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Cyclic Quads and Parallel Lines
gracemoon124   16
N 2 hours ago by ohiorizzler1434
Source: 2015 British Mathematical Olympiad?
Let $ABCD$ be a cyclic quadrilateral. Let $F$ be the midpoint of the arc $AB$ of its circumcircle which does not contain $C$ or $D$. Let the lines $DF$ and $AC$ meet at $P$ and the lines $CF$ and $BD$ meet at $Q$. Prove that the lines $PQ$ and $AB$ are parallel.
16 replies
gracemoon124
Aug 16, 2023
ohiorizzler1434
2 hours ago
Radical Center on the Euler Line (USEMO 2020/3)
franzliszt   37
N 2 hours ago by Ilikeminecraft
Source: USEMO 2020/3
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly.
Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$.

Proposed by Anant Mudgal
37 replies
franzliszt
Oct 24, 2020
Ilikeminecraft
2 hours ago
Functional equation with powers
tapir1729   13
N 2 hours ago by ihategeo_1969
Source: TSTST 2024, problem 6
Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$,
\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]Jaedon Whyte
13 replies
tapir1729
Jun 24, 2024
ihategeo_1969
2 hours ago
Powers of a Prime
numbertheorist17   34
N 2 hours ago by KevinYang2.71
Source: USA TSTST 2014, Problem 6
Suppose we have distinct positive integers $a, b, c, d$, and an odd prime $p$ not dividing any of them, and an integer $M$ such that if one considers the infinite sequence \begin{align*}
		ca &- db \\
		ca^2 &- db^2 \\
		ca^3 &- db^3 \\
		ca^4 &- db^4 \\
&\vdots
	\end{align*} and looks at the highest power of $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a,b,c,d,p,M$) such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^T$.
34 replies
numbertheorist17
Jul 16, 2014
KevinYang2.71
2 hours ago
IMO 2018 Problem 5
orthocentre   80
N 3 hours ago by OronSH
Source: IMO 2018
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

Proposed by Bayarmagnai Gombodorj, Mongolia
80 replies
orthocentre
Jul 10, 2018
OronSH
3 hours ago
Line passes through fixed point, as point varies
Jalil_Huseynov   60
N 4 hours ago by Rayvhs
Source: APMO 2022 P2
Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.
60 replies
Jalil_Huseynov
May 17, 2022
Rayvhs
4 hours ago
Tangent to two circles
Mamadi   2
N 4 hours ago by A22-
Source: Own
Two circles \( w_1 \) and \( w_2 \) intersect each other at \( M \) and \( N \). The common tangent to two circles nearer to \( M \) touch \( w_1 \) and \( w_2 \) at \( A \) and \( B \) respectively. Let \( C \) and \( D \) be the reflection of \( A \) and \( B \) respectively with respect to \( M \). The circumcircle of the triangle \( DCM \) intersect circles \( w_1 \) and \( w_2 \) respectively at points \( E \) and \( F \) (both distinct from \( M \)). Show that the line \( EF \) is the second tangent to \( w_1 \) and \( w_2 \).
2 replies
Mamadi
May 2, 2025
A22-
4 hours ago
Deduction card battle
anantmudgal09   55
N 5 hours ago by deduck
Source: INMO 2021 Problem 4
A Magician and a Detective play a game. The Magician lays down cards numbered from $1$ to $52$ face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.

Prove that the Detective can guarantee a win if and only if she is allowed to ask at least $50$ questions.

Proposed by Anant Mudgal
55 replies
anantmudgal09
Mar 7, 2021
deduck
5 hours ago
Geometry
Lukariman   7
N 5 hours ago by vanstraelen
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
7 replies
Lukariman
Tuesday at 12:43 PM
vanstraelen
5 hours ago
perpendicularity involving ex and incenter
Erken   20
N 6 hours ago by Baimukh
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
20 replies
Erken
Dec 24, 2008
Baimukh
6 hours ago
Isosceles Triangle Geo
oVlad   4
N 6 hours ago by Double07
Source: Romania Junior TST 2025 Day 1 P2
Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$
4 replies
oVlad
Apr 12, 2025
Double07
6 hours ago
Geometry
Lukariman   1
N 6 hours ago by Primeniyazidayi
Given acute triangle ABC ,AB=b,AC=c . M is a variable point on side AB. The circle circumscribing triangle BCM intersects AC at N.

a)Let I be the center of the circle circumscribing triangle AMN. Prove that I always lies on a fixed line.

