0
votes
0answers
53 views

On a differential equation problem of international mathematical competition for university students

I am trying to solve problem 2 of this competition: http://www.imc-math.org.uk/imc2009/imc2009-day2-solutions.pdf I have other thought but i couldn't fill in the detail. Consider the initial value ...
0
votes
1answer
54 views

determing constant in inequality with nonnegative numbers

Let $ r \geq 1$ be an integer. Prove that there exists a constant $ C_r = C(r)>0$ such that for any non-negative real numbers $ a_1, a_2, \cdots, a_n \in [0, \infty)$ the following inequality ...
4
votes
1answer
147 views

Find all differentiable functions $f:[0;2] \to \Bbb{R}$ such that $\int_{0}^{2}xf(x)dx=f(0)+f(2)$

Find all differentiable functions $f:[0;2] \to \Bbb{R}$, with $f'$ continuous, such that the function $e^{-x}f(x)$ is decreasing on $[0;1]$ and increasing on $[1;2]$, and ...
2
votes
2answers
34 views

Let $f: \Bbb{R} \to [0; \infty)$ .Prove that $\forall n \in \Bbb{N}$ $\forall y \in \Bbb{R}$ $ \exists t=t(n;y)$ such that $\int_{y}^{t}f(x)dx=n$

The problem goes like this: Let $f: \Bbb{R} \to [0; \infty)$ be a continuous function such that $\lim_{x \to \infty}f(x)=\infty$. Prove that $\forall n \in \Bbb{N^{*}}$ and $\forall y \in \Bbb{R}$ ...
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. $$\alpha^{2n}-1=\prod_{k=0}^{k=2n-1}(\alpha-e^{i\frac{2k\pi}{2n}})$$ ...
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 ...
4
votes
0answers
177 views

The Monster PolyLog Integral $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

I am trying to solve this integral $$ \int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x $$ which is from some high school training ...
10
votes
3answers
359 views

Integral $I=\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx$

Hi I am stuck on showing that $$ \int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx=\pi G-\frac{3\zeta(3)}{8} $$ where G is the Catalan constant and $\zeta(3)$ is the Riemann zeta function. Explictly ...
2
votes
1answer
50 views

Integral $6\int_{x=0}^{x=1}\int_{y=x}^{y=1}\int_{z=x}^{z=y} f(x) f(y) f(z)dxdydz=\bigg(\int_0^1 f(t) dt\bigg)^3$

Prove that $$ 6\int_{x=0}^{x=1}\int_{y=x}^{y=1}\int_{z=x}^{z=y} f(x) f(y) f(z)dxdydz=\bigg(\int_0^1 f(t) dt\bigg)^3 $$ assuming $f(x)$ is continuous on [0,1]. This is from an old Putnam exam. I am ...
7
votes
2answers
131 views

Integral $I=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0. $

$$ I(\alpha,\beta)=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0. $$ I am trying to solve this integral. This is from the old high school ...
12
votes
5answers
417 views

Putnam Exam Integral

I am trying to evaluate$$ \lim_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\big(\frac{\pi}{2n}(x_1+x_2+...x_n)\big)dx_1 dx_2...dx_n. $$ This is from an old Putnam mathematics competition. Either ...
6
votes
2answers
133 views

Integrating $ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $

Compute $$ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $$ I am not sure how to start this one...I am thinking of a substitution to get started.
7
votes
2answers
264 views

Integral $ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$

Hello there I am trying to calculate $$ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx $$ NOT using mathematica, matlab, etc. We are given that $\sigma, \omega$ are complex. Note, the ...
11
votes
2answers
405 views

Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.

