Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Find $X, Y \in \mathbb{Z}$ such that $2^a X + (2^b - 1) Y = 1$ (coprimality) I've been wrecking my brain trying to solve this exercise. Is this answer wrong?
$$X= (2^{a})^{b-1}, Y= (-1) (2^b +1) \ [(2^b -1)(2^b +1)]^{a-1}$$
| Supposing $a,b>0$.For sure there exists a number $m\in \Bbb{Z}$ such that:
$$mb\geq a\Rightarrow m\geq \frac{a}{b} \Rightarrow m\geq \left \lceil{\frac{a}{b}}\right \rceil=M $$
Notice that if:
$$Y=-\sum_{i=0}^{M} 2^{bi}=\frac{-2^{b(M+1)}+1}{2^b-1}$$
Then the equation becomes:
$$2^a X-2^{b(M+1)}+1=1$$
Clearly this is sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3209285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Solve $2x^2-5x+2=$ $\frac{5-\sqrt{9+8x}}{4}$ Solve $2x^2-5x+2$= $\frac{5-\sqrt{9+8x}}{4}$
I simply do square both sides solve it and I get two value of x one is 2 and other is $\frac{3-√5}{2}$ but this approach it take more time so is there any approach for solving this equation.
| I don't know another way then going through the algebra, I got the same solutions.
$$\begin{align} 2x^2-5x+2 &= \frac{5-\sqrt{9+8x}}{4} \\
8x^2-20x+3 &= -\sqrt{9+8x} \\
(8x^2-20x+3)^2 &= (-\sqrt{9+8x})^2 \\
64x^4-320x^3+448x^2-120x+9 &= 9 +8x\\
64x^4-320x^3+440x^2-120x &= 0 \\
64x(x-2)(x^2-3x+1) &=0 \\
\end{align}$$
We... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3213006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
When recreating the quadratic formula by completing the square of $ax^2+bx+c=0$ I cannot shorten the right hand side I am attempting to derive the quadratic formula by completing the square on the generic generic expression:
$$ax^2+bc+c=0$$
I'm struggling with the right hand side of the equation which, for the step I... | In your last step $\frac{b^2a - 4a^2c}{4a^3}$ cancel $a$ from numerator and denominator, to get $ \frac{b^2 - 4ac}{4a^2}$
or take the common denominator as $4a^2$ in the beginning itself.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3213556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Elementary Proof Relating to Pythagorean Triples Please could I get some feedback on the correctness and overall quality of the following proof, which is meant to show that all Pythagorean triples contain unique integers.
To Prove: all Pythagorean triples contain unique integers.
First we note that in a right-angled tr... | We are given$\quad A=m^2-n^2\quad B=2mn\quad C=m^2+n^2.$ and we wish to prove that that $A,B,C$ are unique integers.
If $A=B$ then$\quad(m^2-n^2)^2+(m^2-n^2)^2=C^2\Rightarrow C=(m^2-n^2)\sqrt{2}$ so it is not Pythagorean triple where all sides are integers.
If $B=A$ then$\quad (2mn)^2+(2mn)^2=8m^2n^2=C\Rightarrow C=2m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3214816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Having the identity: $\int_0^1 \frac{x^{2n}}{1+x^2} dx= \sum_{k=0}^{\infty} \frac{(-1)^k}{2n+2k+1}$, how do I square the fraction inside the sum? I have the integral:
\begin{align}
\int_0^1 \frac{x^{2n}}{1+x^2} dx &= \int_0^1 x^{2n} \sum_{k=0}^{\infty}(-1)^kx^{2k} dx\\
&= \sum_{k=0}^{\infty} \frac{(-1)^k}{2n+2k+1}
\en... | Really what you want to do is define the function
$$f_n(x)=\int_0^x\frac{t^{2n}}{1+t^2}dt=\int_0^xt^{2n}\sum_{k\geq0}(-1)^kt^{2k}dt\\
=\sum_{k\geq0}\frac{(-1)^k}{2n+2k+1}x^{2n+2k+1}$$
Then notice that the sum you seek is given by
$$g(n)=\int_0^1 f_n(x)\frac{dx}{x}=\sum_{k\geq0}\frac{(-1)^k}{(2n+2k+1)^2}=\frac14\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3216113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find tangents to a circle parallel to the straight line Find tangents to a circle $x^2+y^2=5$ parallel to the straight line $2x-y+1=0$
My solution:
$$x^2+y^2=5$$
$$S=(0,0)$$
$$r=\sqrt{5}$$
$$y=2x-1$$
$$a=2$$
Searching for b
$$y=2x+b$$
Using following formula:
$$d=\frac{Ax_0+By_0+C}{\sqrt{A^2+B^2}}$$
$$-2x+y-b$$
$$\sqrt... | *
*The correct distance formula between a point $P_0(x_0,y_0)$ and the straight line $Ax+Bx+C=0$ is $$d=\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}.\tag{1}$$
*From $(1)$ and since the center of the given cirle ( $ x^2+y^2=5 $ ) is $(x_0,y_0)=(0,0)$, we obtain
$$d=\frac{|C|}{\sqrt{A^2+B^2}}.\tag{2}$$
*The straight line $2x−... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3217139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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general solution to a Non homogeneous differential equation Determine the general solution to the following differential equation:
$$y'' + 9y = t^2e^{3t}$$
The homogeneous solution is
$$y(t)=C_1\cos(3t) +C_2\sin(3t)$$
Solving the non homogeneous solution I let
$$y(t)=(At^2+Bt+C)e^{3t}$$
I found $A=1/36$, $B=-5/216$, ... | $y" + 9y = t^2e^{3t}$
$(D^2 + 9)y = t^2e^{3t}$
Auxiliary equation: $m^2 + 9 = 0$ or $m = \pm3i$
$CF = C_1\cos(3t) + C_2\sin(3t)$
$PI = \frac{1}{D^2 + 9}t^2e^{3t}$
Isolating $e^{3t}$ and replacing $D$ by $D+3 $ (coeff. of t)
$PI = e^{3t}\frac{1}{(D+3)^2 + 9}t^2$
$PI = e^{3t}\frac{1}{D^2+6D+18}t^2$
$PI = e^{3t}\frac{1}{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that for all $n \in \mathbb{N}$, either 3 or 13 divides $3^n + 13n^2 + 38$ Let $a\in \{3,13\}.$ I'm having trouble with this proof. I know that
$$3^{n+1} + 13(n+1)^2 + 38 = (3^n + 13n^2 + 38) + (2\cdot 3^n + 26n + 13)$$
But I can't prove that $a \mid 2\cdot3^n + 26n + 13$. I know that 13 doesn't divide this becau... | Define $f(n) = 3^n + 13n^2 + 38$.
Using algebra it can be shown that for all $n \ge 0$ we have the following identities,
$\tag 1 f(n+1) = 3 f(n) -26 n^2 + 26 n -63$
$\tag 2 f(n+2) = 9 f(n) -104 n^2 + 52 n -252$
$\tag 3 f(n+3) = 27 f(n) -338 n^2 + 78 n -871 =$
$\quad \quad \quad \quad \quad \quad \quad \quad \quad \qua... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many Taylor series terms are needed to accurately approximate $\sqrt{a+x}-\sqrt{a}$? Naive evaluation of $\sqrt{a + x} - \sqrt{a}$ when $|a| >> |x|$ suffers from catastrophic cancellation and loss of significance.
