A function $f (x)$ is defined for all positive integers. If $f (a) + f (b) = f (ab)$ for any two positive integers $a$ and $b$ and $f (3) = 5$, what is $f (27)$?
The function $f (n) = a\cdot n! + b$, where $a$ and $b$ are positive integers, is defined for all positive integers. If the range of $f$ contains two numbers that differ by 20, what is the least possible value of $f (1)$?
The function $f (n) = a ⋅ n! + b$, where a and b are positive integers, is defined for all positive integers. If the range of $f$ contains two numbers that differ by 20, what is the least possible value of $f (1)$?
Let $P(x)$ be the polynomial $x^3 + Ax^2 +Bx+C$ for some constants $A, B,$ and $C$. There exists constant $D$ and $E$ such that for all $x$, $P(x+1)=x^3 + Dx^2 + 54x +37$ and $P(x+2)=x^3 + 26x + Ex+115$. Compute the ordered triple $(A, B, C)$.
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
Proposed by Dorlir Ahmeti, Albania
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here $\mathbb{Z}$ denotes the set of integers.)
Proposed by Liam Baker, South Africa
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$x^2f(x)+f(1-x)=2x-x^4$$
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that the Cauchy equation $$f(x+y)=f(x)+f(y)$$ holds for all $x, q\in\mathbb{Q}$.
Find the range of function $y=x+\sqrt{x^2 -3x+2}$.
Solve $$\Big|\frac{1}{\log_{\frac{1}{2}}x+2}\Big|> \frac{3}{2}$$
If for any non-negative real numbers $x$ and $y$, function $f(x)$ satisfies the properties that $f(x)\ge 0$, $f(1)\ne 0$, and $f(x+y^2)=f(x)+2f^2(y)$ , compute the value of $f(2+\sqrt{3})$.
If the minimal and maximum values of function $$f(x)=-\frac{1}{2}x^2 + \frac{13}{2}$$ in the domain $[a, b]$ are $2a$ and $2b$, respectively, determine the values of $a$ and $b$.
Is function $f(x)=\lg(x+\sqrt{x^2+1})$ an odd or even function?
For any real numbers $x$ and $y$, the following holds $$[f(x+y)]^2 = [f(x)]^2 + [f(y)]^2$$
Find the exact form of $f(x)$.
Let $f(x)$ be a polynomial with respect to $x$ and $$f(x+1)+f(x-1)=2x^2-4x$$ Find $f(x)$.
Find the function $f(x)$ such that $f(0)=1$, $f(\frac{\pi}{2})=2$, and for any $x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)\cos y$$
Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any $m, n\in\mathbb{N}$, $$f(m+n)=f(m)+f(n)+mn$$
Determine $f(n)$.
Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any integer $n \ge 2$, $$f(n)=f(n-1) + 2^{n-1}$$
Determine $f(n)$.
Let real numbers $x$, $y$, and $z$ satisfy $0 < x, y, z < 1$. Prove $$x(1-y)+y(1-z)+z(1-x)< 1$$
Show that $$\frac{(x+a)(x+b)}{(c-a)(c-b)}+\frac{(x+b)(x+c)}{(a-b)(a-c)}+\frac{(x+c)(x+a)}{(b-c)(b-a)}=1$$ without expanding the left side of the equation.
Find the range of function $f(x)=3^{-|\log_2x|}-4|x-1|$.
Let $f(x)=x^3 -x^2 -13x+24$. Find three pairs of $(x,y)$ such that if $y=f(x)$, then $x=f(y)$.
The function $f$ satisfies $f(0)=0$, $f(1)=1$, and $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ for all $x,y\in\mathbb{R}$. Show that $f(x)=x$ for all rational numbers $x$.
Let $f$ be a function such that $$ \sqrt {x - \sqrt { x + f(x) } } = f(x) , $$for $x > 1$. In that domain, $f(x)$ has the form $\frac{a+\sqrt{cx+d}}{b},$ where $a,b,c,d$ are integers and $a,b$ are relatively prime. Find $a+b+c+d.$
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that
$$f(yf(x)-x)=f(x)f(y)+2x$$
for all $x,\ y\in{\mathbb{R}}$.