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Can this sequence be bounded?
darij grinberg   70
N 19 minutes ago by ezpotd
Source: German pre-TST 2005, problem 4, ISL 2004, algebra problem 2
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 $AR \perp BC$.
2 replies
ThisNameIsNotAvailable
Jan 28, 2023
Newmaths
Jan 29, 2023
Intersection of circumcircle and excircle
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ThisNameIsNotAvailable
442 posts
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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 $AR \perp BC$.
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ThisNameIsNotAvailable
442 posts
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Any idea?
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Newmaths
48 posts
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2021 ISL G8?
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