WolframAlpha gives the Taylor series for $\sqrt{a+x}-\sqrt{a}$ as:
$$\frac{x}{2 \sqrt{a}} - \frac{x^2}... | It was pointed out by Robert Israel that the series does badly when $|x| \approx |a|$, but in that case the loss of significance of the naive evaluation is small.
It was also suggested by Winther (and a since-deleted answer) to rewrite as $$\frac{x}{\sqrt{a+x}+\sqrt{a}}$$
The series for the denominator is similar to th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3224745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Explicit formula for a sequence of alternating differences Imagine this sequence: $1, 11, 12, 22, 23, 33, 34, 44$, ...
Obviously the sequence is alternating in adding 10 and adding 1. You can find the formula for the differences easily with $5*(-1)^x+6$, but I can't find a way to find the exact explicit formula for the... | If you try to find a formula for the terms in positions $2,4,6,\ldots$ which are $11,22,33,\ldots$ you get $11 \frac{x}2 = \frac{11}{2}x$
If you try to find a formula for the terms in positions $1,3,5,\ldots$ which are $1,12,23,\ldots$ you get $11 \frac{x-1}2 +1 = \frac{11}{2}x-\frac{9}{2}$
These expressions are o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3224967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Confused on an algebraic step of inductive proof
The part I've boxed is what is throwing me - why is it 6(k+1) instead of 6(k+1)^2?
| Let's add some steps in:
\begin{align*}
1^2 + 2^2 + \cdots + k^2 + (k + 1)^2 &= \frac{k(k+1)(2k+1)}{6} + (k+1)^2 \\
&= \frac{(k+1)}{6} \cdot\frac{k(2k+1)}{1} + (k+1)(k+1) \\
&= \frac{(k+1)}{6}k(2k+1) + \frac{6(k+1)}{6}(k+1) \\
&= \frac{(k+1)}{6}[k(2k+1) + 6(k+1)].
\end{align*}
It's not squared, because a factor of $(k ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3225199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$
Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$
By factor theorem we know that $(x-a)$ is a factor of $f(x)$ if $f(a)=0$.
In this case, $f(x)=x^n -nx +n-1 \implies f(1)=0$
Hence we conclude that $(x-1)$ is a factor. From hereon, how can I say that $(x-1)^2$ is a fa... | By division theorem we have:
$$x^n -nx +n-1 = k(x)(x-1)^2+ax+b $$
So for $x=1$ we have $0 = 0+a+b$ so $a=-b$, so $$x^n -nx +n-1 = k(x)(x-1)^2+a(x-1) $$
$$(x-1)(x^{n-1}+...+x^2+x+1) -n(x-1)= k(x)(x-1)^2+a(x-1) $$
so, after dividing by $x-1$ we get $$(x^{n-1}+...+x^2+x+1) -n= k(x)(x-1)+a $$
Now puting $x=1$ again we have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3228680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
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An indefinite Integral Problem with algebric numerator and trigonometric denominator $$\int \frac{x^2+(n(n-1))}{(x\sin x +n\cos x)^2 } dx$$
I know this is an homework problem, but I really couldn't think of any way to solve it. Like DI Method (No go) , What kind of substitution as denominator is trigonometric whereas N... | $I = \int\frac{x^2 + n(n-1)}{(x\sin x + n\cos x )^2}dx$
Now we'll try to convert it into the form of $\frac{a}{y} + \frac{b}{y^2}$, where $a,b$ are functions of $x$ and $y$ is the denominator.
$$x^2+n(n-1) =(x\sin x + (n-1)\cos x)(x\sin x+n\cos x)-((1-n)\sin x + x\cos x)(n\sin x - x\cos x)$$
$$$$
$I = \int \big[\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3231392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
find limit of a multivariable function I have to show continuity at $(0,0)$ of $f(x,y)=\frac{\sin(x^2) + \sin(y^2)}{\sqrt{x^2 +y^2}}$ for $(x,y)\ne(0,0)$ and $f(0,0)=0$.
I tried to find the limit using polar coordinates
$ \frac{\sin(r^2\cos^2 α) + \sin(r^2\sin^2 α)}{r} $
but it's still $\frac00$
| \begin{align*}T&:=\frac{\sin \ x^2+\sin\ y^2}{\sqrt{x^2+y^2}}
\\&=\frac{\sin \ x^2 }{x^2}\frac{x^2}{
\sqrt{x^2+y^2}} +
\frac{\sin\ y^2}{y^2}\frac{y^2}{
\sqrt{x^2+y^2}}
\\&\leq \frac{\sin \ x^2 }{x^2}\frac{x^2 +y^2}{
\sqrt{x^2+y^2}} +
\frac{\sin\ y^2}{y^2}\frac{x^2+y^2}{
\sqrt{x^2+y^2}} \rightarrow 0 \end{align*}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3235028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Solve recurrence with characteristic polynomial $a_n=7\cdot a_{n-1} -7\cdot a_{n-2}+175\cdot a_{n-3}+450\cdot a_{n-4}+(5+13\cdot n)\cdot9^n $ The equation
$$a_n=7\cdot a_{n-1} -7\cdot a_{n-2}+175\cdot a_{n-3}+450\cdot a_{n-4}+(5+13\cdot n)\cdot9^n \enspace,$$
where
$a_0=148, a_1=144, a_2=-55, a_3=-61$.
I assume that a ... | Considering the homogeneous
$$
a_n-7\cdot a_{n-1}+7\cdot a_{n-2}-175\cdot a_{n-3}-450\cdot a_{n-4}=0
$$
after substituting $a_n = C_0 \beta^n$ we obtain
$$
C_0(\beta -9) (\beta +2) \left(\beta ^2+25\right) \beta ^{n-4}=0
$$
so we have the set of exponent solutions (for $n > 4$ assuming $\beta\ne 0$)
$$
\{-2,9,\pm i 5\}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3239328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Angle formed by orthocenter, incenter and circumcenter of a triangle $>135^\circ$? If $H$ is the orthocenter, $I$ the incenter and $O$ the circumcenter of a triangle , the I need to show that the angle $HIO>135^\circ$
With the assumptions of $OI^2=R^2-2Rr$, $OH^2=9R^2-(a^2+b^2+c^2)$, $HI^2=2r^2-4R^2\cos A\cos B \cos C... | This formula calculates the $θ$ angle formed by Orthocenter(H),Incenter(I) and circumcenter(O) of ABC triangle (sides:a,b,c).
$θ(a,b,c)=acot\left [\left [ \frac{(-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2)}{(-a+b+c)(a-b+c)(a+b-c)}-2abc+(-a+b+c)(a-b+c)(a+b-c) \right ] \sqrt{\frac{(-a+b+c)(a-b+c)(a+b-c)}{4(a+b+c)^3(a-b)^2(a-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3243614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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Square valued integer polynomial For what integer values of $n$ is the expression $n^{6}+n^{4}+1$ a square?
