Challenge Definition and 911 Threads

Evaluate the following: $$\Large \sum_{k=1}^{\infty} (-1)^{\left\lfloor \frac{k+3}{2} \right\rfloor} \frac{1}{k}$$

[SIZE="6"]Infinite Products This weeks challenge is a short one: Please list any sources that you have used to solve this question. Using google or other search engines is forbidden. Wikipedia is allowed but not its search engine. Wolfram alpha and other mathematical software are not...

Prove that $6^{33}>3^{33}+4^{33}+5^{33}$.

Let $P$ be a real function with a continuous third derivative such that $P(x),\,P'(x),\,P''(x),\,P'''(x)$ are greater than zero for all $x$. Suppose that $P(x)>P'''(x)$ for all $x$, prove that $2P(x)>P'(x)$ for all $x$.

Given that $0<k,\,l,\,m,\,n<1$ and $klmn=(1-k)(1-l)(1-m)(1-n)$, show that $(k+l+m+n)-(k+m)(l+n)\ge1$.

Evaluate $\dfrac{1}{\sin^2 \dfrac{\pi}{10}}+\dfrac{1}{\sin^2 \dfrac{3\pi}{10}}$.

Find all integral values of $k$ such that $q(a)=a^3+2a+k$ divides $p(a)=a^{12}-a^{11}+3a^{10}+11a^3-a^2+23a+30$.

[SIZE="5"]Happily Married QUESTION: Please list any sources that you have used to solve this question. Using google or other search engines is forbidden. Wikipedia is allowed but not its search engine. Wolfram alpha and other mathematical software is allowed. Points will be given as...

Show that for all $a>1$, there is a triangle with sides $a^4-1$, $a^4+a^3+2a^2+a+1$, and $2a^3+a^2+2a+1$.

Hi MHB, Problem: Find all real $x$ which satisfy $\dfrac{x^3+a^3}{(x+a)^3}+ \dfrac{x^3+b^3}{(x+b)^3}+\dfrac{x^3+c^3}{(x+c)^3} + \dfrac{3(x-a)(x-b)(x-c)}{2(x+a)(x+b)(x+c)}=\dfrac{3}{2}$. I tried my very best to solve this intriguing problem, but failed. Now I'm even clueless than I was...

For $$a\, ,b\in\mathbb{R}\,$$ and $$b>|a|\,$$ show that: $$\int_0^{\infty}\frac{\sinh ax}{\sinh bx}\, dx = \frac{\pi}{2b}\tan\frac{\pi a}{2b}$$

There is a sequence which has the first 3 terms listed as $1,\,94095,\,5265679\cdots$. The 50th term has all but one digit. If the missing digit is $a$, find the $a$th term from this sequence.

The cubic polynomial $x^3+mx^2+nx+k=0$ has three distinct real roots but the other polynomial $(x^2+x+2014)^3+m(x^2+x+2014)^2+n(x^2+x+2014)+k=0$ has no real roots. Show that $k+2014n+2014^2m+2014^3>\dfrac{1}{64}$.

The positive real $a$ and $b$ satisfy $b^3+a^2\ge b^4+a^3$. Prove that $b^3+a^3\le 2$.

Consider the letter T (written as such: thus we have two line segments). 1) Prove that it is impossible to to place uncountably many copies of the letter T disjointly in the plane ##\mathbb{R}^2##. 2) Prove that it is impossible to place uncountably many homeomorphic copies of the letter T...

If $\dfrac{\sin 4x}{a}=\dfrac{\sin 3x}{b}=\dfrac{\sin 2x}{c}=\dfrac{\sin x}{d}$, show that $2d^3(2c^3-a^2)=c^4(3d-b)$.

The idea of this challenge is to investigate the equation x^y = y^x Prove the following parts: If ##0<x<1## or if ##x=e##, then there is a unique real number ##y## such that ##y^x = x^y##. However, if ##x>1## and ##x\neq e##, then there is precisely one number ##g(x)\neq x## such...

Prove that the average of the numbers $n\sin n^{\circ}$ (where $n=2,\,4,\,6,\, \cdots,\,180$) is $\cot 1^{\circ}$.

Homework Statement Find two matrices E and F such that: EA= \begin{bmatrix} 2 & 1 & 2\\ 0 & 2 & 1\\ 0 & 3 & 0\\ \end{bmatrix} FA= \begin{bmatrix} 0 & 2 & 1\\ 0 & 3 & 0\\ 2 & 7 & 2\\ \end{bmatrix} Homework Equations The Attempt at a Solution So I know how to get...

Given a triangle $PQR$ where $QR=m$, $PQ=PR=n$ and $\angle P=\dfrac{\pi}{7}$. Show that $m^4-3m^2n^2-mn^3+n^4=0$.

In triangle $PQR$, $\tan P,\,\tan Q,\,\tan R$ are integers, find their values.

Given two segments of lengths $x$ and $y$, construct with a straightedge and a compass a segment of length $\sqrt[4]{x^4+y^4}$.

Let $f(x) = 5x^{13}+13x^5+9\cdot a \cdot x$ Find the smallest possible integer, $a$, such that $65$ divides $f(x)$ for every integer $x$.

"A diode laser has a divergence of 5mrad in the p-direction and 1 mrad in the s-direction. Design an optical system in front of the laser which will make the output circular, and calculate the resulting divergence." Attempt; I am taking the course optic and waves, and the instructor did some...

