Given finitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial $C(q)$ of degree two so that the following holds: for any $q\ge 1$ integer, if the half planes cover each point of the plane at least $q$ times, then the set of points covered exactly $q$ times is the union of at most $C(q)$ domains. (translated by L. Erdős)

Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied: (a) $xf(x,y,z) = zf(z,y,x)$, (b) $f(x, ky, k^2z) = kf(x,y,z)$, (c) $f(1, k, k+1) = k+1$. ([i]United Kingdom[/i])

Find the area of the domain of the system of inequality \[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]

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\}$

Let $S$ be the domain in the coordinate plane determined by two inequalities: \[y\geq \frac 12x^2,\ \ \frac{x^2}{4}+4y^2\leq \frac 18.\] Denote by $V_1$ the volume of the solid by a rotation of $S$ about the $x$-axis and by $V_2$, by a rotation of $S$ about the $y$-axis. (1) Find the values of $V_1,\ V_2$. (2) Compare the size of the value of $\frac{V_2}{V_1}$ and 1.

$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n))+f(n)=2n+3$ for all nonnegative integers $n$. Find $f(2014)$.

(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]

Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied: (a) $xf(x,y,z) = zf(z,y,x)$, (b) $f(x, ky, k^2z) = kf(x,y,z)$, (c) $f(1, k, k+1) = k+1$. ([i]United Kingdom[/i])

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

Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$, $(1+f(x)f(y))f(x+y)=f(x)+f(y)$.

For any set $ A $ we say that two functions $ f,g:A\longrightarrow A $ are [i]similar,[/i] if there exists a bijection $ h:A\longrightarrow A $ such that $ f\circ h=h\circ g. $ [b]a)[/b] If $ A $ has three elements, construct a finite, arbitrary number functions, having as domain and codomain $ A, $ that are two by two similar, and every other function with the same domain and codomain as the ones determined is similar to, at least, one of them. [b]b)[/b] For $ A=\mathbb{R} , $ show that the functions $ \sin $ and $ -\sin $ are similar.

Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that: $ \textbf{(A)}\ \text{They do not intersect}$ $ \qquad\textbf{(B)}\ \text{They intersect at 1 point only}$ $\qquad\textbf{(C)}\ \text{They intersect at 2 points only}$ $\qquad\textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }$ ${\qquad\textbf{(E)}\ \text{They coincide} } $

Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions: a)$ \mathbb{Z} \varsubsetneqq I$; b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$); c) $ I$ only has trivial endomorphisms. [i]E. Fried[/i]

In the $xyz$-plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$-plane. (1) Let $P(0,\ 0,\ 2)$. When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$, find the equation of the locus of the point $R$. (2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$. When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$, draw the domain swept by the point $R$ on the $xy$-plane, then find the area.

Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.

Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$, where $0<t<1$. Prove that the area of any triangle inside the region $A$, with two vertices $P(t,t^2)$ and $Q(1,0)$, does not exceed $\frac{1}{4}.$

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 $f_n(x)=x$.

Let $x,\ y$ be real numbers. For a function $f(t)=x\sin t+y\cos t$, draw the domain of the points $(x,\ y)$ for which the following inequality holds. \[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]

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

Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

Suppose that $x$ is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for $0\le x\le \frac{\pi}{16}$

The plane is divided into domains by $ n$ straight lines in general position, where $ n \geq 3$. Determine the maximum and minimum possible number of angular domains among them. (We say that $ n$ lines are in general position if no two are parallel and no three are concurrent.)

Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).

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]$

The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A) }19\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad$