Found problems: 4776
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that
$a_1 + ... + a_n = b_1 + ... + b_n = 1$.
Find the minimal value of
$ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.
Find all monotonic functions $u:\mathbb{R}\rightarrow\mathbb{R}$ which have the property that there exists a strictly monotonic function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+y)=f(x)u(x)+f(y) \]
for all $x,y\in\mathbb{R}$.
[i]Vasile Pop[/i]
The graph of the function $ f(x) \equal{} 2x^3 \minus{} 7$ goes:
$ \textbf{(A)}\ \text{up to the right and down to the left} \\
\textbf{(B)}\ \text{down to the right and up to the left} \\
\textbf{(C)}\ \text{up to the right and up to the left} \\
\textbf{(D)}\ \text{down to the right and down to the left} \\
\textbf{(E)}\ \text{none of these ways.}$
A function $f$ is defined on the natural numbers $\mathbb{N}$ and satisfies the following rules:
(a) $f(1)=1$;
(b) $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$ for all $n\in \mathbb{N}$.
Calculate the maximum value $m$ of the set $\{f(n):n\in \mathbb{N}, 1\le n\le 1989\}$, and determine the number of natural numbers $n$, with $1\le n\le 1989$, that satisfy the equation $f(n)=m$.
A family of finite sets $\left\{ A_{1},A_{2},.......,A_{m}\right\} $is called [i]equipartitionable [/i] if there is a function $\varphi:\cup_{i=1}^{m}$$\rightarrow\left\{ -1,1\right\} $ such that $\sum_{x\in A_{i}}\varphi\left(x\right)=0$ for every $i=1,.....,m.$ Let $f\left(n\right)$ denote the smallest possible number of $n$-element sets which form a non-equipartitionable family. Prove that
a) $f(4k +2) = 3$ for each nonnegative integer $k$,
b) $f\left(2n\right)\leq1+m d\left(n\right)$, where $m d\left(n\right)$ denotes the least positive non-divisor of $n.$
Let $f_1,f_2,\cdots ,f_{10}$ be bijections on $\mathbb{Z}$ such that for each integer $n$, there is some composition $f_{\ell_1}\circ f_{\ell_2}\circ \cdots \circ f_{\ell_m}$ (allowing repetitions) which maps $0$ to $n$. Consider the set of $1024$ functions
\[ \mathcal{F}=\{f_1^{\epsilon_1}\circ f_2^{\epsilon_2}\circ \cdots \circ f_{10}^{\epsilon_{10}}\} \]
where $\epsilon _i=0$ or $1$ for $1\le i\le 10.\; (f_i^{0}$ is the identity function and $f_i^1=f_i)$. Show that if $A$ is a finite set of integers then at most $512$ of the functions in $\mathcal{F}$ map $A$ into itself.
Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which
\[p(x)q(x+1)-p(x+1)q(x)=1.\]
Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$.
Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.
Prove that for every convex function $ f(x)$ defined on the interval $ \minus{}1\leq x \leq 1$ and having absolute value at most $ 1$,
there is a linear function $ h(x)$ such that \[ \int_{\minus{}1}^1|f(x)\minus{}h(x)|dx\leq 4\minus{}\sqrt{8}.\] [L. Fejes-Toth]
Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.
Find the total number of different integers the function
\[ f(x) = \left[x \right] + \left[2 \cdot x \right] + \left[\frac{5 \cdot x}{3} \right] + \left[3 \cdot x \right] + \left[4 \cdot x \right] \]
takes for $0 \leq x \leq 100.$
Let $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$, we have $f(xy)\le f(x)f(y)$. Show that for arbitrary positive real number $x$ and natural number $n$, inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.
Let $f$ be a function from R to R. Suppose we have:
(1) $f(0)=0$
(2) For all $x, y \in (-\infty, -1) \cup (1, \infty)$, we have $f(\frac{1}{x})+f(\frac{1}{y})=f(\frac{x+y}{1+xy})$.
(3) If $x \in (-1,0)$, then $f(x) > 0$.
Prove: $\sum_{n=1}^{+\infty} f(\frac{1}{n^2+7n+11}) > f(\frac12)$ with $n \in N^+$.
Find all functions $f : \mathbb{R} \mapsto \mathbb{R} $ such that
$$ \left(f(x)+y\right)\left(f(x-y)+1\right)=f\left(f(xf(x+1))-yf(y-1)\right)$$
for all $x,y \in \mathbb{R}$
Consider the function $$f(x)=\sum_{k=1}^{m}(x-k)^{4}~, \qquad~ x \in \mathbb{R}$$ where $m>1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.
[color=darkblue]Weightlifter Ruslan has just finished the exercise with a weight, which has $ n$ small weights on one side and $ n$ on the another. At each stage he takes some weights from one of the sides, such that at any moment the difference of the numbers of weights on the sides does not exceed $ k$. What is the minimal number of stages (in function if $ n$ and $ k$), which Ruslan need to take off all weights..[/color]
Let $ f(n)$ be the number of times you have to hit the $ \sqrt {\ }$ key on a calculator to get a number less than $ 2$ starting from $ n$. For instance, $ f(2) \equal{} 1$, $ f(5) \equal{} 2$. For how many $ 1 < m < 2008$ is $ f(m)$ odd?
Determine the values of $a, b, c$, so that the graphical representation of the function $$y = ax^3 + bx^2 + cx$$ has an inflection point at the point of abscissa $ x = 3$, with tangent at the point of equation $x - 4y + 1 = 0.$ Then draw the corresponding graph.
Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j=1}^n a_j \geq \sqrt{n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$ \[ \sum_{j=1}^n a_j^2 > \frac{1}{4} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right). \]
Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.
Show that there is precisely one sequence $ a_1,a_2,...$ of integers which satisfies $ a_1\equal{}1, a_2>1,$ and $ a_{n\plus{}1}^3\plus{}1\equal{}a_n a_{n\plus{}2}$ for $ n \ge 1$.
Find all positive integer solutions of the equation $k!l! = k!+l!+m!.$
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.
The number of functions $g:\mathbb{R}^4\to\mathbb{R}$ such that, $\forall a,b,c,d,e,f\in\mathbb{R}$ :
(i) $g(1,0,0,1)=1$
(ii) $g(ea,b,ec,d)=eg(a,b,c,d)$
(iii) $g(a+e, b, c+f, d)= g(a,b,c,d)+g(e,b,f,d)$
(iv) $g(a,b,c,d)+g(b,a,d,c)=0$
is :
(A)$1$
(B)$0$
(C)$\text{infinitely many}$
(D)$\text{None of these}$
[Hide=Hint(given in question)]
Think of matrices[/hide]