7
votes
2answers
145 views

$f:\mathbb{R}\to \mathbb{R}$ continuous and $\lim_{h \to 0^{+}} \frac{f(x+2h)-f(x+h)}{h}=0$ $\implies f=$ constant.

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function with the property that $$\lim_{h \to 0^{+}} \dfrac{f(x+2h)-f(x+h)}{h}=0$$ for all $x \in \mathbb{R}$. Prove that $f$ is constant.
1
vote
2answers
63 views

Product identities

I need to use the following identities for poisson integral but i can't guz i don't know how to prove them. $$\alpha^{2n}-1=\prod_{k=0}^{k=2n-1}(\alpha-e^{i\frac{2k\pi}{2n}})$$ ...
3
votes
1answer
83 views

Proof that infinitely many $f$ exist if $f(f(x))=f(x)^{2013}$

Suppose $f(x)$ is function from $\mathbb{R}$ to $\mathbb{R}$ such that $f(f(x))=f(x)^{2013}$. Show that there are infinitely many such functions, of which exactly four are polynomials. If $f$ is ...
15
votes
2answers
527 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
1
vote
2answers
141 views

Analysis problem from Romanian Contest - 2 sequences which forms another one

Let $a,b$ be 2 real numbers, and the sequences $(a_n)_{n \geq 1}, (b_n)_{n \geq 1}$ defined by $a_{1}=a$, $b_{1}=b$, $a^2+b^2 <1$ and \begin{cases} ...
18
votes
2answers
856 views

Tough contest problem

I found this problem in a collection of contest problems of a Russian competition in 1995 and wasn't able to solve it. Solve for real $x$: $$ \cos (\cos (\cos (\cos(x))))=\sin (\sin (\sin (\sin ...
7
votes
1answer
176 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
0
votes
1answer
135 views

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$ for $|x-y|\le\alpha$.

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $$|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$$ for $|x-y|\le\alpha$. From the mean value theorem, given any $x,y$ with ...
2
votes
0answers
46 views

Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that ...
1
vote
0answers
66 views

Prove that $0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$

Let $f'$ be integrable. Prove that $$0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$$ Source: ...
2
votes
2answers
88 views

Prove $1<\frac1{e^2(e-1)}\int_e^{e^2}\frac{x}{\ln x}dx<e/2$

Prove the following inequalities: a) $1.43 < \int_0^1e^{x^2}dx < \frac{1+e}2$ b) $2e <\int_0^1 e^{x^2}dx+\int_0^1e^{2-x^2}dx<1+e^2$ c) $1<\frac1{e^2(e-1)}\int_e^{e^2}\frac{x}{\ln ...
0
votes
1answer
129 views

Prove $\text{Beta}(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta, \qquad \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0$

Prove that $$\int_0^1 x^{k}(1-x)^kdx=\frac{k!k!}{(2k+1)!}.$$ (Edit: Actually the proof can be found here http://en.wikipedia.org/wiki/Beta_function ) How would you show this $\text{Beta}(x,y) = ...
7
votes
1answer
141 views

Bound on $|f(x)|^2 + |f'(x)|^2$

Let $f\in C^2(\mathbb{R})$ be a twice differentiable function satisfying $$|f(x)|^2\le a$$ and $$|f'(x)|^2 + |f''(x)|^2\le b$$ for all real $x$, where $a$ and $b$ are positive constants. Prove that ...
2
votes
1answer
68 views

Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots+ b_n)$ convergent?

Let $\sum_{n=1}^{\infty}a_n$ be a convergent series of positive terms (so $a_i > 0$ for all $i$) and set $b_n = 1/(na_n^2)$ for $n\ge1$. Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots + b_n)$ ...
7
votes
1answer
138 views

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer.

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $$a_{n+2} = a_{n+1} - a_na_{n+1}/2$$ for $n$ a positive integer. Find $$\lim_{n\to\infty}na_n$$ if it exists. Well, we can deduce that $\lim a_n=0$ by ...
1
vote
2answers
134 views

Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.

Let $f$ be a continuous, nonnegative, real-valued function and $$\int_0^{\infty}f(x)dx<\infty.$$ Prove that $$\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0.$$ A start: If ...
10
votes
2answers
257 views

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
1
vote
2answers
79 views

Determine the value of $p>0$ for which $\sum_{n=1}^{\infty}(-1)^{\lfloor{\sqrt{n}}\rfloor}/n^p$ converges.

