Found problems: 25757
Champions Tournament Seniors - geometry, 2015.3
Given a triangle $ABC$. Let $\Omega$ be the circumscribed circle of this triangle, and $\omega$ be the inscribed circle of this triangle. Let $\delta$ be a circle that touches the sides $AB$ and $AC$, and also touches the circle $\Omega$ internally at point $D$. The line $AD$ intersects the circle $\Omega$ at two points $P$ and $Q$ ($P$ lies between $A$ and $Q$). Let $O$ and $I$ be the centers of the circles $\Omega$ and $\omega$. Prove that $OD \parallel IQ$.
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?
2007 Stanford Mathematics Tournament, 10
A nondegenerate rhombus has side length $l$, and its area is twice that of its inscribed circle. Find the radius of the inscribed circle.
2022 Kazakhstan National Olympiad, 5
Given a cyclic quadrilateral $ABCD$, let it's diagonals intersect at the point $O$. Take the midpoints of $AD$ and $BC$ as $M$ and $N$ respectively. Take a point $S$ on the arc $AB$ not containing $C$ or $D$ such that $$\angle SMA=\angle SNB$$ Prove that if the diagonals of the quadrilateral made from the lines $SM$, $SN$, $AB$, and $CD$ intersect at the point $T$, then $S$, $O$, and $T$ are collinear.
2025 Macedonian Balkan MO TST, 2
Let $\triangle ABC$ be an acute-angled triangle and $A_1, B_1$, and $C_1$ be the feet of the altitudes from $A, B$, and $C$, respectively. On the rays $AA_1, BB_1$, and $CC_1$, we have points $A_2, B_2$, and $C_2$ respectively, lying outside of $\triangle ABC$, such that
\[\frac{A_1A_2}{AA_1} = \frac{B_1B_2}{BB_1} = \frac{C_1C_2}{CC_1}.\]
If the intersections of $B_1C_2$ and $B_2C_1$, $C_1A_2$ and $C_2A_1$, and $A_1B_2$ and $A_2B_1$ are $A', B'$, and $C'$ respectively, prove that $AA', BB'$, and $CC'$ have a common point.
2012 ELMO Shortlist, 2
In triangle $ABC$, $P$ is a point on altitude $AD$. $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$, and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$.
a) Prove $SX,TY, AD$ are concurrent at a point $Z$.
b) Prove $Z$ is on $QR$ iff $Z=H$, where $H$ is the orthocenter of $ABC$.
[i]Ray Li.[/i]
MOAA Team Rounds, 2022.3
The area of the figure enclosed by the $x$-axis, $y$-axis, and line $7x + 8y = 15$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2009 Kyiv Mathematical Festival, 4
Two convex polygons can be placed into a square with the side $1$ without intersection. Prove that at least one polygon has the perimeter that is less than or equal to $3,5$ .
1997 Kurschak Competition, 2
The center of the circumcircle of $\triangle ABC$ is $O$. The incenter of the triangle is $I$, and the intouch triangle is $A_1B_1C_1$. Let $H_1$ be the orthocenter of $\triangle A_1B_1C_1$. Prove that $O$, $I$, and $H_1$ are collinear.
2011 JHMT, 8
Two parallel lines $\ell_1$ and $\ell_2$ lie on a plane, distance $d$ apart. On $\ell_1$ there are an infinite number of points $A_1, A_2, A_3, ...$ , in that order, with $A_nA_{n+1} = 2$ for all $n$. On $\ell_2$ there are an infinite number of points $B_1, B_2, B_3,...$ , in that order and in the same direction, satisfying $B_nB_{n+1} = 1$ for all $n$. Given that $A_1B_1$ is perpendicular to both $\ell_1$ and $\ell_2$, express the sum $\sum_{i=1}^{\infty} \angle A_iB_iA_{i+1}$ in terms of $d$.
[img]https://cdn.artofproblemsolving.com/attachments/c/9/24b8000e19cffb401234be010af78a6eb67524.png[/img]
2012 Kyoto University Entry Examination, 3
When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$
Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$
30 points
2018 China Girls Math Olympiad, 8
Let $I$ be the incenter of triangle $ABC$. The tangent point of $\odot I$ on $AB,AC$ is $D,E$, respectively. Let $BI \cap AC = F$, $CI \cap AB = G$, $DE \cap BI = M$, $DE \cap CI = N$, $DE \cap FG = P$, $BC \cap IP = Q$. Prove that $BC = 2MN$ is equivalent to $IQ = 2IP$.
