Found problems: 913
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that
\[a_{m^2+n^2}=a_m^2+a_n^2 \]
for all integers $m,n\ge 0$.
For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.
The solutions to the system of equations
\begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*}
are $(x_1,y_1)$ and $(x_2, y_2).$ Find $\log_{30}{(x_1y_1x_2y_2)}.$
Are there such $x,y$ that
$\lg{(x+y)}=\lg x \lg y$ and $\lg{(x-y)}=\frac{\lg x}{\lg y}$ ?
Prove the inequality
\[ \sqrt {a^{1 \minus{} a}b^{1 \minus{} b}c^{1 \minus{} c}} \le \frac {1}{3}
\]
holds for all positive real numbers $ a$, $ b$ and $ c$ with $ a \plus{} b \plus{} c \equal{} 1$.
Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized.
Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$
[i]2010 Saitama University entrance exam/Mathematics[/i]
Last Edited
Prove that $1980^{1981^{1982}} + 1982^{1981^{1980}}$ is divisible by $1981^{1981}$.
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
Evaluate the following integrals.
(1) Meiji University
$\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$
(2) Tokyo University of Science
$\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$
Define for positive integer $n$, a function $f_n(x)=\frac{\ln x}{x^n}\ (x>0).$ In the coordinate plane, denote by $S_n$ the area of the figure enclosed by $y=f_n(x)\ (x\leq t)$, the $x$-axis and the line $x=t$ and denote by $T_n$ the area of the rectagle with four vertices $(1,\ 0),\ (t,\ 0),\ (t,\ f_n(t))$ and $(1,\ f_n(t))$.
(1) Find the local maximum $f_n(x)$.
(2) When $t$ moves in the range of $t>1$, find the value of $t$ for which $T_n(t)-S_n(t)$ is maximized.
(3) Find $S_1(t)$ and $S_n(t)\ (n\geq 2)$.
(4) For each $n\geq 2$, prove that there exists the only $t>1$ such that $T_n(t)=S_n(t)$.
Note that you may use $\lim_{x\to\infty} \frac{\ln x}{x}=0.$
Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$.
Evaluate $\int_0^2 g(t)dt$.
[i]2010 Kumamoto University entrance exam/Science[/i]
If $ \log_{10} (x^2\minus{}3x\plus{}6)\equal{}1$, the value of $ x$ is:
$ \textbf{(A)}\ 10\text{ or }2 \qquad\textbf{(B)}\ 4\text{ or }\minus{}2 \qquad\textbf{(C)}\ 3\text{ or }\minus{}1 \qquad\textbf{(D)}\ 4\text{ or }\minus{}1\\
\textbf{(E)}\ \text{none of these}$
Prove that there exists a constant $ c>0$ such that in every nontrivial finite group $ G$ there exists a sequence of length at most $ c\ln |G|$ with the property that each element of $ G$ equals the product of some subsequence. (The elements of $ G$ in the sequence are not required to be distinct. A [i]subsequence[/i] of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $ 4,4,2$ is a subesequence of $ 2,4,6,4,2,$ but $ 2,2,4$ is not.)
The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?
$ \textbf{(A)}\ y = \log_{10}\left(\frac{x + 90}{9}\right)\qquad\textbf{(B)}\ y = \log_{x}10\qquad\textbf{(C)}\ y = \frac{1}{x + 1}\qquad\textbf{(D)}\ y = 10^{-x}\qquad\textbf{(E)}\ y = 10^{x} $
Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.
Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$
Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$
Find $x$ such that \[2010^{\log_{10}x}=11^{\log_{10}(1+3+5+\cdots +4019).}\]
Let $S=\{1,2,3,\ldots,2012\}.$ We want to partition $S$ into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of $2.$ Find the number of such partitions.
Let $x,y$ and $z$ be real numbers satisfying the system \begin{align*}
\log_2(xyz-3+\log_5 x) &= 5 \\
\log_3(xyz-3+\log_5 y) &= 4 \\
\log_4(xyz-3+\log_5 z) &= 4.
\end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.
Prove that
\[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]
Suppose that $n$ people $A_1$, $A_2$, $\ldots$, $A_n$, ($n \geq 3$) are seated in a circle and that $A_i$ has $a_i$ objects such that
\[ a_1 + a_2 + \cdots + a_n = nN \]
where $N$ is a positive integer. In order that each person has the same number of objects, each person $A_i$ is to give or to receive a certain number of objects to or from its two neighbours $A_{i-1}$ and $A_{i+1}$. (Here $A_{n+1}$ means $A_1$ and $A_n$ means $A_0$.) How should this redistribution be performed so that the total number of objects transferred is minimum?
Solve in the real numbers the equation $ 3^{\sqrt[3]{x-1}} \left( 1-\log_3^3 x \right) =1. $
[i]Ion Gușatu[/i]
Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?