Found problems: 1513
Let $f(x)$ be a polynomial such that $2f(x) + f(2 - x) = 5 + x$ for any real number x. Find the value of $f(0) + f(2)$.
A. $4$ B. $0$ C.$ 2$ D. $3$ E. $1$
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: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]
[i]Proposed by Bruno Le Floch, France[/i]
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.
Determine all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ satisfying $xf(xf(2y))=y+xyf(x)$ for all $x,y>0$.
Let $F$ be an increasing real function defined for all $x, 0 \le x \le 1$, satisfying the conditions
(i) $F (\frac{x}{3}) = \frac{F(x)}{2}$.
(ii) $F(1- x) = 1 - F(x)$.
Determine $F(\frac{173}{1993})$ and $F(\frac{1}{13})$ .
There is a unique continuous function $f$ over the positive real numbers satisfying $f(4) = 1$ and
\[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \]
for all positive $x$. Compute the value of $\int_{0}^{140} (f(x))^3\,dx$.
Let $\alpha\in\mathbb{Q}^+$. Determine all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ that for all $x,y\in\mathbb{Q}^+$ satisfy the equation
\[ f\left(\frac{x}{y}+y\right) ~=~ \frac{f(x)}{f(y)}+f(y)+\alpha x.\]
Here $\mathbb{Q}^+$ denote the set of positive rational numbers.
(Proposed by Walther Janous)
Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .
Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that:
$$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$
for all reals $x, y $.
Find all functions $f : R \to R$ such that $$f(x + y) \ge xf(x) + yf(y)$$, for all $x, y \in R$ .
Find all functions \(f:\mathbb{R}^{+} \to \mathbb{R}^{+}\) such that
\[
f(x) > x \ \ \text{and} \ \ f(x-y+xy+f(y)) = f(x+y) + xf(y)
\]
for arbitrary positive real numbers \(x\) and \(y\).
The function ƒ. which is defined for all real numbers satisfies:
$$f(x+y)+f(x-y)=2f(x)+2f(y)$$
Prove that $f(0) = 0$, $f(-x) = f(x)$, $f(2x) = 4 f (x)$, $$f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).$$
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
[i]Proposed by Giannis Galamatis, Greece[/i]
Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$:
\[ f(3mn \plus{} m \plus{} n) \equal{} 4f(m)f(n) \plus{} f(m) \plus{} f(n).
\]
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\le y\le x^3,$ then $(f(x))^2\le f(y) \le (f(x))^3.$
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
[i]Proposed by Netherlands[/i]
Find all the functions $f : C \to R^+$ such that they fulfill simultaneously the following conditions:
$$(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C$$
$$(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C$$
$$(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C$$
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$
holds for all positive integers $x, y$.
Proposed by Adrian Beker.
Let $n \ge 4$ be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition $[n] = A \cup B \cup C$ into disjoint sets, \[ \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c) = |A| |B| |C|. \]
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[ f(xy) = yf(x) + x + f(f(y) - f(x)) \]
for all $x,y \in \mathbb{R}$.
Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $n\in \mathbb{N}_{0}$: \[f(m+f(n))=f(f(m))+f(n).\]
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
Higher Secondary P2
Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.
Does there exist a function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that:
$f(f(x))=-x$, for all $x\in \mathbb{R}$?