This is a square when $n=2$; when $n$ is odd, this expression is $3 \ mod(8)$ and so cannot be a square.
| The only value that works is $n=2$.
Indeed, if there is another value that works, then there exists integers $n > 2$ and $k>0$ such that
$$n^6 + n^4 +1 = (n^3+k)^2$$
which implies that there exists exists integers $n > 2$ and $k>0$ such that
$$n^4+1 = 2n^3+k^2$$
However, note that on the one hand, for $k=n/2$ that $2n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3244865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $x + y + z = 2$, then show $\frac{x y z}{(1-x)(1-y)(1-z)} \geq 8$, added a second question(Problem 2). Problem number 1:
The problem is that $x, y, z$ are proper fractions, and each one of them is greater than zero.
Given $x + y + z = 2$, prove
$$\frac{x y z}{(1-x)(1-y)(1-z)} \geq 8$$
I have tried to solve this... | We have $x+y+z=2$ so $1-x+1-y+1-z=3-(x+y+z)=1$ using Am-GM inequality twice once for $$x+y+z\geq 3(xyz)^{\frac{1}{3}}$$ and second for $$1-x+1-y+1-z\geq 3((1-x)(1-y)(1-z))^{\frac{1}{3}}$$,cubing both inequalities and dividing the two should yield the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3245356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\sum_{0\leq k\leq r} \binom{n+k}{k} \binom{m+n+k}{n+k} = \binom{m+n}{n} \binom{m+n+r+1}{m+n+1}$ where $m,n,r\in \mathbb{N} $ $\sum_{0\leq k\leq r} \binom{n+k}{k} \binom{m+n+k}{n+k} = \binom{m+n}{n} \binom{m+n+r+1}{m+n+1}$ where $m,n,r\in \mathbb{N} $.
Exam problem which stayed unproven for me. I tried ind... | We use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ of a series. This way we can write for instance
\begin{align*}
[z^k](1+z)^n=\binom{n }{k}\tag{1}
\end{align*}
We obtain
\begin{align*}
\color{blue}{\sum_{k=0}^r}&\color{blue}{\binom{n+k}{k}\binom{m+n+k}{n+k}}\\
&=\sum_{k=0}^r\frac{(n+k)!}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3245489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove or disprove: If $a+b \leq \frac{1}{2}$, then $\frac{(1-a)(1-b)}{ab} \geq 1$ for positive $a,b$
Let $a,b$ be two positive numbers. Prove or disprove the statement:
If $a+b \leq \frac{1}{2}$, then $\dfrac{1-a}{a} \dfrac{1-b}{b} \geq 1$.
True. Assume $a+b \leq \frac{1}{2}$. Then
$$\dfrac{1-a}{a} \dfrac{1-b}{b}=\df... | $$\frac{1- a}{a}\times \frac{1- b}{b}> \frac{(\,a+ b\,)- a}{a}\times \frac{(\,a+ b\,)- b}{b}= 1$$
$$\because\,1> a+ b$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3245841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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What is the probability that the first 1 comes in even trials of a six faced fair dice? A six faced fair dice is thrown until 1 comes,
then what is the probability that 1 comes in even no. of trials?
I tried my best and my efforts include:
I find the
probability of getting a 1 on 2nd throw $p(2) = (5/6) (1/6)$
probabi... | Observe that if we define $\Pr(2k)$ to be the probability that the first 1 appears on the $2k$th throw, then
$$\Pr(2k) = \left(\frac{5}{6}\right)^{2k - 1}\left(\frac{1}{6}\right)$$
Hence, the probability that the first one appears on an even-numbered throw is
$$\sum_{k = 1}^{\infty} \left(\frac{5}{6}\right)^{2k - 1}\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3246798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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A limit with sum $S_n=1+\frac{n-1}{n+2}+\frac{n-1}{n+2}\cdot \frac{n-2}{n+3}+\cdots +\frac{n-1}{n+2}\cdot \frac{n-2}{n+3}\cdots \frac{1}{2n}$ $$S_n=1+\frac{n-1}{n+2}+\frac{n-1}{n+2}\cdot \frac{n-2}{n+3}+\cdots +\frac{n-1}{n+2}\cdot \frac{n-2}{n+3}\cdots \frac{1}{2n}$$ Then $S_n/\sqrt{n}$ tends to $\frac{\pi}{2}$.
H... | The logarithm of a term is roughly
$$\int_{n-k-\frac12}^{n-\frac12}\log t dt - \int_{n+\frac32}^{n+k+\frac32}\log t dt$$
Find the leading order behaviour of that, then approximate the sum $S_n by an integral.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3247505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove for all positive a,b,c that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$ Prove for all positive a,b,c $$\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$$
My Try
I tried taking common denominator of the expression,
$\frac{a^2b+ab^2+b^2c+c^2b+ac^2+a^2c}{abc}$
How to proceed? Is there a way to write them as per... | Notice that $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \ge 3 \left( \frac{a}{b} \frac{b}{c} \frac{c}{a}\right)^{1/3}=3~~~~ \mbox{by AM-GM}$$
and $$\frac{b}{a}+ \frac{c}{b}+ \frac{a}{c} \ge 3 \left( \frac{b}{a} \frac{c}{b} \frac{a}{c}\right)^{1/3}=3~~~~ \mbox{by AM-GM}.$$ Adding these two we get the required inequality.
In... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that $x^2-8x+17>0$ for all real values of x Could someone please help me with this question. I thought I had to use the discriminant, which I found to be -4 but now I'm not sure if i'm just meant to factorise, as the discriminant would show there are no real solutions.
Thank you :)
| If the discriminant is negative then there are no values where the expression is equal to zero and as values can't "jump" from positive to negative without "passing through" zero then all values must be either all positive or all negative.
And as for $x=0$ this takes a positive value then all values are positive.
In ge... | {
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Double Integration Problem $\int_{0}^{1} \int_0^1 \frac{1}{1+y(x^2-x)}dydx$
Compute
$$I = \int_{0}^{1} \int_0^1 \frac{1}{1+y(x^2-x)}dydx$$
Here are my steps:
$$\begin{split}
I &=\int_{0}^{1} \left(\int_0^1 \frac{dy}{1+y(x^2-x)}\right)dx\\
&=\int_{0}^{1} \left[\frac{\ln(1+y(x^2-x))}{x^2-x}\right]_0^1dx\\
&=\int_... | Switching the order of integration, we get $$4 \int_{y=0}^1 \frac{1}{\sqrt{y(4-y)}} \tan^{-1} \sqrt{\frac{y}{4-y}}\, dy.$$ Then the substitution $$u = \tan^{-1} \sqrt{y/(4-y)}, \quad du = \frac{1}{1+\frac{y}{4-y}} \cdot \frac{2}{(4-y) \sqrt{y(4-y)}} \, dy = \frac{1}{2 \sqrt{y (4-y)}} \, dy,$$ gives $$4 \left[\left(\ta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Simplifying $\sum_{cyc}\tan^{-1}\left(\sqrt{\frac{x(x+y+z)}{yz}}\right)$. I get $0$, but the answer is $\pi$. So the question is
$$ \tan^{-1}\left(\sqrt{\frac{x(x+y+z)}{yz}}\right)+\tan^{-1}\left(\sqrt{\frac{y(x+y+z)}{xz}}\right)+\tan^{-1}\left(\sqrt{\frac{z(x+y+z)}{yx}}\right) =\ ? $$
So my take on the question is to... | Let $x$, $y$ and $z$ be positive numbers. We consider
a triangle $ABC$ with side lengths $a=BC=y+z$, $b=CA=x+z$ and $c=AB=x+y$. The semi-perimeter $s=x+y+z$ inradius $r$. Now, by Heron’s formula we have
$$\eqalign{\cot(A/2)&=\frac{s-a}{r}=\frac{s(s-a)}{{\rm Area}(ABC)}=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}\cr
&=\sqrt{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3255546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Summation Involving Product Of Two Identical Polynomials. Recently I stuck, to a problem. However I rarely think that there is some proper formula for this problem, but here I am in search of algorithm's or theorem that relate to this problem or can solve this problem.
We have three integers positive integers $a, b, ... | This is not a full answer because the bounds for $a$ and $b$ given by OP imply that there are situations where $a>>n$ and $b>>n.$ (For example, $n=10, a=1000, b=2000.$) The following asymptotic formula works well if $n>>a$ and $n>>b.$ Given the square of bounds for the parameters for $a$ and $b$ relative to $n$, the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Solve $(y^2 + xy)(x^2 - x + 1) = 3x - 1$ over the integers.
Solve $$(y^2 + xy)(x^2 - x + 1) = 3x - 1$$ over the integers.
There are many solutions to this problem, and perhaps I chose the worst one possible. I hope that someone could come up with a better answer.
This problem is adapted from a recent competition (whi... | We have that $$(y^2 + xy)(x^2 - x + 1) = 3x - 1 \implies x^2 - x + 1 \mid 3x - 1$$
$$\implies x^2 - x + 1 \le |3x - 1| \iff x \in [-2, 0] \cup [2 - \sqrt 2, 2 + \sqrt 2]$$
However, $x$ is an integer $\implies x \in \{0, \pm 1, \pm 2, 3\}$
We can set up a table for different values of $x$ and $y^2 + xy$.
$$\begin{matrix... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculate the maximum value of $\sqrt{\frac{3yz}{3yz + x}} + \sqrt{\frac{3zx}{3zx + 4y}} + \sqrt{\frac{xy}{xy + 3z}}$ where $x + 2y + 3z = 2$
$x$, $y$ and $z$ are positives such that $x + 2y + 3z = 2$. Calculate the maximum value of $$ \sqrt{\frac{3yz}{3yz + x}} + \sqrt{\frac{3zx}{3zx + 4y}} + \sqrt{\frac{xy}{xy + 3z}... | Let $x=\frac{2}{3}a$, $y=\frac{1}{3}b$ and $z=\frac{2}{9}c$.
Thus, $a+b+c=3$ and by AM-GM we obtain:
$$\sqrt{\frac{3yz}{3yz + x}} + \sqrt{\frac{3zx}{3zx + 4y}} + \sqrt{\frac{xy}{xy + 3z}}=\sum_{cyc}\sqrt{\frac{bc}{bc+3a}}=$$
$$=\sum_{cyc}\sqrt{\frac{bc}{(a+b)(a+c)}}\leq\frac{1}{2}\sum_{cyc}\left(\frac{b}{a+b}+\frac{c}{... | {
"language": "en",
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"source": "stackexchange",
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Fibonacci power series and fraction decomposition In a recent problem I have established that if $|x|<\frac{1}{2}$ then
$$f(x)=\sum_{n=1}^{\infty}a_{n}x^{n-1}=\frac{-1}{x^2+x-1}$$
where $a_{1}=a_{2}=1$ and $a_{n+1}=a_{n}+a_{n-1}$
(To work out the sum to the fraction, simply note that if $f(x)$ represents the sum then ... | On the second line of your equalities, you multiplied denominators but not numerator by $$\frac2{-1\pm\sqrt5}.$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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least value of $\lfloor \frac{a+b}{c}\rfloor+\lfloor \frac{c+b}{a}\rfloor+\lfloor \frac{a+c}{b}\rfloor$
If $a,b ,c>0$ . Then least value of $$\bigg\lfloor \frac{a+b}{c}\bigg\rfloor+\bigg\lfloor \frac{c+b}{a}\bigg\rfloor+\bigg\lfloor \frac{a+c}{b}\bigg\rfloor$$
Where $\lfloor x\rfloor$ is floor function of $x$
Plan
Us... | $$= 3 \bigg\lfloor \frac{a+b}{c} \bigg\rfloor$$
And given all variables are positive, the minimum occurs when $a+b$ is as close to $0$ as possible and $c$ is as large as possible.
| {
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Find all possible integers $n$ such that $\sqrt{n + 2} + \sqrt{n + \sqrt{n + 2}}$ is an integer.
Find all possible integers $n$ such that $m = \sqrt{n + 2} + \sqrt{n + \sqrt{n + 2}}$ is an integer.
Guess what? This problem is adapted from a recent competition. There have been a solution below for you to check out. I ... | We have that $n+2 = a^2, n + a = b^2$, where $a$ and $b$ are non-negative integers.
If $a > 2$, then $$a^2 = n+2 < n+a = a^2 + a - 2 < a^2 + 2a + 1.$$
This means that $n+a$ cannot be a perfect square as it's bounded between 2 consecutive perfect squares, which is a contradiction. Hence $ 0 \leq a \leq 2$.
If $ a= 0$,... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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quadratic equation solving mistake I'm a student who started self learning quadratic equations for a youth university program. I'm busy at trying to solve such equation:
$$
(1 - 4x)^2 + 9x + 7 = 2(x+3)(1-x) + (x+4)^2
$$
this is my current progress:
\begin{align}
(1 - 4x)^2 + 9x + 7 &= 2(x+3)(1-x)+ (x+4)^2\\
(1 - 4x)(1 ... | $$
(1 - 4)^2 + 9 + 7 = 2(1+3)(1-1) + (1+4)^2
$$
seems true ($25$ in both members), and with a little more effort
$$
\left(1 - 4\frac{\overline{14}}{17}\right)^2 + 9\frac{\overline{14}}{17} + 7 = 2\left(\frac{\overline{14}}{17}+3\right)\left(1-\frac{\overline{14}}{17}\right) + \left(\frac{\overline{14}}{17}+4\right)^2
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3269580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Finding general solution of $3{\times}3$ system I am given the following:
$$
x'=
\begin{bmatrix}
2 &0 &0 \\
-7&9 &7 \\
0&0 &2
\end{bmatrix}
x
$$
Solving $\det(A-\lambda I)$, I get $\lambda = 2,2,9$. Solving $\det(A-2\lambda)$, I get
\begin{bmatrix}
0&0 &0 \\
-7&7 &7 \\
0&0 &0
\end{bmatrix}
So we have g... | You are right that $v_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ is an eigenvector of $A$ with eigenvalue $2$; i.e $ v_1 \in \ker(A-2I)$. But like you mentioned, the dimension of this kernel is $2$, so you need to find another linearly independent eigenvector. It is easy to verify that $\xi = \begin{pmatrix} 1 \\ ... | {
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How to I find a formula for $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \cdots + \frac{1}{n(n+1)}$ using the given method? The following is problem $6$ $(iii)$ from chapter $2$ of Spivak's Calculus:
The formula for $1^{2} + \cdots + n^{2}$ may be derived as follows. We begin with the formula $$ (k+1)^{3} - k^{3} = 3k^{... | It is a perfectly fine answer. The above method which is being illustrated is known as Telescopic method of summation. You can read more about it here.