b)Let J be the center of the circle circumscribing triangle MBC. Prove that line segment IJ has a constant length.
1 reply
Lukariman
Yesterday at 4:02 PM
Primeniyazidayi
6 hours ago
tangency points of common tangents of two conics
PROF65   1
N Feb 4, 2023 by cosmicgenius
Let $c,c'$ be two conics . If the four common tangents touch $c,c'$ at $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ then prove that :
$A_1,B_1,C_1,D_1,A_2,B_2,C_2$ and $D_2$ are conconic i.e. they belong to the same conic .
1 reply
PROF65
Feb 3, 2023
cosmicgenius
Feb 4, 2023
tangency points of common tangents of two conics
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PROF65
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Let $c,c'$ be two conics . If the four common tangents touch $c,c'$ at $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ then prove that :
$A_1,B_1,C_1,D_1,A_2,B_2,C_2$ and $D_2$ are conconic i.e. they belong to the same conic .
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cosmicgenius
1488 posts
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Throw the whole thing onto $\mathbb{CP}^2$. Then, $c$ and $c'$ are the projective varieties of some homogeneous degree $2$ polynomials in $3$ variables $\mathcal E_1$, $\mathcal E_2$, respectively, that are unique up to scaling. Similarly, the common tangents $A_1A_2$, $B_1B_2$, $C_1C_2$, and $D_1D_2$ are the projective varieties of some homogeneous linear polynomials $\ell_1$, $\ell_2$, $\ell_3$, and $\ell_4$, respectively. Finally, the chords $A_1C_1$, $B_1D_1$, $A_2C_2$, and $B_2D_2$ are again the projective varieties of some homogeneous linear polynomials $m_1$, $m_2$, $m_3$, and $m_4$.

By scaling, appropriately, we can find nonzero constants $\lambda_{113}$, $\lambda_{124}$, $\lambda_{213}$, and $\lambda_{224}$, such that
\[
\mathcal E_1 = \lambda_{113} \ell_1\ell_3 + m_1^2 = \lambda_{124} \ell_2 \ell_4 + m_2^2 
\quad \text{and} \quad
\mathcal E_2 = \lambda_{213} \ell_1\ell_3 + m_3^2 = \lambda_{224} \ell_2 \ell_4 + m_4^2,
\]since $c$ is in double-contact with $A_1A_2 \cup C_1C_2$ with double-contact chord $A_1C_1$, etc. It follows that
\[ (m_1-m_2)(m_1+m_2) = \lambda_{124} \ell_2 \ell_4 - \lambda_{113} \ell_1\ell_3.\]But the projective variety generated by the two sides is a degenerate conic formed by $2$ lines, by looking at the LHS, which also passes through the four intersection points of $\ell_1 \ell_2$ and $\ell_3 \ell_4$, by looking at the RHS. But clearly it is neither $\ell_1 \ell_2$ nor $\ell_3 \ell_4$, so it must be the last two sides of the complete quadrangle with the aforementioned $4$ points as vertices.

By repeating the same analysis on the equation for $\mathcal E_2$, we have that $(m_1-m_2)(m_1+m_2)$ and $(m_3-m_4)(m_3+m_4)$ have the same variety. By flipping signs, we can assume WLOG that $m_1-m_2$ and $m_3-m_4$ have the same variety, i.e. there exists some nonzero constant $\alpha$ such that $m_1 - m_2 = \alpha m_3 - \alpha m_4$. Rewrite this as $m_1 - \alpha m_3 = m_2 - \alpha m_4$.

Now we claim that the projective variety of the homogeneous degree $2$ polynomial,
\[ \mathcal M = (m_1 - \alpha m_3)^2 - \mathcal E_1 - \alpha^2 \mathcal E_3 = (m_2 - \alpha m_4)^2 - \mathcal E_1 - \alpha^2 \mathcal E_3\]passes through $A_1$, $B_1$, $C_1$, $D_1$, $A_2$, $B_2$, $C_2$, and $D_2$. Indeed by expansion, we have
\[ \mathcal M = - 2 \alpha m_1 m_3 - \lambda_{113} \ell_1\ell_3 - \alpha^2 \lambda_{213} \ell_1\ell_3 = (-2\alpha) m_1 m_3 + (-\lambda_{113} - \alpha^2 \lambda_{213}) \ell_1\ell_3, \]so the variety of $\mathcal M$ passes through the four intersection points of the varieties of $m_1m_3$ and $\ell_1\ell_3$, namely, $A_1$, $C_1$, $A_2$, and $C_2$. Similarly, $\mathcal M$ passes through the four intersection points of the varieties of $m_2m_4$ and $\ell_2\ell_4$, so we are done. $\blacksquare$
This post has been edited 1 time. Last edited by cosmicgenius, Feb 4, 2023, 9:33 PM
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