Circle touching circumcircle

by srirampanchapakesan, Mar 23, 2025, 1:08 PM

P,Q are isogonal conjugates wrt triangle ABC and midpoint X of PQ lies on circumcircle of ABC.

P1,P2,P3 are the circumcenters of triangles BPC,CPA,APB. Similarly Q1,Q2,Q3

Prove that the circumcircles of triangles ABC , P1P2P3 and Q1Q2Q3 touch each other at X.

geometry

circumcircle

"A perfect AIME problem"

by XAN4, Mar 23, 2025, 1:05 PM

Here is a compilcated problem of calculation. I'd really like to know how you solve it.
Find the minimum $n\in\mathbb Z^+$ such that there exists exactly $n$ different functions $f$ such that $f:[1,5]\rightarrow[1,5]$ satisfying $f^n(x)\geq x$ .

AIME

combinatorics

Mathematics

by slimshady360, Mar 23, 2025, 10:34 AM

Solve this

Attachments:

IMO

Interesting inequality

by sqing, Mar 23, 2025, 3:42 AM

Let $a,b,c\geq 0,(ab+c)(ac+b)\neq 0$ and $a+b+c=3 .$ Prove that
$\frac{1}{ab+kc}+\frac{1}{ac+kb} \geq\frac{4}{3k}$ Where $k\geq 3.$
$\frac{1}{ab+2c}+\frac{1}{ac+2b} \geq\frac{16}{25}$ $\frac{1}{ab+3c}+\frac{1}{ac+3b} \geq\frac{4}{9}$ $\frac{1}{ab+4c}+\frac{1}{ac+4b} \geq\frac{1}{3}$

inequalities proposed

inequalities

1/sqrt(5) ???

by navi_09220114, Mar 22, 2025, 1:10 PM

Two circles $\omega_1$ and $\omega_2$ are externally tangent at a point $A$ . Let $\ell$ be a line tangent to $\omega_1$ at $B\neq A$ and $\omega_2$ at $C\neq A$ . Let $BX$ and $CY$ be diameters in $\omega_1$ and $\omega_2$ respectively. Suppose points $P$ and $Q$ lies on $\omega_2$ such that $XP$ and $XQ$ are tangent to $\omega_2$ , and points $R$ and $S$ lies on $\omega_1$ such that $YR$ and $YS$ are tangent to $\omega_1$ .

a) Prove that the points $P$ , $Q$ , $R$ , $S$ lie on a circle $\Gamma$ .

b) Prove that the four segments $XP$ , $XQ$ , $YR$ , $YS$ determine a quadrilateral with an incircle $\gamma$ , and its radius is $\displaystyle\frac{1}{\sqrt{5}}$ times the radius of $\Gamma$ .

Proposed by Ivan Chan Kai Chin

geometry

Equilateral Triangle and Euler Line

by RetroTurtle, Jul 12, 2024, 4:06 AM

Let $D$ , $E$ , and $F$ be points on the perpendicular bisectors of $BC$ , $CA$ , and $AB$ of triangle $ABC$ such that $DEF$ is equilateral. Show that the center of $DEF$ lies on the Euler line of $ABC$ .

Three similar rectangles

by MarkBcc168, Jun 22, 2024, 3:57 PM

Let $ABC$ be a triangle. Construct rectangles $BA_1A_2C$ , $CB_1B_2A$ , and $AC_1C_2B$ outside $ABC$ such that $\angle BCA_1=\angle CAB_1=\angle ABC_1$ . Let $A_1B_2$ and $A_2C_1$ intersect at $A'$ and define $B',C'$ similarly. Prove that line $AA'$ bisects $B'C'$ .

Linus Tang

geometry

rectangle

Lengths of altitudes

by srirampanchapakesan, Apr 16, 2023, 1:01 PM

h1, h2 h3 are the lengths of the altitudes of a triangle. Prove that h1+h2+h3 = (s^2+4Rr+r^2)/2R, with s being semiperimeter, R the circumradius and r the inradius.

geometry

Isosceles right triangle from square and equilateral triangles

by buratinogigle, Jul 20, 2021, 10:19 AM

Let $ABCD$ be a square inscribed in an equilateral triangle $PQR$ . Construct another equilateral triangle $AMD$ inside the square. Line $BM$ meets $PR$ at $N$ . Prove that $DMN$ is the isosceles right triangle.

Attachments:

geometry

square

Equilateral Triangle

isosceles right triangle

IMO ShortList 2002, geometry problem 7

by orl, Sep 28, 2004, 1:00 PM

The incircle $\Omega$ of the acute-angled triangle $ABC$ is tangent to its side $BC$ at a point $K$ . Let $AD$ be an altitude of triangle $ABC$ , and let $M$ be the midpoint of the segment $AD$ . If $N$ is the common point of the circle $\Omega$ and the line $KM$ (distinct from $K$ ), then prove that the incircle $\Omega$ and the circumcircle of triangle $BCN$ are tangent to each other at the point $N$ .

Attachments:

This post has been edited 1 time. Last edited by orl, Oct 25, 2004, 12:16 AM

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yo dawg i heard you imo
by Mewto55555, Aug 1, 2013, 10:25 PM
Good job on 5 problems on USAMO. I know that is a pretty bad score for someone as awesome as you on a normal USAMO, but no one got all 6 this year so it's GREAT!

VICTOR WANG 42 ON IMO 2013 LET'S GO.

See you at MOP this year (probably ).
by yugrey, May 3, 2013, 10:32 PM
you can just write "Solution
" instead of "Click to reveal hidden text
"

by math154, Feb 6, 2013, 6:33 PM
Hello. May I ask a stupid question: how to turn "Hidden Text" into hidden "Solution" tag?
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by math154, Nov 23, 2011, 10:26 PM
what i got uncontribbed for posting a nice geo problem?
by yugrey, Nov 23, 2011, 1:32 AM
Can I be a contributor?
by Binomial-theorem, Oct 5, 2011, 12:39 AM
by gauss1181, Mar 8, 2009, 5:09 PM
i hate you
by alsyy, Mar 8, 2009, 4:37 PM
Fine. I don't care.
by math154, Mar 8, 2009, 1:34 AM
Math154: It was West Somethign Middle School from Columbia that got 2nd.
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Oh dang, I already feel like deleting this blog. Maybe I should not do that for a while, though.
by math154, Mar 6, 2009, 2:50 AM
math154 mew and tinytim tied for first at chapter...
by AIME15, Mar 6, 2009, 1:06 AM
Thanks! Good luck to you too.
by math154, Mar 5, 2009, 1:00 AM
good luck on saturday
by gauss1181, Mar 5, 2009, 12:51 AM
Yeah gauss, our chapter sends the top 50 individuals and the top 10 teams.
by math154, Mar 4, 2009, 10:49 PM
Me wants be contrib.
by AIME15, Mar 4, 2009, 5:07 PM
I'll contrib in the future when I feel like it. I'm not gonna forget you, though, so don't worry.
by math154, Mar 2, 2009, 11:55 PM
third! contrib?
by nikeballa96, Mar 2, 2009, 10:40 PM
2nd Shout!
by gauss1181, Feb 28, 2009, 5:25 PM
1st Shout!
by mathemagician1729, Feb 28, 2009, 4:02 AM