back to index  |  new

398
For every positive integer $n$, let $\text{mod}_5 (n)$ be the remainder obtained when $n$ is divided by 5. Define a function $f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively as follows: \[f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4. \end{cases}\] What is $f(2015,2)$?

466
Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$ ?

613
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50 < f(7) < 60$, $70 < f(8) < 80$, $5000k < f(100) < 5000(k+1)$ for some integer $k$. What is $k$?

Let $f(x) = \sqrt{2^2-x^2}$. Find the value of $f(f(f(f(f(-1)))))$.

Let $f$ be a real-valued function such that $f(x) + 2f(\frac{2002}{x}) = 3x$ for all $x > 0$. Find $f(2)$.

Let function $f(x)$ is defined as the following: $$ f(x)= \left\{ \begin{array}{ll} x+2 &, \text{if } x \le -1\\ 2x &, \text{if } -1 < x < 2\\ \displaystyle\frac{x^2}{2} &, \text{if } x \ge 2 \end{array} \right. $$ (A) Compute $f(f(f(-\frac{7}{4})))$ (B) If $f(a)=3$, find the value of $a$

Let function $f(x)$ satisfy $f(a)+f(b)=f(ab)$, and $f(2)=2$ and $f(3)=3$. Compute $f(72)$.

A function $f$ has its domain equal to the set of integers $\{0, 1, ..., 11\}$, and $f(n)\ge 0$ for all such $n$, and $f$ satisfies: $f(0) = 0$, $f(6) = 1$. If $x \ge 0$, $y\ge 0$, and $x + y\le 11$, then $f(x + y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$. Find $f(2)^2 + f(10)^2$.

Let even function $f(x)$ and odd function $g(x)$ satisfy the relationship of $f(x)+g(x)=\sqrt{1+x+x^2}$. Find $f(3)$.

Let $f\Big(\dfrac{1}{x}\Big)=\dfrac{1}{x^2+1}$. Compute $$f\Big(\dfrac{1}{2013}\Big)+f\Big(\dfrac{1}{2012}\Big)+f\Big(\dfrac{1}{2011}\Big)+\cdots +f\Big(\dfrac{1}{2}\Big)+f(1)+f(2)+\cdots +f(2011)+f(2012)+f(2013)$$

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 $f (n) = a ⋅ n! + b$, 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 $f (1)$?

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 $f(2+\sqrt{3})$.

Find the number of functions $f(x)$ from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1, 2, 3, 4, 5\}$.