Question

Characterize all continuous functions such that $\int_0^1 \left(f\left(\sqrt[i]{x}\right)\right)^{k - i}\,dx = {i\over k}$

score 78 · Answer 1 · 2016-04-09 20:18:44

78

Views

Characterize all continuous functions such that $\int_0^1 \left(f\left(\sqrt[i]{x}\right)\right)^{k - i}\,dx = {i\over k}$

Published on 09 Apr 2016 - 20:18

score 255 · Answer 2 · 2016-04-10 03:26:12

255

Views

Find all functions $f$, defined over real numbers that satisfy $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$

Published on 10 Apr 2016 - 3:26

score 236 · Answer 3 · 2016-04-10 09:56:30

Find all functoins $f: \mathbb R \rightarrow \mathbb R$ such that $\forall x,y \in \mathbb R f(xy+f(x))=xf(y)+f(x)$

score 53 · Answer 4 · 2016-04-11 03:12:58

53

Views

Find all functions $f$ and $g$ for which $f(x+y) = g(xy)$.

Published on 11 Apr 2016 - 3:12

score 250 · Answer 5 · 2016-04-11 10:22:42

250

Views

All-Russian Olympiad question (composite of quadratics)

Published on 11 Apr 2016 - 10:22

score 128 · Answer 6 · 2016-04-12 00:18:57

Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$

score 354 · Answer 7 · 2016-04-12 03:10:01

354

Views

Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$

Published on 12 Apr 2016 - 3:10

score 80 · Answer 8 · 2016-04-13 02:51:10

80

Views

If $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$, find $f$

Published on 13 Apr 2016 - 2:51

score 65 · Answer 9 · 2016-04-13 12:37:04

65

Views

Functional equation $g(2x )= 1/2 g(x)$

Published on 13 Apr 2016 - 12:37

score 161 · Answer 10 · 2016-04-14 03:08:16

161

Views

Functional Equation $f(f(x))=af(x)+bx$

Published on 14 Apr 2016 - 3:08

score 162 · Answer 11 · 2016-04-14 09:09:15

162

Views

Is the gamma function a solution to this problem?

Published on 14 Apr 2016 - 9:09

score 304 · Answer 12 · 2016-04-14 16:55:56

304

Views

Method of solving the functional equation $f(2x)=f(x)$ using Lagrange's Mean Value Theorem

Published on 14 Apr 2016 - 16:55

score 73 · Answer 13 · 2016-04-14 17:50:05

73

Views

Find all functions $f$ over reals such that $f(xy) \geq f(x+y)$ for all $x,y \in \mathbb{R}$.

Published on 14 Apr 2016 - 17:50

score 1.1k · Answer 14 · 2016-04-15 02:03:53

1.1k

Views

Find all continuous functions over reals such that $f(x)+f(y) = f(x+y)-xy-1$ for all $x,y \in \mathbb{R}$

Published on 15 Apr 2016 - 2:03

score 283 · Answer 15 · 2016-04-15 06:59:10

283

Views

There is no function from $\mathbb{R} \to (0, \infty)$ satisfying $f'(x)=f(f(x))$

Published on 15 Apr 2016 - 6:59

List Question

Characterize all continuous functions such that $\int_0^1 \left(f\left(\sqrt[i]{x}\right)\right)^{k - i}\,dx = {i\over k}$

Find all functions $f$, defined over real numbers that satisfy $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$

Find all functoins $f: \mathbb R \rightarrow \mathbb R$ such that $\forall x,y \in \mathbb R f(xy+f(x))=xf(y)+f(x)$

Find all functions $f$ and $g$ for which $f(x+y) = g(xy)$.

All-Russian Olympiad question (composite of quadratics)

Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$

Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$

If $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$, find $f$

Functional equation $g(2x )= 1/2 g(x)$

Functional Equation $f(f(x))=af(x)+bx$

Is the gamma function a solution to this problem?

Method of solving the functional equation $f(2x)=f(x)$ using Lagrange's Mean Value Theorem

Find all functions $f$ over reals such that $f(xy) \geq f(x+y)$ for all $x,y \in \mathbb{R}$.

Find all continuous functions over reals such that $f(x)+f(y) = f(x+y)-xy-1$ for all $x,y \in \mathbb{R}$

There is no function from $\mathbb{R} \to (0, \infty)$ satisfying $f'(x)=f(f(x))$

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