Determine the value of $p>0$ for which $$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor{\sqrt{n}}\rfloor}}{n^p}$$ converges. By considering $\lfloor{\sqrt{n}}\rfloor$, we see the series is $$\sum_{k\ge1} ...
1
vote
1answer
87 views

Find functions such that under the Cartesian coordinate system $F(x, y) = f(x) g(y)$ but under the polar coordinate system $F(x, y) = h(r)$.

Find all non-constant function $F(x, y)\in C^2(\mathbb{R}^2)$ such that under the Cartesian coordinate system $F(x, y) = f(x)  g(y)$ but under the polar coordinate system $F(x, y) = h(r)$. My ...
1
vote
1answer
86 views

Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge.

Let $\{a_n\}$ be any sequence of positive real numbers. Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge. For example, ...
4
votes
1answer
83 views

Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that $\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$. My ...
11
votes
4answers
366 views

Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds: $$f(x+y)=f(x^{2}+y^{2}).$$
4
votes
2answers
390 views

Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
2
votes
1answer
78 views

assume $f:[a,b] \to \mathbb R$ such that $f '(a)=f '(b)$ how prove $\exists t\in(a,b) : f(t)-f(a)=f '(t)(t-a)$?

let $f:[a,b] \to \mathbb R$ such that $f '(a)=f '(b)$ how prove $$\exists t\in(a,b) : f(t)-f(a)=f '(t)(t-a)$$ Thanks in advance
4
votes
1answer
121 views

Prove that the polynomial $P(x_1,x_2…,x_n)=0$ given a set of conditions.

Let $P(x_1,...,x_n)\in\mathbb{R}[x_1,...,x_n]$ (i.e. $P$ is a polynomial of real coefficients in $x_1,..,x_n$). We are given that $\left(\frac{\partial^2}{\partial ...
1
vote
0answers
75 views

Pi identity with sum and product

Please prove this identity $$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
3
votes
1answer
53 views

$\lim_{x\to\infty} x^{1+1/x}-x-\log x$ and $\lim_{x\to\infty}\frac{x^{1+1/x}-x}{\log x}$

Evaluate $$\lim_{x\to\infty} x^{1+1/x}-x-\log x$$ and $$\lim_{x\to\infty}\frac{x^{1+1/x}-x}{\log x}$$ Would knowing one necessarily give the other?
2
votes
1answer
34 views

Prove that $\lim_{t\to0}(\log t)(1-(2t)^{t/2})=0$

Please prove that $(\log t)(1-(2t)^{t/2})$ tends to 0 as t tends to 0. http://www.wolframalpha.com/input/?i=lim+t-%3E0+%281-%282t%29^%28t%2F2%29%29logt It seems the limit converges to 0 pretty ...
7
votes
4answers
419 views

Given that $f(1)= 2013,$ find the value of $f(2013)$?

Suppose that $f$ is a function defined on the set of natural numbers such that $$f(1)+ 2^2f(2)+ 3^2f(3)+...+n^2f(n) = n^3f(n)$$ for all positive integers $n$. Given that $f(1)= 2013$, find the value ...
1
vote
1answer
54 views

how to prove this question about limit and derivative

Suppose $f:(a,b)\to\mathbb R$ that $ f $ satisfies: $$f\in C^1$$ $$\lim_{x\to a ^ +}f^2(x)=0$$ $$\lim_{x\to b ^ -}f^2(x)=e-1$$ if $\forall x \in(a,b) : 2f(x)f '(x)-f^2(x)\ge1 $, then how to ...
6
votes
1answer
69 views

question about limit and series

consider following hypotheses $ m\in\mathbb N$ $c\in \mathbb C\,$ ,$\, \; a_j\in \mathbb C$ $a_j\in \mathbb C\;$ , $\;|a_j|=1,\;\forall\;1\le j\le m$ if ...
6
votes
2answers
138 views

If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $[a,b]\subset f([c,d])$, how to prove there is some $[r,s]$ such that $f([r,s])=[a,b]$?

Let $f:\mathbb R\to\mathbb R$ satisfy the following: $f$ is continuous there exist closed intervals $[a,b]$ and $[c,d]$ such that $[a,b]\subset f([c,d])$ How to prove that there ...
3
votes
3answers
68 views

If $f:[a,b]\to[a,b]$ is increasing, continuous, and $f(a)=a$, how to prove $f(E)=E$ where $E=\{x:a\le x\le b,f(x)\ge x\}$?

