(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$, where $m$ is an integer and $r \in [0,1)$. Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(xy)=\max\{f(x+y),f(x) f(y)\} \] for all real numbers $x$ and $y$.

Find the minimum value of \[|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}|\] for real numbers $x$.

An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ$. Let $p$ be the probability that the numbers $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n$, where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000$, find $m + n$.

Let $\mathcal{X}$ be the collection of all non-empty subsets (not necessarily finite) of the positive integer set $\mathbb{N}$. Determine all functions $f: \mathcal{X} \to \mathbb{R}^+$ satisfying the following properties: (i) For all $S$, $T \in \mathcal{X}$ with $S\subseteq T$, there holds $f(T) \le f(S)$. (ii) For all $S$, $T \in \mathcal{X}$, there hold \[f(S) + f(T) \le f(S + T),\quad f(S)f(T) = f(S\cdot T), \] where $S + T = \{s + t\mid s\in S, t\in T\}$ and $S \cdot T = \{s\cdot t\mid s\in S, t\in T\}$. [i]Proposed by Li4, Untro368, and Ming Hsiao.[/i]

Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.

Let $(f_i)$ be a sequence of functions defined by: $f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14$. ($n\in \mathbb{N}, n\ge2$) (a) Prove that $f_n(x) \le f_{n-1}(x)$ for all x where both functions are defined. (b) Find for each $n$ the points of $x$ inside the domain for which $f_n(x)=x$.

Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

Let $ R$ be a bounded domain of area $ t$ in the plane, and let $ C$ be its center of gravity. Denoting by $ T_{AB}$ the circle drawn with the diameter $ AB$, let $ K$ be a circle that contains each of the circles $ T_{AB} \;(A,B \in R)$. Is it true in general that $ K$ contains the circle of area $ 2t$ centered at $ C$? [i]J. Szucs[/i]

The function $ f$ has the property that for each real number $ x$ in its domain, $ 1/x$ is also in its domain and \[ f(x) \plus{} f\left(\frac {1}{x}\right) \equal{} x. \]What is the largest set of real numbers that can be in the domain of $ f$? $ \textbf{(A) } \{ x | x\ne 0\} \qquad \textbf{(B) } \{ x | x < 0\} \qquad \textbf{(C) }\{ x | x > 0\}\\ \textbf{(D) } \{ x | x\ne \minus{} 1 \text{ and } x\ne 0 \text{ and } x\ne 1\} \qquad \textbf{(E) } \{ \minus{} 1,1\}$

A function $ f$ has domain $ [0,2]$ and range $ [0,1]$. (The notation $ [a,b]$ denotes $ \{x: a\le x\le b\}$.) What are the domain and range, respectively, of the function $ g$ defined by $ g(x)\equal{}1\minus{}f(x\plus{}1)$? $ \textbf{(A)}\ [\minus{}1,1],[\minus{}1,0] \qquad \textbf{(B)}\ [\minus{}1,1],[0,1] \qquad \textbf{(C)}\ [0,2],[\minus{}1,0] \qquad \textbf{(D)}\ [1,3],[\minus{}1,0] \qquad \textbf{(E)}\ [1,3],[0,1]$

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.