https://en.wikipedia.org/wiki/Telescoping_series
https://brilliant.org/wiki/telescoping-series/
| {
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Solve the following equation: $\sqrt {\sin x - \sqrt {\cos x + \sin x} } = \cos x$ Solve the following equation:
\begin{array}{l}{\sqrt{\sin x-\sqrt{\cos x+\sin x}}=\cos x} \\ \text{my try as follows:}\\{\sin x-\sqrt{\cos x+\sin x}=\cos ^{2} x} \\ {\sin x-\cos ^{2} x=\sqrt{\cos x+\sin x}} \\ {\sin ^{2} x+\cos ^{4} x-2... | In any solution, $\cos(x)$ is nonnegative, since it equals some square root.
In any solution, $\sin(x)$ is nonnegative, since the equation includes the square root of $\sin(x)$ minus some nonnegative quantity.
So $\cos(x)+\sin(x)\geq\sin(x)$, and therefore $$\sqrt{\cos(x)+\sin(x)}\geq\sqrt{\sin(x)}\geq\sin(x)$$
where t... | {
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"url": "https://math.stackexchange.com/questions/3278414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why can I simplify radicals? (eg, $\sqrt{153} = \sqrt{3}\cdot \sqrt{3}\cdot \sqrt{17}$) I know it might sound like a ridiculously easy question to answer, but I just can't put two and two together for some reason.
Say for example you have:
$$\sqrt{153}$$
You can break it down to
$$\sqrt{3}\cdot \sqrt{3}\cdot \sqrt{17... | Ignoring some (ahem) complexities, we can say, for non-negative(!) $x$,
$\sqrt{x}\;$ is the non-negative(!) number that multiplies by itself to give $\;x$.
With that in mind, consider what happens when we multiply $\sqrt{x}\cdot\sqrt{y}$ by itself:
$$\begin{align}
\left(\;\color{red}{\sqrt{x}}\cdot\color{blue}{\sqrt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Wolfram Alpha can't find an explicit form (or even an explicit form of an approximation of) this innocuous series? I suppose my question could be restated as "why?" to the above statement.
I'll rephrase my question if this breaks any rules asking it this way, but the expression is simply:
$$\sum_{m=2}^\infty \frac{1}{... | $$
\begin{align}
\sum_{m=2}^\infty\frac1{m^{2n}(m^2-1)}
&=\sum_{m=2}^\infty\sum_{k=n+1}^\infty m^{2k}\\
&=\sum_{k=n+1}^\infty\sum_{m=2}^\infty m^{2k}\\
&=\sum_{k=n+1}^\infty(\zeta(2k)-1)\\
&=\bbox[5px,border:2px solid #C0A000]{\frac34-\sum_{k=1}^n(\zeta(2k)-1)}\tag1
\end{align}
$$
since
$$
\begin{align}
\sum_{k=1}^\inf... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluate the value of series $$\sum^\infty_{k=0 }\frac{2^k}{5^{2^k}+1}$$
The hint is expanding it to double series.
So, $$\frac{2^k}{5^{2^k}+1}=\frac{2^k}{5^{2^k}}\frac{1}{1+5^{-2^k}}=\frac{2^k}{5^{2^k}}\sum_{n=0}^\infty(-1)^n5^{-n2^k}=\sum_{n=0}^\infty(-1)^n\frac{2^k}{5^{2^k(n+1)}}$$
Thus $$\sum^\infty_{k=0 }\frac{2^k... | Telescopic Solution
Notice that
$$
\frac{2^k}{5^{2^k}-1}-\frac{2^k}{5^{2^k}+1}=\frac{2^{k+1}}{5^{2^{k+1}}-1}
$$
Therefore, we get a telescoping series:
$$
\begin{align}
\sum_{k=0}^\infty\frac{2^k}{5^{2^k}+1}
&=\sum_{k=0}^\infty\left(\frac{2^k}{5^{2^k}-1}-\frac{2^{k+1}}{5^{2^{k+1}}-1}\right)\\[6pt]
&=\frac14
\end{align}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3281454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Division of 11 people into 3 groups with at least 2 people in each
How many ways can 11 people be divided into three teams where each
team must have at least two members?
We are supposed to use multinomial coefficients and number of integer solutions. I have tried this
The number of ways to divide 5 people into thr... | Looking at the number of persons in the groups we discern the following possibilities:
*
*for $(2,2,7)$ there are $\frac{11!}{2!2!7!}\cdot\frac12$ possibilities.
*for $(2,3,6)$ there are $\frac{11!}{2!3!6!}$ possibilities.
*for $(2,4,5)$ there are $\frac{11!}{2!4!5!}$ possibilities.
*for $(3,3,5)$ there are $\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3282948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solve $3x(2+\sqrt{9x^2 + 3}) + (4x-2)(\sqrt{x^2 - x +1}+1) = 0$ for $x\in \mathbb{R}$
Solve the equation in $ \mathbb{R}$: $$3x(2+\sqrt{9x^2 + 3}) + (4x-2)(\sqrt{x^2 - x +1}+1) = 0$$
I've been tried to solve this question for 3 hours, but can't find out any answers.
Just like I running in the maze, if I represent ... | It is a nicely set up problem:
$$(3x)(2+\sqrt{9x^2 + 3}) + (4x-2)(\sqrt{x^2 - x +1}+1) = 0 \iff \\
(6x)(1+\sqrt{\frac94x^2 + \frac34}) = (2-4x)(1+\sqrt{x^2 - x +1}) \\
\begin{cases}6x=2-4x\\ \frac94x^2 + \frac34=x^2-x+1\end{cases} \Rightarrow \begin{cases}x=\frac15\\ 5x^2+4x-1=0 \Rightarrow x=\frac15;-1\end{cases} \Rig... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Is it possible to improve on the bounds for $\varphi(N)/N$, if $N = q^k n^2$ is an odd perfect number with special prime $q$? Let $N = q^k n^2$ be an odd perfect number given in Eulerian form. That is, $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
From a comment underneath ... | Eureka!!!
Let $N = q^k n^2$ be an odd perfect number with special/Euler prime $q$.
