Functions

Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of $$(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?$$

If $n\heartsuit m=n^3m^2$ , what is $\frac{2\heartsuit 4}{4\heartsuit 2}$ ?

Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$ . What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$ ?

Suppose $[a\ b]$ denotes the average of $a$ and $b$ , and $\{a\ b\ c\}$ denotes the average of $a, b$ , and $c$ . What is $\{\{1\ 1\ 0\}\ [0\ 1]\ 0\}$ ?

For a real number $x$ , define $\heartsuit(x)$ to be the average of $x$ and $x^2$ . What is $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$ ?

For positive numbers $x$ and $y$ the operation $x\spadesuit y$ is defined as $$x\spadesuit y = x -\dfrac{1}{y}$$ What is $2\spadesuit (2\spadesuit 2)$ ?

For real numbers $a$ and $b$ , define $a \$b$ $=(a-b)^2$ . What is $(x-y)^2\$(y-x)^2$ ?

Define the operation $\star$ by $a \star b = (a+b)b.$ What is $(3 \star 5) - (5 \star 3)?$

Define $a@b = ab - b^{2}$ and $a\#b = a + b - ab^{2}$ . What is $\frac {6@2}{6\#2}$ ?

Define $x\otimes y=x^3-y$ . What is $h\otimes (h\otimes h)$ ?

For real numbers $x$ and $y$ , define $x \spadesuit y = (x+y)(x-y)$ . What is $3 \spadesuit (4 \spadesuit 5)$ ?

For real numbers $a$ and $b$ , define $a \diamond b = \sqrt{a^2 + b^2}$ . What is the value of $(5 \diamond 12) \diamond ((-12) \diamond (-5))$ ?

For each pair of real numbers $a \neq b$ , define the operation $\star$ as $$(a \star b) = \frac{a+b}{a-b}.$$ What is the value of $((1 \star 2) \star 3)$ ?

For any three real numbers $a$ , $b$ , and $c$ , with $b\neq c$ , the operation $\otimes$ is defined by: $$\otimes(a,b,c)=\frac{a}{b-c}$$ What is $\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))$ ?

Given that a, b, and c are non-zero real numbers, define $(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ , find $(2, 12, 9)$ .

For the nonzero numbers a, b, and c, define $D(a,b,c)=\frac{abc}{a+b+c}$ Find $D(2,4,6)$ .

The function $f$ is defined by $$f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|$$ for all real numbers $x$ , where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$ . What is the range of $f$ ?

Let $@$ denote the "averaged with" operation: $a @ b = (a+b)/2$ . Which of the following distributive laws hold for all numbers $x, y,$ and $z$ ? $$\text{I. x @ (y + z) = (x @ y) + (x @ z)}$$ $$\text{II. x + (y @ z) = (x + y) @ (x + z)}$$ $$\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}$$

Let $f(x) = 100x^3 - 300x^2 + 200x$ . For how many real numbers $a$ does the graph of $y = f(x - a)$ pass through the point $(1, 25)$ ?

Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$ . Furthermore, suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$ . For which of the following numbers $x$ is $f(x)<0$ ?

The graph of $$f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|$$ is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$ .)

A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$ . (Here $\cdot$ represents multiplication). The solution to the equation …

Let $f$ be a function with the following properties: (i) $f(1) = 1$ , and (ii) $f(2n) = n \cdot f(n)$ for any positive integer $n$ . What is the value of $f(2^{100})$ ?

Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$ . Find the sum of all values of $z$ for which $f(3z) = 7$ . \[

Consider functions $f$ that satisfy $$|f(x)-f(y)|\leq \frac{1}{2}|x-y|$$ for all real numbers $x$ and $y$ . Of all such functions that also satisfy the equation $f(300) = f(900)$ , what is the greatest possible value of $$f(f(800))-f(f(400))?$$