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An I for an I
Eyed   67
N 17 minutes ago by AR17296174
Source: 2020 ISL G8
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.

Show that $A,X,Y$ are collinear.
67 replies
Eyed
Jul 20, 2021
AR17296174
17 minutes ago
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IMO 2018 Problem 1
juckter   169
N an hour ago by Thelink_20
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece
169 replies
juckter
Jul 9, 2018
Thelink_20
an hour ago
J
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Croatian mathematical olympiad, day 1, problem 2
Matematika   6
N 2 hours ago by Cqy00000000
There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).
6 replies
Matematika
Apr 10, 2011
Cqy00000000
2 hours ago
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Game
Pascual2005   27
N 2 hours ago by HamstPan38825
Source: Colombia TST, IMO ShortList 2004, combinatorics problem 5
$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win?

Proposed by A. Slinko & S. Marshall, New Zealand
27 replies
Pascual2005
Jun 7, 2005
HamstPan38825
2 hours ago
J
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Lines concur on bisector of BAC
Invertibility   2
N 4 hours ago by NO_SQUARES
Source: Slovenia 2025 TST 3 P2
Let $\Omega$ be the circumcircle of a scalene triangle $ABC$. Let $\omega$ be a circle internally tangent to $\Omega$ in $A$. Tangents from $B$ touch $\omega$ in $P$ and $Q$, such that $P$ lies in the interior of $\triangle{}ABC$. Similarly, tangents from $C$ touch $\omega$ in $R$ and $S$, such that $R$ lies in the interior of $\triangle{}ABC$.

Prove that $PS$ and $QR$ concur on the bisector of $\angle{}BAC$.
2 replies
Invertibility
4 hours ago
NO_SQUARES
4 hours ago
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Why is the old one deleted?
EeEeRUT   16
N 5 hours ago by ravengsd
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
16 replies
EeEeRUT
Apr 16, 2025
ravengsd
5 hours ago
J
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angle chasing with 2 midpoints, equal angles given and wanted
parmenides51   5
N 5 hours ago by breloje17fr
Source: Ukrainian Geometry Olympiad 2017, IX p1, X p1, XI p1
In the triangle $ABC$, ${{A}_{1}}$ and ${{C}_{1}} $ are the midpoints of sides $BC $ and $AB$ respectively. Point $P$ lies inside the triangle. Let $\angle BP {{C}_{1}} = \angle PCA$. Prove that $\angle BP {{A}_{1}} = \angle PAC $.
5 replies
parmenides51
Dec 11, 2018
breloje17fr
5 hours ago
J
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Problem 4 of Finals
GeorgeRP   2
N 6 hours ago by Assassino9931
Source: XIII International Festival of Young Mathematicians Sozopol 2024, Theme for 10-12 grade
The diagonals \( AD \), \( BE \), and \( CF \) of a hexagon \( ABCDEF \) inscribed in a circle \( k \) intersect at a point \( P \), and the acute angle between any two of them is \( 60^\circ \). Let \( r_{AB} \) be the radius of the circle tangent to segments \( PA \) and \( PB \) and internally tangent to \( k \); the radii \( r_{BC} \), \( r_{CD} \), \( r_{DE} \), \( r_{EF} \), and \( r_{FA} \) are defined similarly. Prove that
\[ r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}. \]
2 replies
GeorgeRP
Sep 10, 2024
Assassino9931
6 hours ago
J
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Interesting functional equation with geometry
User21837561   3
N Yesterday at 6:34 PM by Double07
Source: BMOSL 2024 G7
For an acute triangle $ABC$, let $O$ be the circumcentre, $H$ be the orthocentre, and $G$ be the centroid.
Let $f:\pi\rightarrow\mathbb R$ satisfy the following condition:
$f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$
Prove that $f$ is constant.
3 replies
User21837561
Yesterday at 8:14 AM
Double07
Yesterday at 6:34 PM
J
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greatest volume
hzbrl   1
N Yesterday at 6:32 PM by hzbrl
Source: purple comet
A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume $n \pi$. Find $n$.
1 reply
hzbrl
Thursday at 9:56 AM
hzbrl
Yesterday at 6:32 PM
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(n+1)2^n, (n+3)2^{n+2} not perfect squares for the same n
parmenides51   3
N Yesterday at 6:31 PM by AylyGayypow009
Source: Greece JBMO TST 2015 p3
Prove that there is not a positive integer $n$ such that numbers $(n+1)2^n, (n+3)2^{n+2}$ are both perfect squares.
3 replies
parmenides51
Apr 29, 2019
AylyGayypow009
Yesterday at 6:31 PM
J
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IMO 2010 Problem 3
canada   59
N Yesterday at 6:23 PM by pi271828
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$

Proposed by Gabriel Carroll, USA
59 replies
canada
Jul 7, 2010
pi271828
Yesterday at 6:23 PM
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J
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Is the geometric function injective?
Project_Donkey_into_M4   1
N Apr 20, 2025 by Funcshun840
Source: Mock RMO TDP and Kayak 2018, P3
A non-degenerate triangle $\Delta ABC$ is given in the plane, let $S$ be the set of points which lie strictly inside it. Also let $\mathfrak{C}$ be the set of circles in the plane. For a point $P \in S$, let $A_P, B_P, C_P$ be the reflection of $P$ in sides $\overline{BC}, \overline{CA}, \overline{AB}$ respectively. Define a function $\omega: S \rightarrow \mathfrak{C}$ such that $\omega(P)$ is the circumcircle of $A_PB_PC_P$. Is $\omega$ injective?

Note: The function $\omega$ is called injective if for any $P, Q \in S$, $\omega(P) = \omega(Q) \Leftrightarrow P = Q$
1 reply
Project_Donkey_into_M4
Apr 20, 2025
Funcshun840
Apr 20, 2025
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Is the geometric function injective?
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Source: Mock RMO TDP and Kayak 2018, P3
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Project_Donkey_into_M4
151 posts
Project_Donkey_into_M4
#1 h  Apr 20, 2025, 6:23 PM
Y by
A non-degenerate triangle $\Delta ABC$ is given in the plane, let $S$ be the set of points which lie strictly inside it. Also let $\mathfrak{C}$ be the set of circles in the plane. For a point $P \in S$, let $A_P, B_P, C_P$ be the reflection of $P$ in sides $\overline{BC}, \overline{CA}, \overline{AB}$ respectively. Define a function $\omega: S \rightarrow \mathfrak{C}$ such that $\omega(P)$ is the circumcircle of $A_PB_PC_P$. Is $\omega$ injective?

Note: The function $\omega$ is called injective if for any $P, Q \in S$, $\omega(P) = \omega(Q) \Leftrightarrow P = Q$
This post has been edited 1 time. Last edited by Project_Donkey_into_M4, Apr 20, 2025, 6:24 PM
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Funcshun840
22 posts
Funcshun840
#2 h  Apr 20, 2025, 9:04 PM
Y by
I’m confused, doesn’t the injectivity of the function follow from the fact that the center of $\omega(P)$ is the isogonal conjugate of $P$?
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