Intersection of circumcircle and excircle

Let $a_0$ , $a_1$ , $a_2$ , ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$ , where $a_0$ and $a_1$ are two different positive reals.

Can this sequence $a_0$ , $a_1$ , $a_2$ , ... be bounded?

Proposed by Mihai Bălună, Romania

70 replies

darij grinberg

Jan 19, 2005

ezpotd

19 minutes ago

weird conditions in geo

Davdav1232 0

21 minutes ago

Source: Israel TST 7 2025 p1

Let $\triangle ABC$ be an isosceles triangle with $AB = AC$ . Let $D$ be a point on $AC$ . Let $L$ be a point inside the triangle such that $\angle CLD = 90^\circ$ and
$CL \cdot BD = BL \cdot CD.$ Prove that the circumcenter of triangle $\triangle BDL$ lies on line $AB$ .

0 replies

Davdav1232

21 minutes ago

0 replies

find angle

TBazar 4

N 31 minutes ago by vanstraelen

Given $ABC$ triangle with $AC>BC$ . We take $M$ , $N$ point on AC, AB respectively such that $AM=BC$ , $CM=BN$ . $BM$ , $AN$ lines intersect at point $K$ . If $2\angle AKM=\angle ACB$ , find $\angle ACB$

4 replies

TBazar

Today at 6:57 AM

vanstraelen

31 minutes ago

Polys with int coefficients

adihaya 4

N 41 minutes ago by sangsidhya

Source: 2012 INMO (India National Olympiad), Problem #3

Define a sequence $<f_0 (x), f_1 (x), f_2 (x), \dots>$ of functions by $f_0 (x) = 1$ $f_1(x)=x$ $(f_n(x))^2 - 1 = f_{n+1}(x) f_{n-1}(x)$ for $n \ge 1$ . Prove that each $f_n (x)$ is a polynomial with integer coefficients.

4 replies

adihaya

Mar 30, 2016

sangsidhya

41 minutes ago

Italian WinterCamps test07 Problem4

mattilgale 89

N an hour ago by cj13609517288

Source: ISL 2006, G3, VAIMO 2007/5

Let $ABCDE$ be a convex pentagon such that
$\angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.$ The diagonals $BD$ and $CE$ meet at $P$ . Prove that the line $AP$ bisects the side $CD$ .

Proposed by Zuming Feng, USA

89 replies

mattilgale

Jan 29, 2007

cj13609517288

an hour ago

Simple triangle geometry [a fixed point]

darij grinberg 49

N an hour ago by cj13609517288

Source: German TST 2004, IMO ShortList 2003, geometry problem 2

Three distinct points $A$ , $B$ , and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$ . Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$ . Suppose $\Gamma$ meets the segment $PB$ at $Q$ . Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$ .

49 replies

darij grinberg

May 18, 2004

cj13609517288

an hour ago

Kosovo MO 2010 Problem 5

Com10atorics 19

N an hour ago by CM1910

Source: Kosovo MO 2010 Problem 5

Let $x,y$ be positive real numbers such that $x+y=1$ . Prove that
$\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$ .

19 replies

Com10atorics

Jun 7, 2021

CM1910

an hour ago

Hard combi

EeEApO 1

N an hour ago by EeEApO

In a quiz competition, there are a total of $100$ questions, each with $4$ answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?

Now, suppose my spouse and I move into a new home. Every year, we have twins. Starting at the age of $16$ , each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?

1 reply

EeEApO

3 hours ago

EeEApO

an hour ago

Problem on symmetric polynomial

ayan_mathematics_king 5

N an hour ago by bjump

If $a^3+b^3+c^3=(a+b+c)^3$ , prove that $a^5+b^5+c^5=(a+b+c)^5$ where $a,b,c \in \mathbb{R}$

5 replies

ayan_mathematics_king

Jul 28, 2019

bjump

an hour ago

Simple inequality

sqing 57

N 2 hours ago by Sh309had

Source: Shortlist BMO 2018, A1

Let $a, b, c$ be positive real numbers such that $abc = \frac {2} {3}.$ Prove that:

$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$

57 replies

sqing

May 3, 2019

Sh309had

2 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 $AR \perp BC$ .