Challenge Definition and 942 Threads

Prove that $\sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{2006}}}}<2$.

Assuming the axiom of choice, show that a group $G$ has trivial automorphism group if and only if $G$ has less than three elements.

$a_1=2 ,$ and $a_{n+1}=\dfrac{a_n+4}{2a_n+3},\,\, n\in N$ find :$a_n$

For any positive integer $n\;,$ prove that$\sqrt{4n+1}<\sqrt{n}+\sqrt{n+1}<\sqrt{4n+2}$. Hence or otherwise, prove that $\left\lfloor{\sqrt{n}+\sqrt{n+1}}\right\rfloor=\left\lfloor{\sqrt{4n+1}}\right\rfloor$ for any positive integer $n$.

Evaluate $2\cos^3 \dfrac{\pi}{7}-\cos^2 \dfrac{\pi}{7}-\cos \dfrac{\pi}{7}$.

Place the digits 1 through 8 in the boxes so that no two consecutive digits are adjacent (not vertically, horizontally or diagonally). . . \begin{array}{cccccccccc}&& * & - & * & - & * \\ && | && | && | \\ * &-& * &-& * &-& * &-& * \\ | && | && | && | && | \\ * &-& * &-& * &-& * &-& * \\ && | &&...

Prove that $\left\lfloor{\sqrt{n}+\sqrt{n+1}}\right\rfloor=\left\lfloor{\sqrt{4n+2}}\right\rfloor$ for any positive integer $n$.

Hi, I am doing an exercise practice samples for the upcoming quiz, and stumbled across two questions I'm having trouble solving... First question is to integrate integral e-x2 dx ...where the solution is equal to pi1/2 Also... As for the second question (of a different equation) how can one...

Dear Readers, I'm facing a pretty interesting challenge, which is: "How to connect multiple smartphone mic's in the quickest and simplest way?" Here is a use case. I go into a meeting with a group of random (strangers) that all have smartphones, Android, iOS, Windows. Now I want to record this...

Find the minimum of $\sqrt{a^2-12a+40}+\sqrt{b^2-8b+20}+\sqrt{a^2+b^2}$.

One thing I find consistently disturbing about the way printing shows up on web forums is that it's hard to perceive the customary double-space between two sentences.as being much of a physical division. The period is usually a tiny speck, so it doesn't, by itself, do the job of separation...

Hello, Before anyone thinks this is a coursework question, it is not. It is a challenge problem, which I found online, and seems worth discussing. (Question) In a 400-m relay race the anchorman (the person who runs the last 100 m) for team A can run 100 m in 9.8 s. His rival, the anchorman...

Find the maximum value of the function $\sqrt{x^4-9x^2-12x+61}-\sqrt{x^4-x^2+1}$.

$\sqrt{\text{mbh}_{29}}$ Challenge: Sn = 3, 293, 7862, 32251, 7105061, 335283445, 12826573186, ?, ?, 44164106654163 S1 through S7 begin an infinite integer sequence, not found in OEIS. 1) Find S8 and S9. 2) Does S10 belong to Sn? 3) If S10 is incorrect, what is the correct value of S10...

Solve for real (if there is any) of the equation $\left\lfloor{a}\right\rfloor+\left\lfloor{2a}\right\rfloor+\left\lfloor{4a}\right\rfloor+\left\lfloor{8a}\right\rfloor+\left\lfloor{16a}\right\rfloor=300$.

There are real numbers $l,\,m,\,n$ such that $l\ge m\ge n >0$. Prove that $\dfrac{l^2-m^2}{n}+\dfrac{n^2-m^2}{l}+\dfrac{l^2-n^2}{m}\ge 3l-4m+n$.

Like I mentioned in the title, this is probably one of the greatest challenge problems (I've seen so far) that designed for, hmm, well, for a challenge!:o Let $x_1$ be the largest solution to the equation $\dfrac{6}{x-6}+ \dfrac{8}{x-8}+\dfrac{20}{x-20}+\dfrac{22}{x-22}=x^2-14x-4$ Find the...

