Found problems: 25757
2024 Malaysian IMO Training Camp, 8
Given a triangle $ABC$, let $I$ be the incenter, and $J$ be the $A$-excenter. A line $\ell$ through $A$ perpendicular to $BC$ intersect the lines $BI$, $CI$, $BJ$, $CJ$ at $P$, $Q$, $R$, $S$ respectively. Suppose the angle bisector of $\angle BAC$ meet $BC$ at $K$, and $L$ is a point such that $AL$ is a diameter in $(ABC)$.
Prove that the line $KL$, $\ell$, and the line through the centers of circles $(IPQ)$ and $(JRS)$, are concurrent.
[i]Proposed by Chuah Jia Herng & Ivan Chan Kai Chin[/i]
2018 Hanoi Open Mathematics Competitions, 5
Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose that $AH,AO$ and $AM$ are the altitude, the bisector and the median derived from $A$, respectively. If $HO = 3 MO$, then the length of $BC$ is
[img]https://cdn.artofproblemsolving.com/attachments/e/c/26cc00629f4c0ab27096b8bdc562c56ff01ce5.png[/img]
A. $3$ B. $\frac72$ C. $4$ D. $\frac92$ E. $5$
2013 Bangladesh Mathematical Olympiad, 9
Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AB, CD, EF$ is a diameter. Extended
$AB$ and $DC$ meet at $Z, CD$ and $FE$ at $X, EF$ and $BA$ at $Y. AC$ and $BF$ meets at $P, CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R.$ If $O$ is the point of intersection of $YQ$ and $ZR,$ find the $\angle XOP.$
2013 Princeton University Math Competition, 5
Circle $w$ with center $O$ meets circle $\Gamma$ at $X,Y,$ and $O$ is on $\Gamma$. Point $Z\in\Gamma$ lies outside $w$ such that $XZ=11$, $OZ=15$, and $YZ=13$. If the radius of circle $w$ is $r$, find $r^2$.
2024 China Western Mathematical Olympiad, 5
Given hexagon $ \mathcal{P}$ inscribed in a unit square, such that each vertex is on the side of the square. It’s known that all interior angles of the hexagon are equal. Find the maximum possible value of the smallest side length of $\mathcal{P}$.
2012 AMC 12/AHSME, 12
A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$. Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$. What is the side length of this square?
$ \textbf{(A)}\ \frac{\sqrt{10}+5}{10}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{5}\qquad\textbf{(C)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(D)}\ \frac{2\sqrt{19}-4}{5}\qquad\textbf{(E)}\ \frac{9-\sqrt{17}}{5} $
2018 Slovenia Team Selection Test, 5
Let $n$ be a positive integer. We are given a regular $4n$-gon in the plane. We divide its vertices in $2n$ pairs and connect the two vertices in each pair by a line segment. What is the maximal possible amount of distinct intersections of the segments?
2024 Harvard-MIT Mathematics Tournament, 4
Let $ABCD$ be a square, and let $l$ be a line passing through the midpoint of segment $AB$ that intersects segment $BC$. Given that the distances from $A$ and $C$ to $l$ are $4$ and $7$, respectively, compute the area of $ABCD$.
2009 Stanford Mathematics Tournament, 7
Four disks with disjoint interiors are mutually tangent. Three of them are equal in size and the fourth one is smaller. Find the ratio of the radius of the smaller disk to one of the larger disks.
2023 USAJMO Solutions by peace09, 2
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.
[i]Proposed by Holden Mui[/i]
1982 Brazil National Olympiad, 5
Show how to construct a line segment length $(a^4 + b^4)^{1/4}$ given segments lengths $a$ and $b$.
Geometry Mathley 2011-12, 2.3
Let $ABC$ be a triagle inscribed in a circle $(O)$. A variable line through the orthocenter $H$ of the triangle meets the circle $(O)$ at two points $P , Q$. Two lines through $P, Q$ that are perpendicular to $AP , AQ$ respectively meet $BC$ at $M, N$ respectively. Prove that the line through $P$ perpendicular to $OM$ and the line through $Q$ perpendicular to $ON$ meet each other at a point on the circle $(O)$.
