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Sharygin 2025 CR P7
Gengar_in_Galar   5
N an hour ago by Blackbeam999
Source: Sharygin 2025
Let $I$, $I_{a}$ be the incenter and the $A$-excenter of a triangle $ABC$; $E$, $F$ be the touching points of the incircle with $AC$, $AB$ respectively; $G$ be the common point of $BE$ and $CF$. The perpendicular to $BC$ from $G$ meets $AI$ at point $J$. Prove that $E$, $F$, $J$, $I_{a}$ are concyclic.
Proposed by:Y.Shcherbatov
5 replies
Gengar_in_Galar
Mar 10, 2025
Blackbeam999
an hour ago
60^o angle wanted, equilateral on a square
parmenides51   5
N an hour ago by sunken rock
Source: 2019 Austrian Mathematical Olympiad Junior Regional Competition , Problem 2
A square $ABCD$ is given. Over the side $BC$ draw an equilateral triangle $BCS$ on the outside. The midpoint of the segment $AS$ is $N$ and the midpoint of the side $CD$ is $H$. Prove that $\angle NHC = 60^o$.
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(Karl Czakler)
5 replies
parmenides51
Dec 18, 2020
sunken rock
an hour ago
Random walk
EthanWYX2009   0
an hour ago
As shown in the graph, an ant starts from $4$ and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching $1$ without passing through $6.$
0 replies
EthanWYX2009
an hour ago
0 replies
Lemma on tangency involving a parallelogram with orthocenter
Gimbrint   0
2 hours ago
Source: Own
Let $ABC$ be an acute triangle ($AB<BC$) with circumcircle $\omega$ and orthocenter $H$. Let $M$ be the midpoint of $AC$. Line $BH$ intersects $\omega$ again at $L\neq B$, and line $ML$ intersects $\omega$ again at $P\neq L$. Points $D$ and $E$ lie on $AB$ and $BC$ respectively, such that $BEHD$ is a parallelogram.

Prove that $BP$ is tangent to the circumcircle of triangle $BDE$.
0 replies
Gimbrint
2 hours ago
0 replies
Consecutive squares are floors
ICE_CNME_4   10
N 2 hours ago by JARP091

Determine how many positive integers \( n \) have the property that both
\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]are consecutive perfect squares.
10 replies
ICE_CNME_4
Yesterday at 1:50 PM
JARP091
2 hours ago
Sharygin 2025 CR P10
Gengar_in_Galar   2
N 2 hours ago by Kappa_Beta_725
Source: Sharygin 2025
An acute-angled triangle with one side equal to the altitude from the opposite vertex is cut from paper. Construct a point inside this triangle such that the square of the distance from it to one of the vertices equals the sum of the squares of distances to to the remaining two vertices. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines.
Proposed by: M.Evdokimov
2 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
2 hours ago
Sharygin 2025 CR P12
Gengar_in_Galar   8
N 2 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Circles $\omega_{1}$ and $\omega_{2}$ are given. Let $M$ be the midpoint of the segment joining their centers, $X$, $Y$ be arbitrary points on $\omega_{1}$, $\omega_{2}$ respectively such that $MX=MY$. Find the locus of the midpoints of segments $XY$.
Proposed by: L Shatunov
8 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
2 hours ago
Sharygin 2025 CR P17
Gengar_in_Galar   6
N 3 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Let $O$, $I$ be the circumcenter and the incenter of an acute-angled scalene triangle $ABC$; $D$, $E$, $F$ be the touching points of its excircle with the side $BC$ and the extensions of $AC$, $AB$ respectively. Prove that if the orthocenter of the triangle $DEF$ lies on the circumcircle of $ABC$, then it is symmetric to the midpoint of the arc $BC$ with respect to $OI$.
Proposed by: P.Puchkov,E.Utkin
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
3 hours ago
Sharygin 2025 CR P21
Gengar_in_Galar   4
N 3 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur.
Proposed by: G.Galyapin
4 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
3 hours ago
Sharygin 2025 CR P18
Gengar_in_Galar   6
N 3 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right
Proposed by: I.Kukharchuk
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
3 hours ago
a