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Eventually constant sequence with condition
PerfectPlayer   4
N an hour ago by kujyi
Source: Turkey TST 2025 Day 3 P8
A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given.
Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$
\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]Then for every integer $n\geq s,$ the condition
\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]is satisfied. Prove that this sequence must be eventually constant.
4 replies
PerfectPlayer
Mar 18, 2025
kujyi
an hour ago
Pentagon with given diameter, ratio desired
bin_sherlo   3
N an hour ago by tugra_ozbey_eratli
Source: Türkiye 2025 JBMO TST P7
$ABCDE$ is a pentagon whose vertices lie on circle $\omega$ where $\angle DAB=90^{\circ}$. Let $EB$ and $AC$ intersect at $F$, $EC$ meet $BD$ at $G$. $M$ is the midpoint of arc $AB$ on $\omega$, not containing $C$. If $FG\parallel DE\parallel CM$ holds, then what is the value of $\frac{|GE|}{|GD|}$?
3 replies
bin_sherlo
May 11, 2025
tugra_ozbey_eratli
an hour ago
Number Theory
fasttrust_12-mn   15
N an hour ago by Pal702004
Source: Pan African Mathematics Olympiad P1
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
15 replies
fasttrust_12-mn
Aug 15, 2024
Pal702004
an hour ago
pairs (m, n) such that a fractional expression is an integer
cielblue   2
N an hour ago by cielblue
Find all pairs $(m,\ n)$ of positive integers such that $\frac{m^3-mn+1}{m^2+mn+2}$ is an integer.
2 replies
cielblue
May 24, 2025
cielblue
an hour ago
interesting geo config (2/3)
Royal_mhyasd   3
N 2 hours ago by Royal_mhyasd
Source: own
Let $\triangle ABC$ be an acute triangle and $H$ its orthocenter. Let $P$ be a point on the parallel through $A$ to $BC$ such that $\angle APH = |\angle ABC-\angle ACB|$. Define $Q$ and $R$ as points on the parallels through $B$ to $AC$ and through $C$ to $AB$ similarly. If $P,Q,R$ are positioned around the sides of $\triangle ABC$ as in the given configuration, prove that $P,Q,R$ are collinear.
3 replies
Royal_mhyasd
Yesterday at 11:36 PM
Royal_mhyasd
2 hours ago
interesting geo config (1\3)
Royal_mhyasd   2
N 2 hours ago by Royal_mhyasd
Source: own
Let $\triangle ABC$ be an acute triangle with $AC > AB$, $H$ its orthocenter and $O$ it's circumcenter. Let $P$ be a point on the parallel through $A$ to $BC$ such that $\angle APH = \angle ABC - \angle ACB$ and $P$ and $C$ are on different sides of $AB$. Denote by $S$ the intersection of the circumcircle of $\triangle ABC$ and $PA'$, where $A'$ is the reflection of $H$ over $BC$, $M$ the midpoint of $PH$, $Q$ the intersection of $OA$ and the parallel through $M$ to $AS$, $R$ the intersection of $MS$ and the perpendicular through $O$ to $PS$ and $N$ a point on $AS$ such that $NT \parallel PS$, where $T$ is the midpoint of $HS$. Prove that $Q, N, R$ lie on a line.

fiy it's 2am and i'm bored so i decided to look further into this interesting config that i had already made some observations on, maybe this problem is trivial from some theorem so if that's the case then i'm sorry lol :P i'll probably post 2 more problems related to it soon, i'd say they're easier than this though
2 replies
Royal_mhyasd
Yesterday at 11:18 PM
Royal_mhyasd
2 hours ago
Find all sequences satisfying two conditions
orl   35
N 2 hours ago by wangyanliluke
Source: IMO Shortlist 2007, C1, AIMO 2008, TST 1, P1
Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 + n}$ satisfying the following conditions:
\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]

\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]
Author: Dusan Dukic, Serbia
35 replies
orl
Jul 13, 2008
wangyanliluke
2 hours ago
Gcd of N and its coprime pair sum
EeEeRUT   20
N 2 hours ago by Adywastaken
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
20 replies
EeEeRUT
Apr 16, 2025
Adywastaken
2 hours ago
geometry problem with many circumcircles
Melid   0
2 hours ago
Source: own
In scalene triangle $ABC$, which doesn't have right angle, let $O$ be its circumcenter. Circle $BOC$ intersects $AB$ and $AC$ at $A_{1}$ and $A_{2}$ for the second time, respectively. Similarly, circle $COA$ intersects $BC$ and $BA$ at $B_{1}$ and $B_{2}$, and circle $AOB$ intersects $CA$ and $CB$ at $C_{1}$ and $C_{2}$ for the second time, respectively. Let $O_{1}$ and $O_{2}$ be circumcenters of triangle $A_{1}B_{1}C_{1}$ and $A_{2}B_{2}C_{2}$, respectively. Prove that $O, O_{1}, O_{2}$ are collinear.
0 replies
Melid
2 hours ago
0 replies
Rootiful sets
InternetPerson10   38
N 2 hours ago by cursed_tangent1434
Source: IMO 2019 SL N3
We say that a set $S$ of integers is rootiful if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
38 replies
InternetPerson10
Sep 22, 2020
cursed_tangent1434
2 hours ago
weird conditions in geo
Davdav1232   2
N 3 hours ago by teoira
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 \( AB \).
2 replies
Davdav1232
May 8, 2025
teoira
3 hours ago
Complicated FE
XAN4   2
N Apr 24, 2025 by cazanova19921
Source: own
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
2 replies
XAN4
Apr 23, 2025
cazanova19921
Apr 24, 2025
Complicated FE
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G H BBookmark kLocked kLocked NReply
Source: own
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XAN4
61 posts
#1
Y by
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
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jasperE3
11395 posts
#2
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pogress
This post has been edited 1 time. Last edited by jasperE3, Apr 23, 2025, 9:34 PM
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cazanova19921
556 posts
#3
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XAN4 wrote:
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.

I suppose you have a proof to the result you claimed since it’s your own FE ( I know you don’t).


General solution : let $a$ be an additive function from $\mathbb{R}$ to itself and set $g=\exp \circ a \circ \log $, then $f=g+\frac1{g}$
This post has been edited 1 time. Last edited by cazanova19921, Apr 24, 2025, 3:03 AM
Reason: Typo
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