Question

$f:\mathbb N_0\to\mathbb N_0$ with $2f\left(m^2+n^2\right)=f(m)^2+f(n)^2$ and $f\left(m^2\right)\geqslant f\left(n^2\right)$ when $m\geqslant n$

score 139 · Answer 1 · 2020-05-22 12:50:24

139

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$f:\mathbb N_0\to\mathbb N_0$ with $2f\left(m^2+n^2\right)=f(m)^2+f(n)^2$ and $f\left(m^2\right)\geqslant f\left(n^2\right)$ when $m\geqslant n$

Published on 22 May 2020 - 12:50

score 1.7k · Answer 2 · 2020-05-23 04:45:51

The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(x) f(y) = f(x + y) + xy$ for all real numbers $x$ and $y$. Find all possible functions $f$.

score 53 · Answer 3 · 2026-03-30 09:35:58

53

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KL Divergence of (y|z) in terms of KL Divergence of (x|y) and (x|z)

Published on 30 Mar 2026 - 9:35

score 413 · Answer 4 · 2020-05-24 08:36:49

413

Views

$f(m^2 + f(n)) = f(m)^2 + n$ for all natural numbers

Published on 24 May 2020 - 8:36

score 299 · Answer 5 · 2020-05-25 01:01:30

299

Views

If $\cos 17x = f(\cos x)$, then show that $\sin 17 x=f(\sin x)$

Published on 25 May 2020 - 1:01

score 56 · Answer 6 · 2020-05-26 17:13:59

56

Views

Trying to solve $\frac{f(x) f(y) - f(xy)}{3} = x + y + 2$ for $f(x)$

Published on 26 May 2020 - 17:13

score 191 · Answer 7 · 2020-05-27 18:29:57

191

Views

What are the possible functions satisfying functional equation $f(x) f(y) = f(x + y) + xy$?

Published on 27 May 2020 - 18:29

score 60 · Answer 8 · 2020-05-27 19:04:43

60

Views

Help working with the functional equation $f(a) f(b) = f(a+b) + ab$

Published on 27 May 2020 - 19:04

score 194 · Answer 9 · 2020-05-28 18:14:51

194

Views

Is this a correct solution to this problem? 2018 AIME II Problem 10

Published on 28 May 2020 - 18:14

score 123 · Answer 10 · 2020-05-28 19:11:50

123

Views

Functional equation with the property $P(x+1)=P(x)+2x+1$

Published on 28 May 2020 - 19:11

score 248 · Answer 11 · 2020-05-29 03:10:39

248

Views

Given, $f(x+y) =xf(x) +2f(y)-x-y$. If $f(2019)=\frac{a}{b}$, then $a+b=?$

Published on 29 May 2020 - 3:10

score 89 · Answer 12 · 2020-05-30 00:01:56

89

Views

Find $19\Diamond 98$, given rules $(xy)\Diamond y=x(y\Diamond y)$, $(x\Diamond 1)\Diamond x=x\Diamond 1$, and $1\Diamond 1=1$

Published on 30 May 2020 - 0:01

score 179 · Answer 13 · 2020-05-30 10:10:24

179

Views

Find all continuous $f$ that $f(xy) = xf(y) + yf(x)$.

Published on 30 May 2020 - 10:10

score 355 · Answer 14 · 2020-05-30 16:54:42

355

Views

$af(a)+bf(b)+2ab$ is a perfect square for all $a,b \in \mathbb N$

Published on 30 May 2020 - 16:54

score 176 · Answer 15 · 2020-05-31 03:15:27

176

Views

Functional equation involving three different functions: $ f ( x + y ) = g ( x ) + h ( y ) $

Published on 31 May 2020 - 3:15

List Question

$f:\mathbb N_0\to\mathbb N_0$ with $2f\left(m^2+n^2\right)=f(m)^2+f(n)^2$ and $f\left(m^2\right)\geqslant f\left(n^2\right)$ when $m\geqslant n$

The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(x) f(y) = f(x + y) + xy$ for all real numbers $x$ and $y$. Find all possible functions $f$.

KL Divergence of (y|z) in terms of KL Divergence of (x|y) and (x|z)

$f(m^2 + f(n)) = f(m)^2 + n$ for all natural numbers

If $\cos 17x = f(\cos x)$, then show that $\sin 17 x=f(\sin x)$

Trying to solve $\frac{f(x) f(y) - f(xy)}{3} = x + y + 2$ for $f(x)$

What are the possible functions satisfying functional equation $f(x) f(y) = f(x + y) + xy$?

Help working with the functional equation $f(a) f(b) = f(a+b) + ab$

Is this a correct solution to this problem? 2018 AIME II Problem 10

Functional equation with the property $P(x+1)=P(x)+2x+1$

Given, $f(x+y) =xf(x) +2f(y)-x-y$. If $f(2019)=\frac{a}{b}$, then $a+b=?$

Find $19\Diamond 98$, given rules $(xy)\Diamond y=x(y\Diamond y)$, $(x\Diamond 1)\Diamond x=x\Diamond 1$, and $1\Diamond 1=1$

Find all continuous $f$ that $f(xy) = xf(y) + yf(x)$.

$af(a)+bf(b)+2ab$ is a perfect square for all $a,b \in \mathbb N$

Functional equation involving three different functions: $ f ( x + y ) = g ( x ) + h ( y ) $

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