A line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ repsectively. Another line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $E$ and $F$ repsectively. Line $CF$ meets $\Gamma_1$ and $\Gamma_2$ again at $P$ and $Q$ respectively. $M$ and $N$ are midpoints of arc $PB$ and arc $QB$ repsectively. Show that if $CD = EF$, then $C,F,M,N$ are concyclic.
2020 Stars of Mathematics, 2
Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular.
[i]Freddie Illingworth & Dominic Yeo[/i]
2020-IMOC, G6
Let $ABC$ be a triangle, and $M_a, M_b, M_c$ be the midpoints of $BC, CA, AB$, respectively. Extend $M_bM_c$ so that it intersects $\odot (ABC)$ at $P$. Let $AP$ and $BC$ intersect at $Q$. Prove that the tangent at $A$ to $\odot(ABC)$ and the tangent at $P$ to $\odot (P QM_a)$ intersect on line $BC$.
(Li4)
2019 HMNT, 9
Will stands at a point $P$ on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of $n^o$ and $(n + 1)^o$ with the tangent at $P$, where $n$ is a positive integer less than $90$. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach $P$ again. ($P$ is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room?
[img]https://cdn.artofproblemsolving.com/attachments/a/9/5548d7b34551369d1b69eae682855bcc406f9e.jpg[/img]
1983 AIME Problems, 4
A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle.
[asy]
size(150); defaultpen(linewidth(0.65)+fontsize(11));
real r=10;
pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C;
path P=circle(O,r);
C=intersectionpoint(B--(B.x+r,B.y),P);
draw(Arc(O, r, 45, 360-17.0312));
draw(A--B--C);dot(A); dot(B); dot(C);
label("$A$",A,NE);
label("$B$",B,SW);
label("$C$",C,SE);
[/asy]
2018 EGMO, 1
Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$.
Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.
2006 Sharygin Geometry Olympiad, 20
Four points are given $A, B, C, D$. Points $A_1, B_1, C_1,D_1$ are orthocenters of the triangles $BCD, CDA, DAB, ABC$ and $A_2, B_2, C_2,D_2$ are orthocenters of the triangles $B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1$ etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.
2023 JBMO Shortlist, G6
Let $ABC$ be an acute triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $M$ be the midpoint of $OD$. The points $O_b$ and $O_c$ are the circumcenters of triangles $AOC$ and $AOB$, respectively. If $AO=AD$, prove that points $A$, $O_b$, $M$ and $O_c$ are concyclic.
[i]Marin Hristov and Bozhidar Dimitrov, Bulgaria[/i]
1993 IberoAmerican, 2
Show that for every convex polygon whose area is less than or equal to $1$, there exists a parallelogram with area $2$ containing the polygon.
2004 National Olympiad First Round, 33
Let $ABCD$ be a trapezoid such that $|AB|=9$, $|CD|=5$ and $BC\parallel AD$. Let the internal angle bisector of angle $D$ meet the internal angle bisectors of angles $A$ and $C$ at $M$ and $N$, respectively. Let the internal angle bisector of angle $B$ meet the internal angle bisectors of angles $A$ and $C$ at $L$ and $K$, respectively. If $K$ is on $[AD]$ and $\dfrac{|LM|}{|KN|} = \dfrac 37$, what is $\dfrac{|MN|}{|KL|}$?
$
\textbf{(A)}\ \dfrac{62}{63}
\qquad\textbf{(B)}\ \dfrac{27}{35}
\qquad\textbf{(C)}\ \dfrac{2}{3}
\qquad\textbf{(D)}\ \dfrac{5}{21}
\qquad\textbf{(E)}\ \dfrac{24}{63}
$
2014 Sharygin Geometry Olympiad, 21
Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.
2001 National Olympiad First Round, 1
Let $A,B,C$ be points on $[OX$ and $D,E,F$ be points on $[OY$ such that $|OA|=|AB|=|BC|$ and $|OD|=|DE|=|EF|$. If $|OA|>|OD|$, which one below is true?
$\textbf{(A)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)>\text{Area}(DBF)$
$\textbf{(B)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)=\text{Area}(DBF)$
$\textbf{(C)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)<\text{Area}(DBF)$
$\textbf{(D)}$ If $m(\widehat{XOY})<45^\circ$ then $\text{Area}(AEC)<\text{Area}(DBF)$, and if $45^\circ < m(\widehat{XOY})<90^\circ$ then $\text{Area}(AEC)>\text{Area}(DBF)$
$\textbf{(E)}$ None of above
2002 AMC 10, 13
The sides of a triangle have lengths of $ 15$, $ 20$, and $ 25$. Find the length of the shortest altitude.
$ \text{(A)}\ 6 \qquad
\text{(B)}\ 12 \qquad
\text{(C)}\ 12.5 \qquad
\text{(D)}\ 13 \qquad
\text{(E)}\ 15$
2022 Bolivia IMO TST, P3
On $\triangle ABC$, let $M$ the midpoint of $AB$ and $N$ the midpoint of $CM$. Let $X$ a point such that $\angle XMC=\angle MBC$ and $\angle XCM=\angle MCB$ with $X,B$ in opposite sides of line $CM$. Let $\Omega$ the circumcircle of triangle $\triangle AMX$
[b]a)[/b] Show that $CM$ is tangent to $\Omega$
[b]b)[/b] Show that the lines $NX$ and $AC$ meet at $\Omega$
2008 Harvard-MIT Mathematics Tournament, 1
Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$, and also that $ AX \equal{} \frac {\sqrt {2}}{2}$. Determine the value of $ CX^2$.
2020 Novosibirsk Oral Olympiad in Geometry, 2
A $2 \times 2$ square was cut out of a sheet of grid paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.
2010 JBMO Shortlist, 4
Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.
2005 Harvard-MIT Mathematics Tournament, 9
Let $AC$ be a diameter of a circle $ \omega $ of radius $1$, and let $D$ be a point on $AC$ such that $CD=\frac{1}{5}$. Let $B$ be the point on $\omega$ such that $DB$ is perpendicular to $AC$, and $E$ is the midpoint of $DB$. The line tangent to $\omega$ at $B$ intersects line $CE$ at the point $X$. Compute $AX$.
2006 Singapore Senior Math Olympiad, 2
Let $ABCD$ be a cyclic quadrilateral, let the angle bisectors at $A$ and $B$ meet at $E$, and let the line through $E$ parallel to side $CD$ intersect $AD$ at $L$ and $BC$ at $M$. Prove that $LA + MB = LM$.