Found problems: 4776
Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let \[S=\sum_{d|2008}\phi(d),\] in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.
What is the sum of the squares of the roots of the equation $x^2 -7 \lfloor x\rfloor +5=0$ ?
Let $A$ be a real number and $(a_{n})$ be a sequence of real numbers such that $a_{1}=1$ and \[1<\frac{a_{n+1}}{a_{n}}\leq A \mbox{ for all }n\in\mathbb{N}.\]
$(a)$ Show that there is a unique non-decreasing surjective function $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $1<A^{k(n)}/a_{n}\leq A$ for all $n\in \mathbb{N}$.
$(b)$ If $k$ takes every value at most $m$ times, show that there is a real number $C>1$ such that $Aa_{n}\geq C^{n}$ for all $n\in \mathbb{N}$.
Prove that for all integers $ a > 1$ and $ b > 1$ there exists a function $ f$ from the positive integers to the positive integers such that $ f(a\cdot f(n)) \equal{} b\cdot n$ for all $ n$ positive integer.
For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 \minus{} n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.
Let $A \text{ :}= \mathbb{Q}\setminus \{0,1\}$ denote the set of all rationals other than $0$ and $1$. A function $f:A\to \mathbb{R}$ has the property that for all $x\in A$, \[f(x)+f\left(1-\dfrac{1}{x}\right)=\log |x|.\] Compute the value of $f(2007)$.
Let $n \geq 3$ be a positive integer. A [i]chipped $n$-board[/i] is a $2 \times n$ checkerboard with the bottom left square removed. Lino wants to tile a chipped $n$-board and is allowed to use the following types of tiles:
[list]
[*] Type 1: any $1 \times k$ board where $1 \leq k \leq n$
[*] Type 2: any chipped $k$-board where $1 \leq k \leq n$ that must cover the left-most tile of the $2 \times n$ checkerboard.
[/list]
Two tilings $T_1$ and $T_2$ are considered the same if there is a set of consecutive Type 1 tiles in both rows of $T_1$ that can be vertically swapped to obtain the tiling $T_2$. For example, the following three tilings of a chipped $7$-board are the same:
[img]http://i.imgur.com/8QaSgc0.png[/img]
For any positive integer $n$ and any positive integer $1 \leq m \leq 2n - 1$, let $c_{m,n}$ be the number of distinct tilings of a chipped $n$-board using exactly $m$ tiles (any combination of tile types may be used), and define the polynomial $$P_n(x) = \sum^{2n-1}_{m=1} c_{m,n}x^m.$$
Find, with justification, polynomials $f(x)$ and $g(x)$ such that $$P_n(x) = f(x)P_{n-1}(x) + g(x)P_{n-2}(x)$$ for all $n \geq 3$.
Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.
Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$, forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}} $)?
(translated by L. Erdős)
Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove that $T_n - n$ is always even.
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$;
(ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$.
Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list]
[i]Proposed by Hans Zantema, Netherlands[/i]
Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and
\[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \]
$(a)$ Prove that $f$ has a fixed point different from $1$.
$(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.
Consider the expansion
\[(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.\]
[b](a)[/b] Determine the greatest common divisor of the coefficients $a_3, a_8, a_{13}, \ldots , a_{1983}.$
[b](b)[/b] Prove that $10^{340 }< a_{992} < 10^{347}.$
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y\]
for all $x,y\in\mathbb{R}$
Let $0<a<1$. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous at $x = 0$ such that $f(x) + f(ax) = x,\, \forall x \in \mathbb{R}$
Let $\mathbb{R}$ denote the set of real numbers. Suppose a function $f: \mathbb{R} \to \mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x\in \mathbb{R}$. Show that
[b](i)[/b] $f$ is one-one,
[b](ii)[/b] $f$ cannot be strictly decreasing, and
[b](iii)[/b] if $f$ is strictly increasing, then $f(x)=x$ for all $x \in \mathbb{R}$.
Let $y = f (x)$ be a convex function defined on $[0,1]$, $f (0) = 0,$ $f (1) = 0$. It is also known that the area of the segment bounded by this function and the segment $[0, 1]$ is equal to $1$. Find and draw the set of points of the coordinate plane through which the graph of such a function can pass. (A function is called convex if all points of the line segment connecting any two points on its graph are located no higher than the graph of this function.)
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions :
1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$
2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.
The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations
\[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.
A function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is said to be [i]continuous in each variable separately [/i] if, for each fixed value $y_0$ of $y$, the function $f(x, y_0)$ is contnuous in the usual sense as a function in $x,$ and similarly $f(x_0 , y)$ is continuous as a function of $y$ for each fixed $x_0$.
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous in each variable separately. Show that there exists a sequence of continuous functions $g_n: \mathbb{R}^{2} \rightarrow \mathbb{R}$ such that
$$f(x,y) =\lim_{n\to \infty}g_{n}(x,y)$$
for all $(x,y)\in \mathbb{R}^{2}.$
Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) \equal{} az^{2n \plus{} 1} \plus{} bz^{2n} \plus{} \bar bz \plus{} \bar a$ has a root on unit circle
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that
\[ x f(x \plus{} xy) \equal{} x f(x) \plus{} f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]