## Functional inequalities (1)

Problem 1. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that

$\displaystyle \frac{1}{2}f(xy)+\frac{1}{2}f(xz)-f(x)f(yz)\geq\frac{1}{4}\,\,\forall x,y,z\in\mathbb{R}.$

Problem 2. Let $f:(0;+\infty)\to (0;+\infty)$ be a function such that

$f(2x)\geq x+f(f(x))\,\,\forall x\in (0;+\infty).$ Prove that $f(x)\geq x\,\,\forall x\in (0;+\infty).$

Problem 3. Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that

$f(x+19)-19\leq f(x)\leq f(x+94)-94\,\,\forall x\in\mathbb{R}.$ Prove that $f(x+1)=f(x)+1\,\,\forall x\in\mathbb{R}.$

Problem 4. Find all functions $f:[1;+\infty)\to [1;+\infty)$ such that

$f(x)\leq 2x+2\,\,\text{and}\,\, xf(x+1)=f^2(x)-1\,\,\forall x\in [1;+\infty).$

Problem 5. Find all functions $f:\mathbb{N}\to \mathbb{N}$ such that

$mf(n)+nf(m)=(m+n)f(m^2+n^2)\,\,\forall m,n\in \mathbb{N}.$

Problem 6. Find all injective mappings $f:\mathbb{N}^*\to\mathbb{N}^*$ such that for all positive integers $n$ the following relation holds: $f(f(n)) \leq \dfrac {n+f(n)}{2}.$

Problem 7. Find all surjective mappings $f:\mathbb{N}^*\to\mathbb{N}^*$ such that for all positive integers $n$ the following relation holds: $f(n) \geq n+(-1)^n.$ Continue reading “Functional inequalities (1)”

## Phương trình hàm-07/03/2016

Bài 1. Cho hàm số $f:(0;\infty)\to (0;\infty)$ thoả mãn điều kiện
$f(3x)\geq f(\dfrac{1}{2}f(2x))+2x\,\,\forall x>0.$ Chứng minh rằng $f(x)\geq x\,\,\forall x>0$.
Bài 2. Tìm tất cả các hàm số $f:(0;+\infty)\to\ (0;\infty)$ sao cho
$f(f(x))=6x-f(x)\,\,\forall x\in (0;+\infty).$
Bài 3. Tìm tất cả các song ánh $f:(0;+\infty)\to\ (0;\infty)$ sao cho
$f(f(x))=6x+f(x)\,\,\forall x\in (0;+\infty).$
Bài 4. Tìm tất cả các hàm số $f:[1,\infty)\to [1,\infty)$ sao cho
a) $f(x)\leq 2(x+1)\,\,\forall x\geq 1$
b) $f(x+1)=\dfrac{f^2(x)-1}{x}\,\,\forall x\geq 1$.
Bài 5. Tìm tất cả các hàm số $f:\mathbb{R}\to\mathbb{R}$ thoả mãn
$f(x^2+y+f(y))=f^2(x)+2y\,\,\forall x,y\in\mathbb{R}.$
Bài 6. Tìm tất cả $f:(0;+\infty)\to\mathbb{R}$ thỏa mãn đồng thời hai điều kiện
1) $f(x)$ liên tục trên $(0;+\infty)$;
2) $f(x)=f\left(\dfrac{3x+1}{x+1}\right)\,\,\forall x\in (0;+\infty)$.
Bài 7. Tìm tất cả các hàm số $f:\mathbb{R}\to\mathbb{R}$ sao cho nó liên tục trên $\mathbb{R}$
$f(x+f(y+z))+f(y+f(z+x))+f(z+f(x+y))=0\,\,\forall x,y,z\in\mathbb{R}.$ Continue reading “Phương trình hàm-07/03/2016”

## Bài tập Hàm số

1. Find all functions $f$ from the set $\mathbb{R}$ of real numbers into $\mathbb{R}$ which satisfy for all $x, y, z \in \mathbb{R}$ the identity

$f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).$

2. Consider the function $f: \mathbb{N}_0\to\mathbb{N}_0$, where $\mathbb{N}_0$ is the set of all non-negative integers, defined by the following conditions

(i)$f(0)=0$;

(ii)$f(2n)=2f(n)$ and

(iii)$f(2n+1)=n+2f(n)$ for all $n\geq 0$.

(a)Determine the three sets $L=\{ n | f(n) < f(n+1) \}$, $E=\{n | f(n)=f(n+1)\}$, and $G=\{n | f(n) > f(n+1)\}$.

(b)For each $k\geq 0$, find a formula for $a_k=\max\{f(n) : 0 \leq n \leq 2^k\}$ in terms of $k$.

3. Let $n$ be a positive integer. Find the largest nonnegative real number $f(n)$ (depending on $n$) with the following property: whenever $a_1,a_2,...,a_n$ are real numbers such that $a_1+a_2+\cdots +a_n$ is an integer, there exists some $i$ such that  $\left|a_i-\dfrac{1}{2}\right|\ge f(n)$.

4. Let ${\bf R}$ denote the set of all real numbers. Find all functions $f$ from ${\bf R}$ to ${\bf R}$ satisfying

(i)There are  only finitely many $s$ in ${\bf R}$ such that $f(s)=0$,

And

(ii)$f(x^4+y)=x^3f(x)+f(f(y))$ for all $x,y$ in ${\bf R}$.

5. Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that $a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)$

for all $x,y\in(0,1].$

6. Consider function $f: \mathbb{R}\to\mathbb{R}$ which satisfies the conditions for any mutually distinct real numbers $a,b,c,d$ satisfying $\dfrac{a-b}{b-c}+\dfrac{a-d}{d-c}=0$, $f(a),f(b),f(c),f(d)$ are mutully different and

$\dfrac{f(a)-f(b)}{f(b)-f(c)}+\dfrac{f(a)-f(d)}{f(d)-f(c)}=0.$ Prove that function $f$ is linear.

