## 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)”

## China Team Selection Test 2016 (3)

Đây là phần cuối, mời các bạn xem 2 phần trước ở https://nttuan.org/2016/04/11/topic-771/https://nttuan.org/2016/04/09/topic-769/

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Ngày thứ nhất

Bài 13. Cho số nguyên $n$ lớn hơn $1$, $\alpha$ là số thực thỏa mãn $0<\alpha < 2$, $a_1,\ldots ,a_n,c_1,\ldots ,c_n$ là các số nguyên dương. Với $y>0$, đặt $f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}.$ Với số dương $x$ thỏa mãn $x\ge f(y)$ (với $y$ nào đấy), chứng minh $f(x)\le 8^{\frac{1}{\alpha}}\cdot x$.

Bài 14. Trong mặt phẳng tọa độ, những điểm với cả hai tọa độ là số hữu tỷ sẽ được gọi là các điểm hữu tỷ. Với mỗi số nguyên dương $n$, liệu có thể dùng $n$ màu để tô tất cả các điểm hữu tỷ (mỗi điểm tô bởi $1$ màu) sao cho mỗi đoạn với các đầu mút là các điểm hữu tỷ chứa các điểm hữu tỷ mang mỗi màu?

Bài 15. Cho tứ giác nội tiếp $ABCD$$AB>BC$, $AD>DC$, $I,J$ là tâm nội tiếp của $\triangle ABC$,$\triangle ADC$ tương ứng. Đường tròn đường kính $AC$ cắt đoạn $IB$ tại $X$, và phần kéo dài của $JD$ tại $Y$. Chứng minh nếu $B,I,J,D$ cùng nằm trên một đường tròn thì $X$$Y$ đối xứng với nhau qua $AC$. Continue reading “China Team Selection Test 2016 (3)”

## 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 $x,y$.