Nguyễn Văn Linh
2011 AMC 12/AHSME, 20
Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $ X \ne E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA+XB+XC$?
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 14\sqrt{3} \qquad
\textbf{(C)}\ \frac{195}{8} \qquad
\textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad
\textbf{(E)}\ \frac{69\sqrt{2}}{4} $
1996 Balkan MO, 3
In a convex pentagon $ABCDE$, the points $M$, $N$, $P$, $Q$, $R$ are the midpoints of the sides $AB$, $BC$, $CD$, $DE$, $EA$, respectively. If the segments $AP$, $BQ$, $CR$ and $DM$ pass through a single point, prove that $EN$ contains that point as well.
[i]Yugoslavia[/i]
2008 All-Russian Olympiad, 7
In convex quadrilateral $ ABCD$, the rays $ BA,CD$ meet at $ P$, and the rays $ BC,AD$ meet at $ Q$. $ H$ is the projection of $ D$ on $ PQ$. Prove that there is a circle inscribed in $ ABCD$ if and only if the incircles of triangles $ ADP,CDQ$ are visible from $ H$ under the same angle.
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.
2020 Junior Balkan Team Selection Tests-Serbia, 1#
Given is triangle $ABC$ with arbitrary point $D$ on $AB$ and central of inscribed circle $I$. The perpendicular bisector of $AB$ intersects $AI$ and $BI$ at $P$ and $Q$, respectively. The circle $(ADP)$ intersects $CA$ at $E$, and the circle $(BDQ)$ intersects $BC$ at $F$ and $(ADP)$ intersects $(BDQ)$ at $K$. Prove that $E, F, K, I$ lie on one circle.
2007 Today's Calculation Of Integral, 182
Find the area of the domain of the system of inequality
\[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]
1979 Vietnam National Olympiad, 6
$ABCD$ is a rectangle with $BC / AB = \sqrt2$. $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.
2022 IFYM, Sozopol, 2
Let $k$ be the circumcircle of the acute triangle $ABC$. Its inscribed circle touches sides $BC$, $CA$ and $AB$ at points $D, E$ and $F$ respectively. The line $ED$ intersects $k$ at the points $M$ and $N$, so that $E$ lies between $M$ and $D$. Let $K$ and $L$ be the second intersection points of the lines $NF$ and $MF$ respectively with $k$. Let $AK \cap BL = Q$. Prove that the lines $AL$, $BK$ and $QF$ intersect at a point.
1993 Putnam, B5
Show that given any $4$ points in the plane we can find two whose distance apart is not an odd integer.
2022 Taiwan TST Round 1, 1
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
2013 Dutch IMO TST, 5
Let $ABCDEF$ be a cyclic hexagon satisfying $AB\perp BD$ and $BC=EF$.Let $P$ be the intersection of lines $BC$ and $AD$ and let $Q$ be the intersection of lines $EF$ and $AD$.Assume that $P$ and $Q$ are on the same side of $D$ and $A$ is on the opposite side.Let $S$ be the midpoint of $AD$.Let $K$ and $L$ be the incentres of $\triangle BPS$ and $\triangle EQS$ respectively.Prove that $\angle KDL=90^0$.
2019 ELMO Problems, 4
Carl is given three distinct non-parallel lines $\ell_1, \ell_2, \ell_3$ and a circle $\omega$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $\ell$ and a point $P$, constructs a new line passing through $P$ parallel to $\ell$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $\omega$ whose sides are parallel to $\ell_1,\ell_2,\ell_3$ in some order.
[i]Proposed by Vincent Huang[/i]
Cono Sur Shortlist - geometry, 2020.G4
Let $ABC$ be a triangle with circumcircle $\omega$. The bisector of $\angle BAC$ intersects $\omega$ at point $A_1$. Let $A_2$ be a point on the segment $AA_1$, $CA_2$ cuts $AB$ and $\omega$ at points $C_1$ and $C_2$, respectively. Similarly, $BA_2$ cuts $AC$ and $\omega$ at points $B_1$ and $B_2$, respectively. Let $M$ be the intersection point of $B_1C_2$ and $B_2C_1$. Prove that $MA_2$ passes the midpoint of $BC$.
[i]proposed by Jhefferson Lopez, Perú[/i]