From the equation and lower bound for $\varphi(N)/N$
$$\frac{120}{217\zeta(3)} < \frac{\varphi(N)}{N} = \frac{\varphi(q^k)}{q^k}\cdot\frac{\varphi(n)}{n}$$
and the equation
$$\frac{\varphi(q^k)}{q^k} = \frac{q - 1}{q},$$
we get the lower... | {
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"source": "stackexchange",
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What is the problem here (all integers are irrational proof...I think so)? Let us assume $a$ is an integer which is rational which implies $a=p/q$ (where $p$ and $q$ are integers and $q$ not equal to $0$). If $p$ and $q$ are not coprime, let us simplify the fraction so this it is (I don't know how to talk like mathemat... | The problem in the proof is that $a^2|b^2\nRightarrow a^2|b$. For instance, take $a=2$ and $b=6$. Clearly, $4|36$ but $4\nmid 6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3285940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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The integral $\int_{0}^{\infty} \frac{\cot^{-1}\sqrt{1+x^2}}{\sqrt{1+x^2}}dx =\frac{\pi}{2}\ln (1+\sqrt{2})$ At Mathematica the numerical value of the integral
$$\int_{0}^{\infty} \frac{\cot^{-1}\sqrt{1+x^2}}{\sqrt{1+x^2}} dx$$
equals 1.3844.., which is nothing but $\frac{\pi}{2}\ln (1+\sqrt{2})=z$. Also, one of its t... | Consider the following integral:
$$I(a)=\int_0^\infty \frac{\operatorname{arccot}(a\sqrt{1+x^2})}{\sqrt{1+x^2}}dx\Rightarrow I'(a)=-\int_0^\infty \frac{1}{1+a^2+a^2x^2}dx$$
$$=-\frac{1}{a\sqrt{1+a^2}}\arctan\left(\frac{ax}{\sqrt{1+a^2}}\right)\bigg|_0^\infty=-\frac{\pi}{2}\frac{1}{a\sqrt{1+a^2}} $$
Since $I(\infty)=0$ ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $x^{3} = 6+ 3xy - 3 ( \sqrt{2}+2 )^{{1}/{3}} , y^{3} = 9 + 3xy(\sqrt{2}+2)^{{1}/{3}} - 3(\sqrt{2}+2)^{{2}/{3}}$ Solve the system of equations for $x,y \in \mathbb{R}$
$x^{3} = 6+ 3xy - 3\left ( \sqrt{2}+2 \right )^{\frac{1}{3}} $
$ y^{3} = 9 + 3xy(\sqrt{2}+2)^{\frac{1}{3}} - 3(\sqrt{2}+2)^{\frac{2}{3}}$
I j... | Solving your first equation for $y$ we get
$$y=\frac{x^3+3 \sqrt[3]{2+\sqrt{2}}-6}{3 x}$$
Plugging this in the second equation we get
$$\frac{x^6}{27}-\frac{2}{3}
\left(1+\sqrt[3]{2+\sqrt{2}}\right)
x^3+\frac{-6+\sqrt{2}+12 \sqrt[3]{2+\sqrt{2}}-6
\left(2+\sqrt{2}\right)^{2/3}}{x^3}-2
\left(2+\sqrt{2}\right)... | {
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"url": "https://math.stackexchange.com/questions/3286730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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Splitting a population, probability of 2 people landing in same subpopulation.
$10$ people has been split into $3$ groups $A,B,C$ of $5,3,2$ people respectively. What is the probability that $2$ predetermined people $x,y$ land in the same group?
My attempt:
There is $\binom{10}{5, 3, 2} = \frac{10!}{5!3!2!}$ ways to ... | I agree with your answer. In particular, your term for $x,y \in A$ exceeds the whole book answer and your logic is sound.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove $x+y+z \ge xy+yz+zx$ Given $x,y,z \ge 0$ and $x+y+z=4-xyz$ Then Prove that
$$x+y+z \ge xy+yz+zx$$
My try:
Letting $x=1-a$, $y=1-b$ and $z=1-c$ we get
$$(1-a)+(1-b)+(1-c)+(1-a)(1-b)(1-c)=4$$
$$-(a+b+c)-(a+b+c)+ab+bc+ca-abc=0$$
$$ab+bc+ca-abc=2(a+b+c)$$
Where $a, b,c \le 1$
is there a clue here?
| Suppose $x+y+z < xy+yz+zx$, then from Schur's inequality we have
$$\begin{align}
\frac{9xyz}{x+y+z} &\geqslant 4(xy+yz+zx)- (x+y+z)^2 \\
&> (x+y+z)\left( 4- (x+y+z)\right) \\
&= (x+y+z) \cdot xyz \\
\end{align}$$
This gives $x+y+z< 3$, further we have from $4= x+y+z+xyz \geqslant 4\sqrt{xyz} \implies xyz \leqslant 1$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3292661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Finding constant $a$ so that differential equation becomes correct Question:
Find the value of $a$ so that the function
$$y = \sqrt{x} \ln{x}$$
is a solution to the differential equation
$$y' - \frac{a}{x} \cdot y = \frac{1}{\sqrt{x}}$$
Attempted solution:
My basic approach would be to take the derivative of y, put y a... | It is $$\frac{\ln(x)}{2\sqrt{x}}+\frac{\sqrt{x}}{x}-\frac{a}{x}\sqrt{x}\ln(x)=\frac{1}{\sqrt{x}}$$ multiplying by $\sqrt{x}$ we get
$$\frac{\ln(x)}{2}-a\ln(x)=0$$ so $$a=\frac{1}{2}$$ for $\ln(x)\ne 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How can I prove that starting from $0$, and repeatedly taking $+10,-10,\times10$ or $\div10$, requires at least $9$ operations to get to $2019$? Sometimes, I teach my sister math homework. Today, when she was doing her math problem solving summer homework, I teached her how to do the problems except the final problem t... | You can say that every button $\boxed{+10}$ or $\boxed{-10}$ has a different worth depending on how many buttons $\boxed{\times 10}$ or $\boxed{÷10}$ are pressed at any point after it. Each $\boxed{\times 10}$ increases the values of any previous $\boxed{+10}$ or $\boxed{-10}$ by a factor of $10$, and each $\boxed{÷10}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Show that if $a\sin x + b\cos x + ce^x$ is the zero function for constants $a, b, c$ then $a = b = c = 0$
Suppose that $a\sin x + b\cos x + ce^x$ is the zero function. Prove that $a=b=c=0$.
Does the zero function simply mean that $a\sin x + b\cos x + ce^x = 0$? I am going under the assumption that it is.
Geometrical... | Given
$a\sin x + b\cos x + ce^x = 0, \tag 1$
we multiply through by $e^{-x}$, which yields
$ae^{-x}\sin x + be^{-x}\cos x + c = 0; \tag 2$
letting $x \to \infty$, we find that
$c = 0; \tag 3$
we are left with
$a\sin x + b\cos x = 0; \tag 4$
now set
$x = \dfrac{\pi}{2}, \tag 5$
and obtain
$a = a\sin \dfrac{\pi}{2} + b\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Find all extrema of a complicated trigonometric function Problem
Find all local extrema for
$$f(x) = \frac{\sin{3x}}{1+\frac{1}{2}\cos{3x}}$$
Attempted solution
My basic approach is to take the derivative, set the derivative equal to zero and solve for x.
