Found problems: 913
For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m \minus{} \log k| < \log n$. Find the sum of all possible values of the product $ mn$.
If $\log_{2x}216 = x$, where $x$ is real, then $x$ is:
$ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$
$\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$
$\textbf{(C)}\ \text{An irrational number} \qquad$
$\textbf{(D)}\ \text{A perfect square}\qquad$
$\textbf{(E)}\ \text{A perfect cube} $
Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\log(kx)=2\log(x+2)$ has exactly one real solution.
Evaluate $\int_1^e \frac{\ln x-1}{x^2-(\ln x)^2}dx.$
For every positive integer $ n$ let
\[ a_n\equal{}\sum_{k\equal{}n}^{2n}\frac{(2k\plus{}1)^n}{k}\]
Show that there exists no $ n$, for which $ a_n$ is a non-negative integer.
Find all pairs of positive real numbers $ (a, b)$ such that for every $ n \in\mathbb{N}^*$ and every real root $ x_n$ of the equation $ 4n^2x \equal{} \log_2(2n^2x \plus{} 1)$ we always have $ a^{x_n} \plus{} b^{x_n} \ge 2 \plus{} 3x_n$.
Evaluate the following definite integrals.
(a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$
(b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$
(c) $\int_1^e x\ln \sqrt{x}\ dx$
(d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$
If $a,b,c$ are positive real numbers such that $abc= 1$, Prove that \[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]
Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.
Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.
Let
\[ S_1 \equal{} \{ (x,y)\ | \ \log_{10} (1 \plus{} x^2 \plus{} y^2)\le 1 \plus{} \log_{10}(x \plus{} y)\}
\]and
\[ S_2 \equal{} \{ (x,y)\ | \ \log_{10} (2 \plus{} x^2 \plus{} y^2)\le 2 \plus{} \log_{10}(x \plus{} y)\}.
\]What is the ratio of the area of $ S_2$ to the area of $ S_1$?
$ \textbf{(A) } 98\qquad \textbf{(B) } 99\qquad \textbf{(C) } 100\qquad \textbf{(D) } 101\qquad \textbf{(E) } 102$
Let $f:\mathbb{N}\to\mathbb{N}$ be a function satisfying the following conditions:
(a) $f(1)=1$.
(b) $f(a)\leq f(b)$ whenever $a$ and $b$ are positive integers with $a\leq b$.
(c) $f(2a)=f(a)+1$ for all positive integers $a$.
How many possible values can the $2014$-tuple $(f(1),f(2),\ldots,f(2014))$ take?
Answer the following questions:
(1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$
(2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$
35 points
In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis.
Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down
\begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54
\end{align*}
and finds that this system of equations has a single real number solution $x > 1$. Find $b$.
Determine the value of $ab$ if $\log_8 a + \log_4 b^2 = 5$ and $\log_8 b + \log_4 a^2 = 7$.
What is the value of the expression
\[ \frac {1}{\log_2 100!} \plus{} \frac {1}{\log_3 100!} \plus{} \frac {1}{\log_4 100!} \plus{} \cdots \plus{} \frac {1}{\log_{100} 100!}?
\]$ \textbf{(A)}\ 0.01 \qquad \textbf{(B)}\ 0.1 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 10$
Evaluate the limit:
\[ \lim_{n \to \infty}{n \cdot \sin{1} \cdot \sin{2} \cdot \dots \cdot \sin{n}}.\]
Proposed to "Unirea" Intercounty contest, grade 11, Romania
Find the value of $ k\ (0<k<5)$ such that $ \int_0^{\infty} \frac{x^k}{2\plus{}4x\plus{}3x^2\plus{}5x^3\plus{}3x^4\plus{}4x^5\plus{}2x^6}\ dx$ is minimal.
Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$
[i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.
Let $S$ be a class of functions from $[0,\infty)$ to $[0,\infty)$ that satisfies:
(i) The functions $f_1(x)=e^x-1$ and $f_2(x)=\ln(x+1)$ are in $S;$
(ii) If $f(x)$ and $g(x)$ are in $S,$ the functions $f(x)+g(x)$ and $f(g(x))$ are in $S;$
(iii) If $f(x)$ and $g(x)$ are in $S$ and $f(x)\ge g(x)$ for all $x\ge 0,$ then the function $f(x)-g(x)$ is in $S.$
Prove that if $f(x)$ and $g(x)$ are in $S,$ then the function $f(x)g(x)$ is also in $S.$
You are given three lists A, B, and C. List A contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: \[\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists B or C that has exactly $n$ digits.
Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$, evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$
2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.