Let $f:[a,b]\to[a,b]$ satisfy: $f$ is increasing $f$ is continuous $f(a)=a$ If $E=\{x:a\le x\le b,f(x)\ge x\}$, then how can we prove that $f(E)=E$?
9
votes
1answer
117 views

Geometric inequality with a triangle

The positive real numbers $x,y,z$ are the side lengths of a triangle iff $$x^2 + y^2 + z^2 < 2\sqrt{x^2y^2 + y^2z^2 + z^2x^2}$$
8
votes
3answers
96 views

How to prove there is no positive and continuous function satisfying some conditions

Let $\alpha \in \mathbb R $. How could I prove there isn't any positive and continuous function $f$ such that the following conditions hold? $\int_{0}^1 f(x)dx=1$ $\int_{0}^1 xf(x)dx=\alpha $ ...
5
votes
3answers
138 views

Find $\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}.$

Find $$\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}\;.$$
13
votes
4answers
599 views

Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

For every nonnegative integer $n$ and every real number $ x$ prove the inequality: $$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$
7
votes
1answer
176 views

$f:[a,b]\to(a,b)$ be continuous how prove $f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)(c+\frac{nd}{2})$

let $f:[a,b]\to(a,b)$ be continuous how prove $\forall n\in\mathbb N$ $\exists d\gt0$ ,$\exists c\in(a,b) $ such that $$f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)\left(c+\frac{nd}{2}\right)$$thanks in advance ...
3
votes
1answer
43 views

$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
2
votes
1answer
117 views

$\exists a,b\in \mathbb R^+ $such that $|f(x)|\le a|x|+b$

Assuming $f:\mathbb R\to\mathbb R $ be an uniform continuous function, how to prove $$\exists a,b\in \mathbb R^+~~~~\text{such that}~~~~|f(x)|\le a|x|+b.$$ Thanks in advance!
0
votes
2answers
92 views

Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives.

Let U be an open subset of $\mathbb{R}^n$ and C a compact subset of U. Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives. Prove that f is Lipschitz on C. Thoughts: Let ...
7
votes
1answer
99 views

How to prove that exists distinct $x_1,x_2 \in(a,b)$ such that $f '(x_1)f '(x_2)=1$?

Assume $f:[a,b]\to[a,b]$ be continuous and differentiable on $(a,b)$ and $f(a)=a$, $f(b)=b$. How to prove that exists distinct $x_1,x_2 \in(a,b)$ such that $f '(x_1)f '(x_2)=1$? Thanks in advance.
3
votes
4answers
186 views

how prove $\sum_{n=1}^\infty\frac{a_n}{b_n+a_n} $is convergent?

Let$a_n,b_n\in\mathbb R$ and $(a_n+b_n)b_n\neq 0\quad \forall n\in \mathbb{N}$. The series $\sum_{n=1}^\infty\frac{a_n}{b_n} $ and $\sum_{n=1}^\infty(\frac{a_n}{b_n})^2 $ are convergent. How to prove ...
0
votes
1answer
54 views

how prove $\exists a,b$ that satisfied in following conditions $0 <a\leq b\leq 1, b-a=\frac12, \text{ and }f(a)=f(b)$ [duplicate]

Possible Duplicate: Universal Chord Theorem let $f:[0,1]\mapsto\mathbb R$ be continuous and $f(0)=f(1)$how prove $\exists a,b$ that satisfied in following conditions $$1)0<a\leq b\leq ...
1
vote
2answers
150 views

$\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$

Let $a_n>0,n\in\mathbb{N}$ be a sequence of positive real numbers. There exists a positive real number $c$ such that $\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$ as $n\to\infty$ for all ...
0
votes
1answer
115 views

Convergence of series with modified denominator

Suppose the series with positive terms $\sum_{n=1}^{\infty} a_n$ converges. Let $r_n=\sum_{k=n}^{\infty}a_k$. Prove or disprove that $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverges, and prove or ...
24
votes
2answers
521 views

New twist on a Putnam problem

A recent Putnam problem: Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f$ is identically zero? The ...
10
votes
2answers
160 views

Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?

Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
2
votes
1answer
243 views

how prove this integral inequality?

How prove that for all continuous and decreasing function $f:[0 ,1]\mapsto(0,+\infty)$ $$\frac{\int_{0}^1x(f(x))^2dx}{\int_{0}^1xf(x)dx}\leq \frac{\int_{0}^1(f(x))^2dx}{\int_{0}^1f(x)dx}$$ thanks in ...
4
votes
2answers
186 views

how to prove this question about derivative and differentiation

Let $$ f:\mathbb{R}\to \mathbb{R} $$ such that $f ',f'',f'''$ exist and $\lim_{x\to+\infty} f(x)=t$ exists if $ \lim_{x\to+\infty} f'''(x)=0$. Then prove that $$ \lim_{x\to+\infty} f'(x) = ...