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Find all real functions withf(x^2 + yf(z)) = xf(x) + zf(y)
Rushil 31
N
by Jakjjdm
Source: INMO 2005 Problem 6
Find all functions 
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](//latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
31 replies
+1 w
Prove that x1=x2=....=x2025
Rohit-2006 8
N
by Project_Donkey_into_M4
Source: A mock
The real numbers 
satisfy,
Where {
} is a permutation of . Prove that 
satisfy,
Where {
} is a permutation of . Prove that 
8 replies
function Z to Z..
Jackson0423 2
N
by Rasul_Gasimli
Let 
be a function satisfying
![\[
f(f(x)) = x^2 - 6x + 6
\quad \text{for all} \quad x \in \mathbb{Z}.
\]](//latex.artofproblemsolving.com/8/a/e/8aebed1e260316d675ca7d104d83fabcd8b5717e.png)
Given that
![\[
f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5,
\]](//latex.artofproblemsolving.com/9/f/9/9f92574f9fe14e2e30c3e51e194c661fbd4ece07.png)
find the value of
![\[
f(0) + f(1) + f(2) + \cdots + f(6).
\]](//latex.artofproblemsolving.com/4/0/7/4075d7efa78dda4c46a5cf764fb8852e8c0fb09b.png)
be a function satisfying![\[
f(f(x)) = x^2 - 6x + 6
\quad \text{for all} \quad x \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/8/a/e/8aebed1e260316d675ca7d104d83fabcd8b5717e.png)
Given that![\[
f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5,
\]](http://latex.artofproblemsolving.com/9/f/9/9f92574f9fe14e2e30c3e51e194c661fbd4ece07.png)
find the value of![\[
f(0) + f(1) + f(2) + \cdots + f(6).
\]](http://latex.artofproblemsolving.com/4/0/7/4075d7efa78dda4c46a5cf764fb8852e8c0fb09b.png)
2 replies
Weird exponent, but ok
oVlad 11
N
by wassupevery1
Source: 2021 ISL A7
Let 
be an integer, and let 
be 
non-negative real numbers that satisfy 
for all 
Show that ![\[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\]](//latex.artofproblemsolving.com/9/5/e/95e8548c24bd65d25822b852851e9445f328acf6.png)
Pakawut Jiradilok and Wijit Yangjit, Thailand
be an integer, and let 
be 
non-negative real numbers that satisfy 
for all 
Show that ![\[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\]](http://latex.artofproblemsolving.com/9/5/e/95e8548c24bd65d25822b852851e9445f328acf6.png)
Pakawut Jiradilok and Wijit Yangjit, Thailand
11 replies
Monic Polynomial
IstekOlympiadTeam 22
N
by zuat.e
Source: Romanian Masters 2017 D1 P2
Determine all positive integers 
satisfying the following condition: for every monic polynomial 
of degree at most with integer coefficients, there exists a positive integer 
and 
distinct integers 
such that ![\[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]](//latex.artofproblemsolving.com/e/e/8/ee8adc79dc258b021529245fec16e081da2d1b0a.png)
.
Note. A polynomial is monic if the coefficient of the highest power is one.
satisfying the following condition: for every monic polynomial 
of degree at most with integer coefficients, there exists a positive integer 
and 
distinct integers 
such that ![\[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]](http://latex.artofproblemsolving.com/e/e/8/ee8adc79dc258b021529245fec16e081da2d1b0a.png)
.Note. A polynomial is monic if the coefficient of the highest power is one.
22 replies
Weird Inequality Problem
Omerking 3
N
by Primeniyazidayi
Following inequality is given:
Find the range of values that can be taken by :
Where
are positive reals.
Find the range of values that can be taken by :
Where
are positive reals.
3 replies
Divisibility of 121
kalra 1
N
by maxamc
Source: Own.
would be divisible by 
for any integer values of 
, provided 
is not a factor of . How to prove this?
1 reply
and 
Prove that
be reals such that 
Prove that
Functional equation
Amin12 46
N
by FredAlexander
Source: Iranian 3rd round 2016 first Algebra exam
Find all function 
such that for all 
,
such that for all 
,
46 replies
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