Equate the limit $$\lim_{n \to \infty} \frac1{n} \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i + j}$$ Note : This was a challenge from a user in mathstackexchange. From a glance, there should be many ways to do it, so partly I posed this problem to see how the resident analysts in MHB handle it...

Solve for all real $a$ of the equation below: $\dfrac{1}{\left\lfloor{a}\right\rfloor}+\dfrac{1}{\left\lfloor{2a}\right\rfloor}=a-\left\lfloor{a}\right\rfloor+\dfrac{1}{3}$

Prove that $\cot 7\dfrac{1}{2}^{\circ}=\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$.

Homework Statement A) A rocket fires two engines simultaneously. One produces a thrust of 675N directly forward while the other gives a thrust of 450N at an angle 20.4∘ above the forward direction. a) Find the magnitude of the resultant force which these engines exert on the rocket...

Let $S$ be a nonempty set of natural numbers, equipped with the following membership rules: $$\text{if} ~ ~ x \in S ~ ~ \text{then} ~ ~ 4x \in S \tag{1}$$ $$\text{if} ~ ~ x \in S ~ ~ \text{then} ~ ~ \lfloor \sqrt{x} \rfloor \in S \tag{2}$$ Show that $S = \mathbb{N}$, and find all the natural...

A sequence of integers ${x_i}$ is defined as follows: $x_i=i$ for all $1<i<5$ and $x_i=(x_1x_2\cdots x_{i-1})-1$ for $i>5$. Evaluate $\displaystyle x_1x_2\cdots x_{2011}-\sum_{i=1}^{2011} (x_i)^2$.

Evaluate \lim_{{k}\to{\infty}} \int_{k}^{2k} \frac{k^3x}{x^5+1}\,dx.

Let $ABC$ be an equilateral triangle, and let $K$ be a point in its interior. Let the line $AK,\,BK,\,CK$ meet the sides of $BC,\,CA,\,AB$ in the points $A',\,B',\,C'$ respectively. Prove that $A'B'\cdot B'C'\cdot C'A' \ge A'B\cdot B'C\cdot C'A$.

There is a famous picture. Could you write in LaTeX something similar to this: without using explicit commands that insert whitespace such as \, \: \; \enskip \quad \hskip \mskip \hspace \kern and \mkern?

Prove that $p^4+q^4+r^4-2p^2q^2-2q^2r^2-2r^2p^2<0$ for $p,\,q,\,r$ are the sides of a triangle.

Solve the equation $\sin^7 x+\dfrac{1}{\sin^3 x}=\cos^7 x+\dfrac{1}{\cos^3 x}$.

Prove that $\displaystyle\left(\sum_{k=1}^{n} \sqrt{\dfrac{k-\sqrt{k^2-1}}{\sqrt{k(k+1)}}}\right)^2\le n\sqrt{\dfrac{n}{n+1}}$, where $n$ is a positive integer.

Hi, Let y= |sin(x) + cos(x) + tan(x) + sec(x) + csc(x) + cot(x)| Find the minimum value of "y" for all real numbers. Graphing is not allowed, no devices, calculators whatsoever. Its VERY hard to find where this function = 0 analytically so it is better to take two different...

Prove that $(-1,\,\pm 1)$, $(0,\,\pm 1)$, $(3,\,\pm 11)$ are the only integers solution for the equation $y^2=x^4+x^3+x^2+x+1$.

A polynomial $P(x)=x^3+mx^2+nx+k$ is such that $n<0$ and $mn=9k$. Prove that the polynomial has three distinct real roots.

Find the maximum of $P(x)=\dfrac{x(\sqrt{100-x^2}+\sqrt{81-x^2})}{2}$.