7. Find all complex polynomial $P(x)$ such that for any three integers $a,b,c$ satisfying $a+b+c\not=0,$ $\dfrac{P(a)+P(b)+P(c)}{a+b+c}$ is an integer.

8. Find all functions $f:\mathbb{Q}^{+}\to\mathbb{Q}^{+}$ such that

$f(x)+f(y)+2xyf(xy)=\dfrac{f(xy)}{f(x+y)}\forall x,y\in\mathbb{Q}^+.$

9. Let $\alpha$ be given positive real number, find all the functions $f:\mathbb{N}^{+}\to\mathbb{R}$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.

10. Given non-zero reals $a$, $b$, find all functions $f: \mathbb{R} \to \mathbb{R}$, such that for every $x, y \in \mathbb{R}$, $y \neq 0$, $f(2x)=af(x)+bx$ and $f(x)f(y)=f(xy)+f\left(\dfrac{x}{y}\right).$

11. Prove that for all integers $a > 1$ and $b > 1$ there exists a function $f$ from the positive integers to the positive integers such that $f(a\cdot f(n))=b\cdot n$ for all $n$ positive integer.

12. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(xf(y)+f(x)) = 2f(x)+xy$ for every reals $x,y$.

13. Let $f(x)$ be a real-valued function defined on the positive reals such that

(1)If $x < y$, then $f(x) < f(y)$,

(2)$f\left(\dfrac{2xy}{x+y}\right) \geq\dfrac{f(x) + f(y)}{2}$ for all $x,y>0$.

Show that $f(x) < 0$ for some value of $x$.

14. Define $f$ on the positive integers by $f(n) = k^2 + k + 1$, where $n=2^k(2l+1)$ for some $k,l$ nonnegative integers. Find the smallest $n$ such that

$f(1) + f(2) + ... + f(n) \geq 123456.$

15. Find all functions $f:\mathbb{N}\to\mathbb{N}$ satisfying, for all $x\in\mathbb{N}$, $f(2f(x)) = x + 1998.$

16.  a) Show that there are no functions $f,g: \mathbb{R}\to\mathbb{R}$ such that $g(f(x)) = x^3$ and $f(g(x)) = x^2$ for all $x\in\mathbb{R}$.

b)Let $S$ be the set of all real numbers greater than $1$. Show that there are functions $f,g:S\to S$ satsfying the condition above.

17. Let $f(x)= x^2-C$ where $C$ is a rational constant. Show that exists only finitely many rationals $x$ such that $\{x,f(x),f(f(x)),\cdots\}$ is finite.

18. Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$, and $f(xy) = f(x)f(y)$for all $x,y$.

19. Find at least one function $f:\mathbb{R}\to\mathbb{R}$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$.

20. Let $f(x)=\dfrac{ax+b}{cx+d}$, $F_n(x)=f(f(f\cdots (f(x))\cdots))$ (with $n\ f's$). Suppose that $f(0)\not =0$, $f(f(0))\not =0$, and for some $n$ we have $F_n(0)=0$, show that $F_n(x)=x$ (for any valid $x$).

21. Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$ $f(x-y+f(y))=f(x)+f(y).$

22. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $\forall x,y,z\in\mathbb{R}$ we have: If $x^3+f(y) \cdot x+f(z)=0,$ then $f(x)^3+y \cdot f(x)+z=0.$

23. Let $S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions

(i) Both functions are strictly increasing, i.e. $f(x) < f(y)$ and $g(x) < g(y)$ for all $x$, $y\in S$ with $x < y$;

(ii) The inequality $f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $x\in S$.

Decide whether there exists a Spanish Couple

a)On the set $S=\mathbb{N}$ of positive integers;

b)On the set $S=\{a-\dfrac{1}{b}: a, b\in\mathbb{N}\}$.

24. For every $n\in\mathbb{N}$ let $d(n)$ denote the number of (positive) divisors of $n$. Find all functions $f:\mathbb{N}\to\mathbb{N}$ with the following properties:

a)$d\left(f(x)\right)=x$ for all $x\in\mathbb{N}$, and

b)$f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x$, $y\in\mathbb{N}$.

25. Consider those functions $f:\mathbb{N}\to\mathbb{N}$ which satisfy the condition $f(m+n) \geq f(m)+f(f(n))-1$ for all $m,n\in\mathbb{N}.$ Find all possible values of $f(2007).$

26. Find all surjective functions $f:\mathbb{N}\to\mathbb{N}$ such that for every $m,n\in\mathbb{N}$ and every prime $p,$ the number $f(m+n)$ is divisible by $p$ if and only if $f(m)+f(n)$ is divisible by $p.$

27. Find all real polynomials $f$ such that $2yf(x+y)+(x-y)(f(x)+f(y)) \geq 0\forall x,y\in\mathbb{R}.$

28. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ with $x,y\in\mathbb{R}$ such that $f(x-f(y))=f(x+y)+f(y).$

29. Show that for positive integer $n$, and for $x\not =0$,

$\left(x^{n-1}\sin\dfrac{1}{x}\right)^{(n)}=\dfrac{(-1)^n}{x^{n+1}}\sin\left(\dfrac{1}{x}+\dfrac{n\pi}{2}\right).$

30. Find all $f:\mathbb{R}\to\mathbb{R}$ such that

$f(xy+f(x))=xf(y)+f(x)$

for every pair of real numbers $x,y$.