Found problems: 4776
The sequence of digits \[123456789101112131415161718192021\ldots\] is obtained by writing the positive integers in order. If the $10^n$th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2) = 2$ because the $100^{\text{th}}$ digit enters the sequence in the placement of the two-digit integer $55$. Find the value of $f(2007)$.
There are $n$ balls numbered from $1$ to $n$, and $2n-1$ boxes numbered from $1$ to $2n-1$. For each $i$, ball number $i$ can only be put in the boxes with numbers from $1$ to $2i-1$. Let $k$ be an integer from $1$ to $n$. In how many ways we can choose $k$ balls, $k$ boxes and put these balls in the selected boxes so that each box has exactly one ball?
In the cartesian plane, consider the curves $x^2+y^2=r^2$ and $(xy)^2=1$. Call $F_r$ the convex polygon with vertices the points of intersection of these 2 curves. (if they exist)
(a) Find the area of the polygon as a function of $r$.
(b) For which values of $r$ do we have a regular polygon?
Let $ S$ be a sequence that:
\[ \left\{
\begin{array}{cc}
S_0\equal{}0\hfill\\
S_1\equal{}1\hfill\\
S_n\equal{}S_{n\minus{}1}\plus{}S_{n\minus{}2}\plus{}F_n& (n>1)
\end{array}
\right.\]
such that $ F_n$ is Fibonacci sequence such that $ F_1\equal{}F_2\equal{}1$. Find $ S_n$ in terms of Fibonacci numbers.
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i
t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
$f$ be a function satisfying $2f(x)+3f(-x)=x^2+5x$. Find $f(7)$
[list=1]
[*] $-\frac{105}{4}$
[*] $-\frac{126}{5}$
[*] $-\frac{120}{7}$
[*] $-\frac{132}{7}$
[/list]
Let $\rho:\mathbb{R}^n\to \mathbb{R}$, $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$, and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula
\[\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.\]
Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$.
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
Show that the power series for the function
$$e^{ax} \cos bx,$$
where $a,b >0$, has either no zero coefficients or infinitely many zero coefficients.
Let $f$ be a function from positive integers to positive integers where $f(n) = \frac{n}{2}$ if $n$ is even and $f(n) = 3n+1$ if $n$ is odd. If $a$ is the smallest positive integer satisfying \[ \underbrace{f(f(\cdots f}_{2013\ f\text{'s}} (a)\cdots)) = 2013, \] find the remainder when $a$ is divided by $1000$.
[i]Based on a proposal by Ivan Koswara[/i]
Suppose $f(x)$ is a function satisfying $\left | f(m+n)-f(m) \right | \leq \frac{n}{m}$ for all positive integers $m$,$n$. Show that for all positive integers $k$:
\[\sum_{i=1}^{k}\left |f(2^k)-f(2^i) \right |\leq \frac{k(k-1)}{2}\].
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that there exists a continuous and bounded function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that verifies the equality
$$ f(x)=\int_0^x f(\xi )g(\xi )d\xi , $$
for any real number $ x. $ Prove that $ f=0. $
[i]Nelu Chichirim[/i]
$n\in{Z^{+}}$ and $A={1,\ldots ,n}$. $f: N\rightarrow N$ and $\sigma: N\rightarrow N$ are two permutations, if there is one $k\in A$ such that $(f\circ\sigma)(1),\ldots ,(f\circ\sigma)(k)$ is increasing and $(f\circ\sigma)(k),\ldots ,(f\circ\sigma)(n)$ is decreasing sequences we say that $f$ is good for $\sigma$. $S_\sigma$ shows the set of good functions for $\sigma$.
a) Prove that, $S_\sigma$ has got $2^{n-1}$ elements for every $\sigma$ permutation.
b)$n\geq 4$, prove that there are permutations $\sigma$ and $\tau$ such that, $S_{\sigma}\cap S_{\tau}=\phi$
.
Find all strictly increasing functions $f : \mathbb{N} \to \mathbb{N} $ such that $\frac {f(x) + f(y)}{1 + f(x + y)}$ is a non-zero natural number, for all $x, y\in\mathbb{N}$.
The function $ƒ$ is defined at $[0,1]$, and $f\{f(x)\} = ƒ(x)$.
$\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]$ Determine $f\left(\frac12 \right).$
$\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]$. Determine $f$.
Prove that the function $g$, with $g(x) = x$,$0 \le x \le k$, $g(x) = k$, $k \le x \le 1$ satisfies the relation $g\{g(x)\} = g(x)$.
For how many ordered pairs of positive integers $ (m,n)$, $ m \cdot n$ divides $ 2008 \cdot 2009 \cdot 2010$ ?
$\textbf{(A)}\ 2\cdot3^7\cdot 5 \qquad\textbf{(B)}\ 2^5\cdot3\cdot 5 \qquad\textbf{(C)}\ 2^5\cdot3^7\cdot 5 \qquad\textbf{(D)}\ 2^3\cdot3^5\cdot 5^2 \qquad\textbf{(E)}\ \text{None}$
We know that $k$ is a positive integer and the equation \[ x^3+y^3-2y(x^2-xy+y^2)=k^2(x-y) \quad (1) \] has one solution $(x_0,y_0)$ with
$x_0,y_0\in \mathbb{Z}-\{0\}$ and $x_0\neq y_0$. Prove that
i) the equation (1) has a finite number of solutions $(x,y)$ with $x,y\in \mathbb{Z}$ and $x\neq y$;
ii) it is possible to find $11$ addition different solutions $(X,Y)$ of the equation (1) with $X,Y\in \mathbb{Z}-\{0\}$ and $X\neq Y$ where $X,Y$ are functions of $x_0,y_0$.
Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions whose graphs both have an axis of symmetry.
[i]D. Tereshin[/i]
Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$
1978 Chuo university entrance exam/Science and Technology
The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.
Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$.
Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$.
Find a closed form for $a_n$.
[i]Proposed by Bobby Shen[/i]
For a positive integer $m$, prove the following ineqaulity.
$0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$
1996 Osaka University entrance exam
The $2^{nd}$ order differentiable function $f:\mathbb R \longrightarrow \mathbb R$ is in such a way that for every $x\in \mathbb R$ we have $f''(x)+f(x)=0$.
[b]a)[/b] Prove that if in addition, $f(0)=f'(0)=0$, then $f\equiv 0$.
[b]b)[/b] Use the previous part to show that there exist $a,b\in \mathbb R$ such that $f(x)=a\sin x+b\cos x$.
Let be two real numbers $ a<b, $ a nonempty and non-maximal subset $ K $ of the interval $ (a,b) $ and three functions
$$ f:(a,b)\longrightarrow\mathbb{R}, g,h:\mathbb{R}\longrightarrow\mathbb{R} $$
satisfying the following relations.
$ \text{(i)} g $ and $ h $ are primitivable.
$ \text{(ii)} g-h $ hasn't any root in $ (a,b). $
$ \text{(iii)} $ The restrictions of $ f $ at $ K $ and $ (a,b)\setminus K $ are equal to $ g,h, $ respectively.
Prove that $ f $ is not primitivable.
What is the smallest positive integer $n$ such that $2013^n$ ends in $001$ (i.e. the rightmost three digits of $2013^n$ are $001$?