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is there a maximal tlaxcalteca arithmetic progression of 11 elements?
parmenides51   1
N 14 minutes ago by venhancefan777
Source: Mathematics Regional Olympiad of Mexico Center Zone 2016 P5
An arithmetic sequence is a sequence of $(a_1, a_2, \dots, a_n) $ such that the difference between any two consecutive terms is the same. That is, $a_ {i + 1} -a_i = d $ for all $i \in \{1,2, \dots, n-1 \} $, where $d$ is the difference of the progression.

A sequence $(a_1, a_2, \dots, a_n) $ is tlaxcalteca if for all $i \in \{1,2, \dots, n-1 \} $, there exists $m_i $ positive integer such that $a_i = \frac {1} {m_i}$. A taxcalteca arithmetic progression $(a_1, a_2, \dots, a_n )$ is said to be maximal if $(a_1-d, a_1, a_2, \dots, a_n) $ and $(a_1, a_2, \dots, a_n, a_n + d) $ are not Tlaxcalan arithmetic progressions.

Is there a maximal tlaxcalteca arithmetic progression of $11$ elements?
1 reply
parmenides51
Nov 12, 2021
venhancefan777
14 minutes ago
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Integer Polynomial with P(a)=b, P(b)=c, and P(c)=a
Brut3Forc3   30
N 2 hours ago by megahertz13
Source: 1974 USAMO Problem 1
Let $ a,b,$ and $ c$ denote three distinct integers, and let $ P$ denote a polynomial having integer coefficients. Show that it is impossible that $ P(a) = b, P(b) = c,$ and $ P(c) = a$.
30 replies
Brut3Forc3
Mar 13, 2010
megahertz13
2 hours ago
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power sum system of equations in 3 variables
Stear14   1
N 3 hours ago by Stear14
Given that
$x^2+y^2+z^2=8\ ,$
$x^3+y^3+z^3=15\ ,$
$x^5+y^5+z^5=100\ .$

Find the value of $\ x+y+z\ .$
1 reply
Stear14
May 25, 2025
Stear14
3 hours ago
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IMO 2006 Slovenia - PROBLEM 5
Valentin Vornicu   70
N 4 hours ago by bjump
Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.
70 replies
Valentin Vornicu
Jul 13, 2006
bjump
4 hours ago
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Multiple of power of two.
dgrozev   4
N 5 hours ago by Assassino9931
Source: Bulgarian TST, 2020, p2
Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$
4 replies
dgrozev
Aug 4, 2020
Assassino9931
5 hours ago
J
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Functional equation involving decimal-place count
saulgodman   0
5 hours ago
Source: Own

Let
\[ S = \left\{\, x \in \mathbb{Q} : x \text{ has a finite decimal expansion} \,\right\}. \]For each \( x \in S \), define
\[ d(x) = \text{the number of digits after the decimal point in the (reduced) decimal form of } x. \]Find all functions \( f\colon S \to \mathbb{Z} \) such that, whenever both \( x+y \in S \) and \( x y \in S \),
\[ f(x) + f(y) = f(x+y) + d(x y). \]
0 replies
saulgodman
5 hours ago
0 replies
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Did you talk to Noga Alon?
pohoatza   36
N 5 hours ago by ezpotd
Source: IMO Shortlist 2006, Combinatorics 3, AIMO 2007, TST 6, P2
Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1,\] where the sum is taken over all convex polygons with vertices in $ S$.

Alternative formulation:

Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A -$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A -$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial
\[ P_A(x) = x^{|A|}(1 - x)^{r(A)}. \]
Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) = 1.$

Proposed by Federico Ardila, Colombia
36 replies
pohoatza
Jun 28, 2007
ezpotd
5 hours ago
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High School Olympiads Tackle IMO Problems
InternationalUniLatex   0
Yesterday at 7:10 PM
Source: IMO 2008 - Problem 2
(i) If $x$, $y$ and $z$ are three real numbers, all different from $1$, such that $xyz = 1$, then prove that $\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$. (With the $\sum$ sign for cyclic summation, this inequality could be rewritten as $\sum \frac {x^{2}}{\left(x - 1\right)^{2}} \geq 1$.)

(ii) Prove that equality is achieved for infinitely many triples of rational numbers $x$, $y$ and $z$.

0 replies
InternationalUniLatex
Yesterday at 7:10 PM
0 replies
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Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y
cretanman   61
N Yesterday at 6:18 PM by HDavisWashu
Source: BMO 2023 Problem 1
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[xf(x+f(y))=(y-x)f(f(x)).\]
Proposed by Nikola Velov, Macedonia
61 replies
cretanman
May 10, 2023
HDavisWashu
Yesterday at 6:18 PM
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Functional equation over the integers
Jutaro   33
N Yesterday at 5:48 PM by lpieleanu
Source: Centroamerican 2020, problem 3
Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then

$$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$
33 replies
Jutaro
Oct 28, 2020
lpieleanu
Yesterday at 5:48 PM
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continuous function
lolm2k   17
N Yesterday at 4:33 PM by hung9A
Let $f: \mathbb R \rightarrow \mathbb R$ be a continuous function such that $f(f(f(x))) = x^2+1, \forall x \in \mathbb R$ show that $f$ is even.
17 replies
lolm2k
Mar 24, 2018
hung9A
Yesterday at 4:33 PM
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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
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Complicated FE
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Source: own
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XAN4
63 posts
XAN4
#1 h  Apr 23, 2025, 11:53 AM
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
jasperE3
#2 h  Apr 23, 2025, 8:32 PM
Y by
pogress
This post has been edited 1 time. Last edited by jasperE3, Apr 23, 2025, 9:34 PM
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cazanova19921
558 posts
cazanova19921
#3 h  Apr 24, 2025, 3:03 AM
Y by
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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