Found problems: 125
Find the volume of the part bounded by $z=x+y,\ z=x^2+y^2$ in the $xyz$ space.
Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that
\[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]
Let $ n\geq 3 $ be an integer and let $ p\geq 2n-3 $ be a prime number. For a set $ M $ of $ n $ points in the plane, no 3 collinear, let $ f: M\to \{0,1,\ldots, p-1\} $ be a function such that
(i) exactly one point of $ M $ maps to 0,
(ii) if a circle $ \mathcal{C} $ passes through 3 distinct points of $ A,B,C\in M $ then $ \sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p $.
Prove that all the points in $ M $ lie on a circle.
$P,Q,R$ are non-zero polynomials that for each $z\in\mathbb C$, $P(z)Q(\bar z)=R(z)$.
a) If $P,Q,R\in\mathbb R[x]$, prove that $Q$ is constant polynomial.
b) Is the above statement correct for $P,Q,R\in\mathbb C[x]$?
The set of all real numbers $ x$ for which
\[ \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))
\]is defined is $ \{x|x > c\}$. What is the value of $ c$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2001^{2002} \qquad \textbf{(C)}\ 2002^{2003} \qquad \textbf{(D)}\ 2003^{2004} \qquad \textbf{(E)}\ 2001^{2002^{2003}}$
Consider the positive reals $ x$, $ y$ and $ z$. Prove that:
a) $ \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2}$ iff $ xy < 1$.
b) $ \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi$ iff $ xyz < x \plus{} y \plus{} z$.
The sequence \[\log_{12}{162},\, \log_{12}{x},\, \log_{12}{y},\, \log_{12}{z},\, \log_{12}{1250}\] is an arithmetic progression. What is $x$?
$ \textbf{(A)} \ 125\sqrt{3} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}$
Let $A=\{1, 3, 5, 7, 9\}$ and $B=\{2, 4, 6, 8, 10\}$. Let $f$ be a randomly chosen function from the set $A\cup B$ into itself. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probablity that $f$ is a one-to-one function on $A\cup B$ given that it maps $A$ one-to-one into $A\cup B$ and it maps $B$ one-to-one into $A\cup B$. Find $m+n$.
Find the area of the domain expressed by the following system inequalities.
\[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]
Find all pairs of positive integers $(m,n)$ such that
\[m+n-\frac{3mn}{m+n}=\frac{2011}{3}.\]
Solve the system
\[(4x)_5+7y=14 \\ (2y)_5 -(3x)_7=74\]
in the domain of integers, where $(n)_k$ stands for the multiple of the number $k$ closest to the number $n$.
Let $ f_1,f_2,\dots,f_n$ be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions $ f_i\overline{f}_k, \;1 \leq i,k \leq n$, are also linearly independent.
[i]L. Lempert[/i]
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 $BC$.
For $ \phi: \mathbb{N} \mapsto \mathbb{Z}$ let us define \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{Z}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}.\] Prove that if $ M_{\phi_1} \equal{} M_{\phi_2} \neq \emptyset,$ then $ \phi_1 \equal{} \phi_2.$ Does this property remain true if \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{N}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}?\]
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]
Find the area of the domain of the system of inequality
\[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]
Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find
\[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]
On the domain $0\leq \theta \leq 2\pi:$
(a) Prove that $\sin^{2}\theta \cdot \sin 2\theta$ takes its maximum at $\frac{\pi}{3}$ and $\frac{4 \pi}{3}$ (and hence its minimum at $\frac{2 \pi}{3}$ and $\frac{ 5 \pi}{3}$).
(b) Show that
$$| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta |$$
takes its maximum at $\frac{4 \pi}{3}$ (the maximum may also be attained at other points).
(c) Derive the inequality:
$$ \sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.$$
Find the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$
Let $f_1(x)=\sqrt{1-x}$, and for integers $n \ge 2$, let $f_n(x)=f_{n-1}(\sqrt{n^2-x})$. If $N$ is the largest value of $n$ for which the domain of $f_n$ is nonempty, the domain of $f_N$ is ${c}$. What is $N+c$?
$ \textbf{(A)}\ -226 \qquad
\textbf{(B)}\ -144 \qquad
\textbf{(C)}\ -20 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 144$
Calculate the following indefinite integrals.
[1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$
[2] $\int x\log_{10} x dx$
[3] $\int \frac{x}{\sqrt{2x-1}}dx$
[4] $\int (x^2+1)\ln x dx$
[5] $\int e^x\cos x dx$
Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ \text{infinitely many}$
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 $m+n$?
$\textbf{(A) }19\qquad
\textbf{(B) }31\qquad
\textbf{(C) }271\qquad
\textbf{(D) }319\qquad
\textbf{(E) }511\qquad$
(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]