I am trying to calculate $$ I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}. $$ Note, we can expand the log in the integral to ...
3
votes
2answers
208 views

Integral, Definite Integral $ \int_{-\infty}^\infty \exp{\big(\alpha x^4+\beta x^3+\gamma x^2 +\delta x+\epsilon}\big)dx, \ \alpha <0. $

Calculate the integral $$ I=\int_{-\infty}^\infty \exp{\big(\alpha x^4+\beta x^3+\gamma x^2 +\delta x+\epsilon}\big)dx, \ \alpha <0. $$ The answer can be expressed analytically in terms of a ...
6
votes
2answers
191 views

Computing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $

Integrate $$ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2) \, d\phi. $$ Something that may help $(1-2x\cos\phi+x^2)=(1-xe^{i\phi})(1-xe^{-i\phi})$. And using the series ...
6
votes
3answers
249 views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...
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 ...
8
votes
2answers
176 views

No cont function $f\colon\mathbb{R}\to\mathbb{R}$ with $f(x)$ rational $\iff f(x+1)$ irrational.

Prove that there are no continuous functions $f\colon \mathbb{R} \to \mathbb{R}$ with the property: For any $x \in \mathbb{R}$, $f(x)$ is a rational number if and only if $f(x+1)$ is an irrational ...
8
votes
1answer
120 views

How to fill up $(0,1)$ with disjoint closed intervals all total measure one

This is a problem which was proposed, but not chosen, for a mathematics competition for University students not long ago, and its solution is missing: Let $\sum_{n=1}^\infty a_n=1$, where ...
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 ...
1
vote
1answer
57 views

Prove that for $n\ge1$, $\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},…,a_2,a_1\rangle\right)^{-1}$

Prove that for $n\ge1$, $$\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},...,a_2,a_1\rangle\right)^{-1}$$ In addition, show 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 ...
7
votes
2answers
137 views

Compute $\int_0^1\int_0^1…\int_0^1\lfloor{x_1+x_2+…+x_n}\rfloor dx_1dx_2…dx_n$

Compute $\int_0^1\int_0^1...\int_0^1\lfloor{x_1+x_2+...+x_n}\rfloor dx_1dx_2...dx_n$ where the integrand consists of the floor (or greatest integer less than or equal) function. The case $n=1,2,3$ ...
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 ...
1
vote
1answer
60 views

Let $f,g$ be two distinct functions from $[0,1]$ to $(0, +\infty)$ such that $\int_{0}^{1} g = \int_{0}^{1} f $.

Let $f,g$ be two continuous, distinct functions from $[0,1]$ to $(0, +\infty)$ such that $\int_{0}^{1} g = \int_{0}^{1} f $. Given $n\in \mathbb{N},$ let $y_n = \int_{0}^{1} \frac{f^{(n+1)}}{g^{(n)}} ...
12
votes
1answer
317 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
2
votes
1answer
196 views

These two sequences have the same limit

Let $a_1$ and $b_1$ be any two positive numbers, and define $\{ a_n\}$ and $\{ b_n\}$ by $$a_n = \frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}},$$ $$b_n = \sqrt{a_{n-1}b_{n-1} }.$$ Prove that the ...
13
votes
1answer
188 views

integral inequality $\int_0^a \left(\frac{f(x)}{2x}\right)^2 dx \le \int_0^a (f'(x))^2 dx$

Let $f:[0,a]\rightarrow\mathbb{R}$ be continuous differentiable function satisfying $f(0)=0$. Prove the following inequality $$\int_0^a \left(\frac{f(x)}{2x}\right)^2 dx \le \int_0^a (f'(x))^2 dx$$ ...
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 ...
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 $f(E)=E$?
12
votes
4answers
382 views

Proving the inequality $\arctan\frac{\pi}{2}\ge1$

Do you see any nice way to prove that $$\arctan\frac{\pi}{2}\ge1 ?$$ Thanks! Sis.
8
votes
3answers
253 views

Proving that $x(1-e^{-1/x})$ is strictly increasing

Prove that the function below is strictly increasing $$f(x)=x(1-e^{-1/x}), \quad x>0$$
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 $$\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}\;.$$