Find $m,\,n$ and $A$ such that $\sqrt{9-8\cos 40^{\circ}}=m+n\cos A^{\circ}$ where $m,\,n\in N$.

given:$a>b>c>0$ prove:$a^{2a}b^{2b}c^{2c}>a^{b+c}b^{c+a}c^{a+b}$

prove: $3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+7\sqrt{1+8\sqrt{1+9--}}}}}}}}$

Let $A$ be the intersection point of the diagonals $PR$ and $QS$ of a convex quadrilateral $PQRS$. The bisector of angle $PRS$ hits the line $QP$ at $B$. If $AP\cdot AR+AP\cdot RS=AQ\cdot AS$, prove that $\angle QBR=\angle RSQ$.

Let $x\ge \dfrac{1}{2}$ be a real number and $n$ a positive integer. Prove that $x^{2n}\ge (x-1)^{2n}+(2x-1)^n$.

Pranav-Arora has sent me an excellent math challenge for this week. The problem statement is easy: 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...

Solve for real solutions of the system of equations below: $a(\sqrt{b}+b)=\sqrt{1-a}(\sqrt{a}+\sqrt{1-a})$ $32a(a^2-1)(2a^2-1)^2+b=0$

Let $a$ and $b$ be two different primes. Prove that $\displaystyle\left\lfloor\dfrac{a}{b} \right\rfloor+\left\lfloor\dfrac{2a}{b} \right\rfloor+\left\lfloor\dfrac{3a}{b} \right\rfloor+\cdots+\left\lfloor\dfrac{(b-1)a}{b} \right\rfloor=\dfrac{(a-1)(b-1)}{2}$.

Let $p,\,q,\,r,\,s,\,t$ be distinct real numbers. Prove that the equation $(x-p)(x-q)(x-r)(x-s)+(x-p)(x-q)(x-r)(x-t)+(x-p)(x-q)(x-s)(x-t)+(x-p)(x-r)(x-s)(x-t)+(x-q)(x-r)(x-s)(x-t)=0$ has 4 distinct real solutions.

If $a,\,b$ are the two largest real roots of the polynomial $f(x)=3x^3-17x+5\sqrt{6}$, and their sum can be expressed as $\dfrac{\sqrt{m}+\sqrt{n}}{k}$ for positive integers $m,\,n,\,k$, find the value for $a+b$.

Let $f(x)$ be a polynomial with real coefficients, satisfying $f(x)-f'(x)-f''(x)+f'''(x)>0$ for all real $x$. Prove that $f(x)>0$ for all real $x$.

Solve the system of equations below: $(a+\sqrt{a^2+1})(b+\sqrt{b^2+1})=1$ $b+\dfrac{b}{\sqrt{a^2-1}}+\dfrac{35}{12}=0$

Show that if $x^4+px^3+2x^2+qx+1$ has a real solution, then $p^2+q^2\ge 8$.

From the binomial theorem, we have $\displaystyle \begin{align*}\left(1+\dfrac{1}{5}\right)^{1000}&={1000 \choose 0}\left(\dfrac{1}{5}\right)^{0}+{1000 \choose 1}\left(\dfrac{1}{5}\right)^{1}+{1000 \choose 2}\left(\dfrac{1}{5}\right)^{2}+\cdots+{1000 \choose...

If ABCDEFG is a regular heptagon prove that $\frac{1}{AB}=\frac{1}{AC}+\frac{1}{AD}$.

Mr. Smith’s dog Rosie takes a flying leap off his bed. The bed is 1m high, and Rosie leaves with a muzzle velocity of 5 m/s [40° above the horizontal]. Sometime after Rosie leaves the bed, Mr. Smith (who is 5m away from the bed) throws a doggie treat to Rosie from ground level with a muzzle...

For all real $x$, prove that $\displaystyle\sum_{k=0}^{\infty} \left\lfloor\dfrac{x+2^k}{2^{k+1}}\right\rfloor=\lfloor x\rfloor$.