2001 Kazakhstan National Olympiad, 6
Each interior point of an equilateral triangle with sides equal to $1$ lies in one of six circles of the same radius $ r $. Prove that $ r \geq \frac {{\sqrt 3}} {{10}} $.
2017 Azerbaijan Team Selection Test, 2
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2012 Online Math Open Problems, 23
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]Author: Ray Li[/i]
2017 Iran Team Selection Test, 6
In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$. Assume that $P$ is not on the same side of $BC$ as $A$. Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ , CF=PB$. Let $K$ be the intersection point of $AP,OH$. Prove that $\angle EKF = 90 ^{\circ}$
[i] Proposed by Iman Maghsoudi[/i]
2010 Czech And Slovak Olympiad III A, 4
A circle $k$ is given with a non-diameter chord $AC$. On the tangent at point $A$ select point $X \ne A$ and mark $D$ the intersection of the circle $k$ with the interior of the line $XC$ (if any). Let $B$ a point in circle $k$ such that quadrilateral $ABCD$ is a trapezoid . Determine the set of intersections of lines $BC$ and $AD$ belonging to all such trapezoids.
2005 Flanders Junior Olympiad, 2
Starting with two points A and B, some circles and points are constructed as shown in
the figure:[list][*]the circle with centre A through B
[*]the circle with centre B through A
[*]the circle with centre C through A
[*]the circle with centre D through B
[*]the circle with centre E through A
[*]the circle with centre F through A
[*]the circle with centre G through A[/list]
[i][size=75](I think the wording is not very rigorous, you should assume intersections from the drawing)[/size][/i]
Show that $M$ is the midpoint of $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/d/4/2352ab21cc19549f0381e88ddde9dce4299c2e.png[/img]
2004 Romania National Olympiad, 2
Let $ABCD$ be a tetrahedron in which the opposite sides are equal and form equal angles.
Prove that it is regular.
2020 Canada National Olympiad, 2
$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.
1980 All Soviet Union Mathematical Olympiad, 289
Given a point $E$ on the diameter $AC$ of the certain circle. Draw a chord $BD$ to maximise the area of the quadrangle $ABCD$.
2003 National Olympiad First Round, 10
Which of the followings is congruent (in $\bmod{25}$) to the sum in of integers $0\leq x < 25$ such that $x^3+3x^2-2x+4 \equiv 0 \pmod{25}$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 17
\qquad\textbf{(D)}\ 22
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2019 Vietnam National Olympiad, Day 2
Let $ABC$ be an acute, nonisosceles triangle with inscribe in a circle $(O)$ and has orthocenter $H$. Denote $M,N,P$ as the midpoints of sides $BC,CA,AB$ and $D,E,F$ as the feet of the altitudes from vertices $A,B,C$ of triangle $ABC$. Let $K$ as the reflection of $H$ through $BC$. Two lines $DE,MP$ meet at $X$; two lines $DF,MN$ meet at $Y$.
a) The line $XY$ cut the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.
b) Two lines $KE,KF$ cuts $(O)$ second time at $S,T$. Prove that $BS,CT,XY$ are concurrent.
2016 ASMT, 6
Let $ABC$ be a triangle with $AB = 5$ and $AC = 4$. Let $D$ be the reflection of $C$ across $AB$, and let $E$ be the reflection of $B$ across $AC$. $D$ and $E$ have the special property that $D, A, E$ are collinear. Finally, suppose that lines $DB$ and $EC$ intersect at a point $F$. Compute the area of $\vartriangle BCF$.
2006 Korea National Olympiad, 4
On the circle $O,$ six points $A,B,C,D,E,F$ are on the circle counterclockwise. $BD$ is the diameter of the circle and it is perpendicular to $CF.$ Also, lines $CF, BE, AD$ is concurrent. Let $M$ be the foot of altitude from $B$ to $AC$ and let $N$ be the foot of altitude from $D$ to $CE.$ Prove that the area of $\triangle MNC$ is less than half the area of $\square ACEF.$