Intersection of circumcircle and excircle

Proposition 12.1.
(a) The isogonal conjugate of the Gergonne point is the insimilicenter of
the circumcircle and the incircle.
(b) The isogonal conjugate of the Nagel point is the exsimilicenter of the circumcircle and
the incircle.
Note: I need synthetic solution.

4 replies

SerdarBozdag

Apr 17, 2021

zuat.e

2 hours ago

Cute NT Problem

M11100111001Y1R 4

N 2 hours ago by RANDOM__USER

Source: Iran TST 2025 Test 4 Problem 1

A number $n$ is called lucky if it has at least two distinct prime divisors and can be written in the form:
$n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}$ where $p_1, \dots, p_k$ are distinct prime numbers that divide $n$ . (Note: it is possible that $n$ has other prime divisors not among $p_1, \dots, p_k$ .) Prove that for every prime number $p$ , there exists a lucky number $n$ such that $p \mid n$ .

4 replies

M11100111001Y1R

Today at 7:20 AM

RANDOM__USER

2 hours ago

USAMO 2003 Problem 4

MithsApprentice 72

N 2 hours ago by endless_abyss

Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$ , respectively. Lines $AB$ and $DE$ intersect at $F$ , while lines $BD$ and $CF$ intersect at $M$ . Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$ .

72 replies

MithsApprentice

Sep 27, 2005

endless_abyss

2 hours ago

Easy but unusual junior ineq

Maths_VC 1

N 2 hours ago by blug

Source: Serbia JBMO TST 2025, Problem 2

Real numbers $x, y$ $\ge$ $0$ satisfy $1$ $\le$ $x^2 + y^2$ $\le$ $5$ . Determine the minimal and the maximal value of the expression $2x + y$

1 reply

Maths_VC

3 hours ago

blug

2 hours ago

Bosnia and Herzegovina JBMO TST 2009 Problem 1

gobathegreat 1

N 2 hours ago by FishkoBiH

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2009

Lengths of sides of triangle $ABC$ are positive integers, and smallest side is equal to $2$ . Determine the area of triangle $P$ if $v_c = v_a + v_b$ , where $v_a$ , $v_b$ and $v_c$ are lengths of altitudes in triangle $ABC$ from vertices $A$ , $B$ and $C$ , respectively.

1 reply

gobathegreat

Sep 17, 2018

FishkoBiH

2 hours ago

USAMO 2001 Problem 2

MithsApprentice 53

N 2 hours ago by lksb

Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$ , respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$ , respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$ , and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$ . Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$ . Prove that $AQ=D_2P$ .

53 replies

1 viewing

MithsApprentice

Sep 30, 2005

lksb

2 hours ago

A=b

k2c901_1 89

N 3 hours ago by reni_wee

Source: Taiwan 1st TST 2006, 1st day, problem 3

Let $a$ , $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$ . Prove that $a=b$ .

Proposed by Mohsen Jamali, Iran

89 replies

k2c901_1

Mar 29, 2006

reni_wee

3 hours ago

Strange angle condition and concyclic points

lminsl 129

N 3 hours ago by Aiden-1089

Source: IMO 2019 Problem 2

In triangle $ABC$ , point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$ . Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$ , respectively, such that $PQ$ is parallel to $AB$ . Let $P_1$ be a point on line $PB_1$ , such that $B_1$ lies strictly between $P$ and $P_1$ , and $\angle PP_1C=\angle BAC$ . Similarly, let $Q_1$ be the point on line $QA_1$ , such that $A_1$ lies strictly between $Q$ and $Q_1$ , and $\angle CQ_1Q=\angle CBA$ .

Prove that points $P,Q,P_1$ , and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine

129 replies

1 viewing

lminsl

Jul 16, 2019

Aiden-1089

3 hours ago

Simple inequality

sqing 12

N 3 hours ago by Rayvhs

Source: MEMO 2018 T1

Let $a,b$ and $c$ be positive real numbers satisfying $abc=1.$ Prove that $\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.$

12 replies

sqing

Sep 2, 2018

Rayvhs

3 hours ago

Random concyclicity in a square config

Maths_VC 2

N 3 hours ago by Maths_VC

Source: Serbia JBMO TST 2025, Problem 1

Let $M$ be a random point on the smaller arc $AB$ of the circumcircle of square $ABCD$ , and let $N$ be the intersection point of segments $AC$ and $DM$ . The feet of the tangents from point $D$ to the circumcircle of the triangle $OMN$ are $P$ and $Q$ , where $O$ is the center of the square. Prove that points $A$ , $C$ , $P$ and $Q$ lie on a single circle.

2 replies

Maths_VC

3 hours ago

Maths_VC

3 hours ago

Serbian selection contest for the IMO 2025 - P3

OgnjenTesic 3

N 3 hours ago by atdaotlohbh

Source: Serbian selection contest for the IMO 2025

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that:
- $f$ is strictly increasing,
- there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$ ,
- for every $x \in \mathbb{Z}$ , there exists $y \in \mathbb{Z}$ such that
$f(y) = \frac{f(x) + f(x + 2024)}{2}.$ Proposed by Pavle Martinović

3 replies

OgnjenTesic

May 22, 2025

atdaotlohbh

3 hours ago

Intersection of circumcircle and excircle

ThisNameIsNotAvailable 2

N Jan 29, 2023 by Newmaths

Let $\omega$ be circumcircle and $\Omega$ be $A$ -excircle of triangle $ABC$ . Let $X, Y$ be the intersection of $\omega$ and $\Omega$ . Let $P$ and $Q$ be the projection of A onto the tangent lines to $\Omega$ at $X, Y$ respectively. The tangent at $P$ of $(APX)$ intersects the tangent at $Q$ of $(AQY)$ at $R$ . Prove that $AR \perp BC$ .