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CSMGO P3: A problem on the infamous line XH
amar_04   12
N 20 minutes ago by WLOGQED1729
Source: https://artofproblemsolving.com/community/c594864h2372843p19407517
Let $\triangle ABC$ be a scalene triangle with the orthocenter $H$. Let $B'$ be the reflection of $B$ over $AC$ and $C'$ be the reflection of $C$ over $AB$. Let the tangents to the circumcircle of $\triangle ABC$ at points $B$ and $C$ meet at a point $X$. Suppose that the lines $B'C'$ and $BC$ meet at a point $T$. Prove that $AT$ is perpendicular to $XH$.
12 replies
amar_04
Feb 16, 2021
WLOGQED1729
20 minutes ago
Three mutually tangent circles
math154   8
N 22 minutes ago by lakshya2009
Source: ELMO Shortlist 2011, G2
Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$.

David Yang.
8 replies
math154
Jul 3, 2012
lakshya2009
22 minutes ago
Line AT passes through either S_1 or S_2
v_Enhance   89
N 33 minutes ago by zuat.e
Source: USA December TST for 57th IMO 2016, Problem 2
Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$. The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$. The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$. Prove that line $AT$ passes through either $S_1$ or $S_2$.

Proposed by Evan Chen
89 replies
v_Enhance
Dec 21, 2015
zuat.e
33 minutes ago
c^a + a = 2^b
Havu   3
N 34 minutes ago by Havu
Find $a, b, c\in\mathbb{Z}^+$ such that $a,b,c$ coprime, $a + b = 2c$ and $c^a + a = 2^b$.
3 replies
Havu
May 10, 2025
Havu
34 minutes ago
Easy geo
kooooo   3
N an hour ago by Blackbeam999
Source: own
In triangle $ABC$, let $O$ and $H$ be the circumcenter and orthocenter, respectively. Let $M$ and $N$ be the midpoints of $AC$ and $AB$, respectively, and let $D$ and $E$ be the feet of the perpendiculars from $B$ and $C$ to the opposite sides, respectively. Show that if $X$ is the intersection of $MN$ and $DE$, then $AX$ is perpendicular to $OH$.
3 replies
kooooo
Jul 31, 2024
Blackbeam999
an hour ago
Interesting
imnotgoodatmathsorry   0
an hour ago
Source: Own.
Problem 1. Let $x,y,z >0$. Prove that:
$\frac{108(x^6+y^6)(y^6+z^6)(z^6+x^6)}{x^9y^9z^9} - (xy+yz+zx)^6 \le 135$
Problem 2. Let $a,b,c >0$. Prove that:
$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$
0 replies
imnotgoodatmathsorry
an hour ago
0 replies
$n^{22}-1$ and $n^{40}-1$
v_Enhance   5
N an hour ago by Kempu33334
Source: OTIS Mock AIME 2024 #13
Let $S$ denote the sum of all integers $n$ such that $1 \leq n \leq 2024$ and exactly one of $n^{22}-1$ and $n^{40}-1$ is divisible by $2024$. Compute the remainder when $S$ is divided by $1000$.

Raymond Zhu

5 replies
v_Enhance
Jan 16, 2024
Kempu33334
an hour ago
Annoying 2^x-5 = 11^y
Valentin Vornicu   38
N an hour ago by Kempu33334
Find all positive integer solutions to $2^x - 5 = 11^y$.

Comment (some ideas)
38 replies
Valentin Vornicu
Jan 14, 2006
Kempu33334
an hour ago
Polish MO Finals 2014, Problem 5
j___d   14
N an hour ago by Kempu33334
Source: Polish MO Finals 2014
Find all pairs $(x,y)$ of positive integers that satisfy
$$2^x+17=y^4$$.
14 replies
j___d
Jul 27, 2016
Kempu33334
an hour ago
IMO LongList 1985 CYP2 - System of Simultaneous Equations
Amir Hossein   15
N an hour ago by Kempu33334
Solve the system of simultaneous equations
\[\sqrt x - \frac 1y - 2w + 3z = 1,\]\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]
15 replies
Amir Hossein
Sep 10, 2010
Kempu33334
an hour ago
Prove that the triangle is isosceles.
TUAN2k8   6
N an hour ago by on_gale
Source: My book
Given acute triangle $ABC$ with two altitudes $CF$ and $BE$.Let $D$ be the point on the line $CF$ such that $DB \perp BC$.The lines $AD$ and $EF$ intersect at point $X$, and $Y$ is the point on segment $BX$ such that $CY \perp BY$.Suppose that $CF$ bisects $BE$.Prove that triangle $ACY$ is isosceles.
6 replies
TUAN2k8
Yesterday at 9:55 AM
on_gale
an hour ago
Radical Condition Implies Isosceles
peace09   10
N an hour ago by Kempu33334
Source: Black MOP 2012
Prove that any triangle with
\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]is isosceles.
10 replies
peace09
Aug 10, 2023
Kempu33334
an hour ago
arbitrary point on Euler line
sarjinius   1
N Apr 22, 2021 by SerdarBozdag
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Suppose that $AB \neq AC$. Let $M$ be an arbitrary point on line segment $OH$. Let $P$ and $Q$ be points on $AB$ and $AC$ respectively such that $MA = MP = MQ$. Prove that the circumcenter of $\triangle MPQ$ lies on the perpendicular bisector of $BC$.
1 reply
sarjinius
Apr 14, 2021
SerdarBozdag
Apr 22, 2021
arbitrary point on Euler line
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sarjinius
245 posts
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Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Suppose that $AB \neq AC$. Let $M$ be an arbitrary point on line segment $OH$. Let $P$ and $Q$ be points on $AB$ and $AC$ respectively such that $MA = MP = MQ$. Prove that the circumcenter of $\triangle MPQ$ lies on the perpendicular bisector of $BC$.
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SerdarBozdag
892 posts
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Y by Mango247
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