Taking the derivative with the quotient rule and a few cases of ... | It suffices to cancel the numerator of the derivative,
$$\cos(3x)(2+\cos(3x))+\sin(3x)\sin(3x)=2\cos(3x)+1=0$$
and
$$3x=2k\pi\pm\frac{2\pi}3.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Prove ${_4 F_3} \left(\frac12, \frac12, 1, 1; \frac34, \frac54, \frac32; \frac14 \right)= \frac14 \left(\frac{\pi^2}{4}+\log^2 (2+\sqrt{3} ) \right)$ I was experimenting with series and numerically found this gem:
$$S=\sum_{n=0}^\infty \frac{2^{2n+1}}{(2n+1)^2 \binom{4n+2}{2n+1}}= \frac14 \left(\frac{\pi^2}{4}+\log^2... | $$I=\int_0^\infty \operatorname{actanh}\left(\frac{2x}{(1+x)^2}\right)\frac{dx}{x}=\frac12 \int_0^\infty \frac{\ln(1+4x+x^2)-\ln(1+x^2)}{x}dx$$
Now we will consider the following integral:
$$I(a)=\frac12 \int_0^\infty \frac{\ln(1+ax+x^2)-\ln(1+x^2)}{x}dx\Rightarrow I'(a)=\frac12 \int_0^\infty \frac{dx}{1+ax+x^2}$$
$$=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Solve $\int_{-\infty}^{\infty} \text{sech}^2(x-a)\text{sech}^2(x+a)\ \mathrm{d}x$ I am wondering if anyone knows a trick on how to solve this integral $$f(a)=\int_{-\infty}^{\infty} \text{sech}^2(x-a)\text{sech}^2(x+a)dx.$$ The answer should be a function of $a$.
Basically I am trying to reproduce some results from a ... | Use the exponential definition, $\operatorname{sech}(x)=\frac{2}{e^{x}+e^{-x}}$, to rearrange the integrand.
$$\begin{aligned}\operatorname{sech}^2(x-a)\operatorname{sech}^2(x+a)&=\left(\frac{2}{e^{x-a}+e^{-(x-a)}}\right)^2\left(\frac{2}{e^{x+a}+e^{-(x+a)}}\right)^2
\\
&=\frac{16}{\left(e^{2x}+e^{-2x}+c\right)^2}
\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Determining whether a base is a Generator in $Z^*_{11}$ as well as differentiating Primitive Root and/or Primitive Element? Given $Z^*_{11}$, specifically looking at the bases $2$ and $3$:
\begin{array} {|r|r|}\hline 2^1 \pmod {11}=2 & 3^1 \pmod {11}=3 \\ \hline 2^2 \pmod {11}=4 & 3^2 \pmod {11}=9 \\ \hline 2^3 \pmod {... | $x$ is a primitive root (synonymously, a primitive element) mod $n$ if and only if
$x$ is a generator of the multiplicative group of integers modulo $n$.
Modulo $11$, $2$ is a primitive root and $3$ is not.
Modulo $7$, on the other hand, $3$ is a primitive root and $2$ is not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3300072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find all $x$ such that $\sin x = \frac{4}{5}$ and $\cos x = \frac{3}{5}$.
Let
$$
\left\{
\begin{array}{c}
\sin x = \frac{4}{5} \\
\cos x = \frac{3}{5}
\end{array}
\right.
$$
Find all of the possible values for $x$.
My try: By dividing the equations we obtain $\tan x = \frac{4}{3}$ and then $$x = \arctan\frac{4}... | Your original (set of) equations implies $\tan x=\frac{4}{3}$ but not the other way around. When you solve $\tan x=\frac{4}{3}$ you get the solutions of your original equation and the solutions $\sin x=\frac{-4}{5}$, $\cos x=\frac{-3}{5}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Does this function hit every odd number for some integers $n$ and $m$? I have recently been looking into a problem and created a function that looked interesting and I wondered if it would hit all odd numbers on a graph.
For integer $n$, define
$$g_n(x) = \frac13\cdot \begin{cases}
(3n-1) \cdot 2^{2x-1} - 1, & n \t... | It can be seen that $f_n = 3n+\frac{cos(\pi n)-3}{2}$. $\frac{cos(\pi n)-3}{2}$ will be equal to $-2$ if $n$ is odd and $-1$ if $n$ is even.
Thus, $f_n = 3n-(
n\pmod 2)-1$.
From that, we get
$$g_n(x) = \frac{(3n-(
n\pmod 2)-1)*2^{2x+1-(((3n-(
n\pmod 2)-1) \space \text{mod} \space 3))}-1}{3}$$
Simplifying this finds $$g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Alternative methods for solving a system of one linear one non linear simultaneous equations Take the equations $$x+y=5$$ $$x^2 + y^2 =13$$
The most basic method to solve this system is to first express the linear equation in terms of one of the variables and then sub that into the non-linear equation.
But I am curious... | The quadratic equation can be used.
Given:
x + y = 5, then y = 5 -4
Given
x^2 + y^2 = 13
then x^2 + (4-x)^2 = 13
and x^2 + x^2 - 10x + 25 -13 = 0
2x^2 + (-10x) + 12 = 0
Then the co-factors are a = 2, b = -10, c = 12
y = [-b (+-) sqrt(b^2 - 4ac)]/[2a] <-- Quadratic Formula
y = [-(-10) (+-) sqrt((-10)^2 - 4(2*12))]/(2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3302101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 10,
"answer_id": 2
} |
What is the coefficient of $x^3$ in expansion of $(x^2 - x + 2)^{10}$
What is the coefficient of $x^3$ in expansion of $(x^2 - x + 2)^{10}$.
I have tried:
$$\frac{10!}{(3!\times7!)} \times (-x + 2)^7 \times (x^2)^3 $$
But got an incorrect answer $-15360$.
| For a problem of this size, I would just ploddingly take a few derivatives:
$$\begin{align}
f(x)=(x^2-x+2)^{10}
&\implies f'(x)=10(x^2-x+2)^9(2x-1)\\
&\implies f''(x)=90(x^2-x+2)^8(2x-1)^2+20(x^2-x+2)^9\\
&\implies f'''(x)=720(x^2-x+2)^7(2x-1)^3+360(x^2-x+2)^8(2x-1)+180(x^2-x+2)^8(2x-1)\\
&\implies f'''(0)=-720\cdot2^7... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
How to show the matrix has Rank $\le 5$
I want to show that the following matrix has Rank $\le 5$.
The matrix is
\begin{bmatrix}
2&1&1&1&0&1&1&1\\
1&2&1&1&1&0&1&1\\
1&1&2&1&1&1&0&1\\
1&1&1&2&1&1&1&0\\
0&1&1&1&2&1&1&1\\
1&0&1&1&1&2&1&1\\
1&1&0&1&1&1&2&1\\
1&1&1&0&1&1&1&2
\end{bmatrix}
I found that there is a submatri... | We will show that the rank of the given matrix is exactly $5$.
The matrix can be written as a block matrix such that (see LU-decomposition),
$$M:=\begin{bmatrix} A & B \\B &A \end{bmatrix}=\begin{bmatrix} I & 0 \\BA^{-1}&I \end{bmatrix}\cdot \begin{bmatrix} A&0\\0&A-BA^{-1}B \end{bmatrix}\cdot\begin{bmatrix} I&A^{-1}B ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Find $n$ if $\frac{9^{n+1}+4^{n+1}}{9^n+4^n} = 6$ Find $n$ if $$\frac{9^{n+1}+4^{n+1}}{9^n+4^n} = 6$$
In this video they show a shortcut and say $n=-1/2$ without any explanation.