If $\alpha_n=\arctan n$, prove that $\alpha_{n+1}-\alpha_n<\dfrac{1}{n^2+n}$ for $n=1,\,2,\,\cdots$.

Prove that for any real numbers $a$ and $b$ in $(0,\,1)$, that $(a+b)^{a+b}\le (2a)^a(2b)^b$.

If $P(0)=3$ and $P(1)=11$ where $P$ is a polynomial of degree 3 with integer coefficients and $P$ has only 2 integer roots, find how many such polynomials $P$ exist?

Evaluate $\displaystyle\int^{\dfrac{\pi}{4}}_0 \dfrac{x}{(\sin x+\cos x)\cos x}\ dx$.

Let $a_1 = 1$, $a_2 = a_3 = 2$, $a_4 = a_5 = a_6 = 3$, $a_7 = a_8 = a_9 = a_{10} = 4$, and so on. That is, $a_n ∶ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, . . . . $ What is $a_{2013}$?

Homework Statement Nothing travels faster than light, which manages to get to the moon from the Earth in 1 second. However, we can still get there in a shorter amount of time. How fast would we have to travel to reach the moon in 0.9 seconds? Homework Equations I know the question is weird but...

If the quadratic equation $x^2+(2 – \tan \theta)x – (1 + \tan \theta) = 0$ has two integral roots, then sum of all possible values of $\theta$ in the interval $(0, 2\pi)$ is $k\pi$. Find $k$.

Show that $\displaystyle \sum_{k=0}^n \sin k=\dfrac{\sin \dfrac{n}{2} \sin\dfrac{n+1}{2}}{\sin \dfrac{1}{2}}$.

Prove that $\dfrac{1}{\sqrt{4x}}\le\left( \dfrac{1}{2} \right)\left( \dfrac{3}{4} \right)\cdots\left( \dfrac{2x-1}{2x} \right)<\dfrac{1}{\sqrt{2x}}$.

Show that $e^\dfrac{1}{e}_{\phantom{i}}+e^{\dfrac{1}{\pi}}_{\phantom{i}} \ge2e^{\dfrac{1}{3}}_{\phantom{i}}$.

For all $x,\,y,\,z \in R$ with $x+y+z=2\pi$, prove that $\cos^2 x+\cos^2 y+\cos^2 z+2\cos x\cos y \cos z=1$

Show that the equation $a^2=b^4+b^2+1$ does not have integer solutions.

Jason has a coin which will come up the same as the last flip $\dfrac{2}{3}$ of the time and the other side $\dfrac{1}{3}$ of the time. He flips it and it comes up heads. He then flips it 2010 more times. What is the probability that the last flip is heads?

Let $x,\,y$ be positive integers such that $\dfrac{7}{10}<\dfrac{x}{y}<\dfrac{11}{15}$. Find the smallest possible value of $y$.

Prove that $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)=\tan 9^{\circ}$

Solve the equation $\sin a \cos b+ \sin b \cos c+ \sin c \cos a=\dfrac{3}{2}$

Show that $\dfrac{1}{2} \cdot \dfrac{3}{4} \cdot \dfrac{5}{6} \cdots \dfrac{1997}{1998} >\dfrac{1}{1999}$, where the use of induction method is not allowed.

Without the help of calculator, show that $\tan 50^{\circ}>1.18$

Evaluate: $$2^{2009}\frac{\displaystyle \int_0^1 x^{1004}(1-x)^{1004}\,dx}{\displaystyle \int_0^1x^{1004}(1-x^{2010})^{1004}\,dx}$$ ...of course without the use of beta or gamma functions. :p

Consider the sequences $(c_n)_n,\,(d_n)_n$ defined by $c_0=0$, $c_1=2$, $c_{n+1}=4c_n+c_{n-1}$, $n \ge 0$, $d_0=0$, $d_1=1$, $d_{n+1}=c_n-d_n+d_{n-1}$, $n \ge 0$. Prove that $(c_n)^3=d_{3n}$ for all $n$.

Let $k$ be an odd positive integer. Solve the equation $\cos kx=2^{k-1} \cos x$.

Let $p,\,q,\,r,\,s\,\in[0,\,\pi]$ and we are given that $2\cos p+6 \cos q+7 \cos r+9 \cos s=0$ and $2\sin p-6 \sin q+7 \sin r-9 \sin s=0$. Prove that $3 \cos (p+s)=7\cos(q+r)$.

Evaluate the following: $$\int_0^{\pi} e^{\cos x} \cos(\sin x)\,\,dx$$

Many of you don't know this, but as a young man, Greg once decided to worship the moon. He was so obsessed by the moon that he once decided to start following it. So at any given moment, he would check where the moon is and then walk in that direction. Greg has special powers so that he can see...

Let $f(x)=x^4+px^3+qx^2+rx+s$, where $p,\,q,\,r,\,s$ are real constants. Suppose $f(3)=2481$, $f(2)=1654$, $f(1)=827$. Determine the value of $\dfrac{f(-5)+f(9)}{4}$.

Show that the equation $8x^4-16x^3+16x^2-8x+p=0$ has at least one non-real root for every real number $p$ and find the sum of all the non-real roots of the equation.