Found problems: 913
Let $N_0=\{0, 1, 2 \cdots \}$. Find all functions: $N_0 \to N_0$ such that:
(1) $f(n) < f(n+1)$, all $n \in N_0$;
(2) $f(2)=2$;
(3) $f(mn)=f(m)f(n)$, all $m, n \in N_0$.
$x^3+(\log_2{5}+\log_3{2}+\log_5{3})x=(\log_2{3}+\log_3{5}+\log_5{2})x^2+1$
Prove that $\ln n \geq k\ln 2$, where $n$ is a natural number and $k$ is the number of distinct primes that divide $n$.
Solve the Dirichlet problem
\[\Delta u=0\text{ for }x^2+y^2<1\]
\[u=1\text{ for }x^2+y^2=1,\;y>0\]
\[u=-1\text{ for }x^2+y^2=1,\;y<0\]
Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$).
Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.
Show that the following inequality hold for all $ k \geq 1$, real numbers $ a_1,a_2,...,a_k$, and positive numbers $ x_1,x_2,...,x_k.$
\[ \ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} .
\]
[i]L. Losonczi[/i]
For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$
If $\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2$, then $y$ equals:
$ \textbf{(A)}\ \frac92\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 81 $
Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements.
Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating
Given $2^x=8^{y+1}$ and $9^y=3^{x-9}$, find the value of $x+y$.
${{ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 21 \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 27 }\qquad\textbf{(E)}\ 30 } $
(1) Show the following inequality for every natural number $ k$.
\[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\]
(2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$.
\[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]
Prove that the inequality $ (x^{k}\minus{}y^{k})^{n}<(x^{n}\minus{}y^{n})^{k}$ holds forall reals $ x>y>0$ and positive integers $ n>k$.
Let $ x_0 \equal{} 1$ and for $ n\ge0,$ let $ x_{n \plus{} 1} \equal{} 3x_n \plus{} \left\lfloor x_n\sqrt {5}\right\rfloor.$ In particular, $ x_1 \equal{} 5,\ x_2 \equal{} 26,\ x_3 \equal{} 136,\ x_4 \equal{} 712.$ Find a closed-form expression for $ x_{2007}.$ ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)
Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.
The values of $ a$ in the equation: $ \log_{10}(a^2 \minus{} 15a) \equal{} 2$ are:
$ \textbf{(A)}\ \frac {15\pm\sqrt {233}}{2} \qquad\textbf{(B)}\ 20, \minus{} 5 \qquad\textbf{(C)}\ \frac {15 \pm \sqrt {305}}{2}$
$ \textbf{(D)}\ \pm20 \qquad\textbf{(E)}\ \text{none of these}$
Round to the nearest tenth: $\log_6 (6^2-6+1) + 3\log_6 (5) - \frac{1}{2}\log_6 (9)$.
Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\]
$\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$
$\textbf{(B) }\text{if and only if }a=b^2\qquad$
$\textbf{(C) }\text{if and only if }b=a^2\qquad$
$\textbf{(D) }\text{if and only if }x=ab\qquad$
$ \textbf{(E) }\text{for none of these}$
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
The value of $ \log_5 \frac {(125)(625)}{25}$ is equal to:
$\textbf{(A)}\ 725 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 3125 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ \text{None of these}$
Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
We call a nonnegative integer $r$-Fibonacci number if it is a sum of $r$ (not necessarily distinct) Fibonacci numbers. Show that there infinitely many positive integers that are not $r$-Fibonacci numbers for any $r, 1 \leq r\leq 5.$
For every complex number $z$ different from 0 and 1 we define the following function
\[ f(z) := \sum \frac 1 { \log^4 z } \]
where the sum is over all branches of the complex logarithm.
a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$.
b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have
\[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]
Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$
[i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]
How many positive integers $ b$ have the property that $ \log_b729$ is a positive integer?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$