Key observation here is that the geometric mean of $9$ and $4$ is $6$.
It seems numerator and denominator are partial sums of geometric ser... | Another idea: $$\frac{9^{n+1}+4^{n+1}}{9^n+4^n}=\frac{5\cdot9^n+4\left(9^n+4^n\right)}{9^n+4^n}=5\frac{9^n}{9^n+4^n}+4\implies$$
$$\frac{9^{n+1}+4^{n+1}}{9^n+4^n}=6\iff5\frac{9^n}{9^n+4^n}=2\implies\frac1{\frac{9^n+4^n}{9^n}}=\frac25\implies\frac1{1+\left(\frac49\right)^n}=\frac25\implies1+\left(\frac49\right)^n=\frac5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 3
} |
Find all possible values of $\gcd(n^2+3, (n+1)^2+3)$. Let $n \in \mathbb{N}$. Find all possible values of $\gcd(n^2+3, (n+1)^2+3)$.
I began this problem giving some values for $n$ and I found that $\gcd(n^2+3, (n+1)^2+3)=1$ for most of $n$ I tried, but if $n=6$, then $\gcd=13$.
Then I tried to prove that only for $n=... | We assume the Lemma that if $\gcd(a,b)=1$ and $c$ is an integer, then $\gcd(a,bc)=\gcd(a,c).$
Also, that in general: $\gcd(a,b)=\gcd(a,b-ak)$ for any integer $k.$
Then $$\begin{align}
\gcd(n^2+3,(n+1)^2+3)&=\gcd(n^2+3,(n+1)^2+3-(n^2+3))\\
&=\gcd(n^2+3,2n+1)\\
&=\gcd(n^2+3-3(2n+1),2n+1)\\
&=\gcd(n(n-6),2n+1)\\
&=\gcd(n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3308652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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Evaluating $\int ^\infty _{0} \frac{x\ln x}{(1+x^2)^2} \,dx$ My first instinct was to evaluate the indefinite form of the integral, which I did by substituting $x=\tan t$, therefore yielding
\begin{align}
\int \frac{x\ln x}{(1+x^2)^2} \,dx
&= \int \frac{\tan t \sec^2 t \ln\tan t}{(1+\tan^2 t)^2} \,dt
&& \text{by substi... | Set $\dfrac1x=y$
$$\int ^\infty _{0} \frac{x\ln x}{(1+x^2)^2} \,dx=I=\int_\infty^0\dfrac{y^4\ln(y^{-1})}{y(1+y^2)^2}\left(\dfrac{dy}{-y^2}\right)=-\int_0^\infty\dfrac{y\ln y}{(1+y^2)^2}\ dy =-I$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3311259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 1
} |
Find the asymptote of the function $f(x) = \sqrt{\frac{x^3}{x - 3}} - x$ We have a function $f(x) = \sqrt{\frac{x^3}{x - 3}} - x$ and when $x$ goes towards $-\infty$, we have an asymptote
$y = -2x - 3/2$.
How we get this asymptote?
| Let
$$
f(x)
=\sqrt{\frac{x^3}{x-3}}-x
=|x|\sqrt{\frac{x}{x-3}}-x
$$
For $x<0$ we have $|x|=-x$, thus
$$
f(x)
=-x\sqrt{\frac{x}{x-3}}-x
=\Bigl(-\sqrt{\frac{x}{x-3}}-1\Bigr)x
=\Bigl(-\sqrt{\frac{1}{1-3/x}}-1\Bigr)x.
$$
For $x<0$;
$$\lim_{x\to-\infty}\frac{f(x)}{x}
=\lim_{x\to-\infty}\Bigl(-\sqrt{\frac{1}{1-3/x}}-1\Bigr)
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3312555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Interesting four-sum inequality $n^2 \left(\sum \limits_{k=1}^n \frac{k+1}{k}\right) \left( \sum \limits_{k=1}^n \frac{k}{k+1}\right) \ge...$
Prove that for all $n \in \mathbb{N}$ the inequality $$ \left(\sum \limits_{k=1}^n (2k-1)\frac{k+1}{k}\right) \left( \sum \limits_{k=1}^n (2k-1)\frac{k}{k+1}\right) \le n^2 \lef... | Here's a probabilistic interpretation and solution to this problem, although the method itself may be negligibly faster than the existing answers. Let $X$ be a discrete random uniform random variable on $\{1, \cdots, n\}$. Then the problem can be rewritten as
\begin{align*}
\mathbb{E}\left[(2X-1)\frac{X+1}{X}\right]\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3312678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
} |
Rules of inequality and modulus inequality. I am very confused whenever I encounter an inequality .
Like suppose
$ \frac{| \sqrt {1 -4x^{2}}|}{|x|} \le 1 $
Now I am confused whether to square or not .
I am very well aware of the rule at we do not square if the sign of $x$ is not known . But what if the range of $x$ ... |
Note: All of what follows is based on the axiom that if $a < b$ (regardless of the signs of either) and $c > 0$ then $ac < bc$ and likewise if $d < 0$ then $ad > bd$.
I'm assuming that is known and familiar.
You can do whatever you want. You can take them out to dinner for all I care. You just have to know what y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3313361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Indices - factorising Simplify: $$\frac {x^5y^2x^3 + x^4y^5 - y^5x^7y^4}{x^4y^3}$$
I know this is probably low level stuff but I need to be able to do this specific type of question and I have no way of checking my work. If anyone could offer a step by step working I'd be appreciative
| An alternate way is to break the problem down into simple workable parts:
$$\frac{x^5y^2x^3+x^4y^5−y^5x^7y^4}{x^4y^3}$$
$$=\frac{x^5y^2x^3}{x^4y^3}+\frac{x^4y^5}{x^4y^3}-\frac{y^5x^7y^4}{x^4y^3}$$
$$=\frac{x^8y^2}{x^4y^3}+\frac{x^4y^5}{x^4y^3}-\frac{x^7y^9}{x^4y^3}$$
$$=\frac{x^4}{y}+y^2-x^3y^6,\quad x\not=0.$$
This is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3316509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How do I evaluate $\lim _{x\to 0}\left(\frac{x-\sin x}{x\sin x}\right)$ without using L'Hopital or series?
How do I evaluate $\lim _{x\to 0}\left(\frac{x-\sin x}{x\sin x}\right)$ without using L'Hopital or series?
I've tried expanding the variable such as $x = 2y$ or $x = 3y$, but seemed to still get stuck.
| First, read this answer. This answer shows that, for $x$ close to $0$, we have the following:
$$\sin x \leq x \leq \tan x$$
Thus, since $\sin x \leq x$ and $\sin x$ has the same sign as $x$ (i.e. either both are positive, both are negative, or both are $0$), we know that $\frac{1}{\sin x} \geq \frac{1}{x}$. Thus, $\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Getting different answers for an integral: $\frac{1}{2}x-\frac{3}{2}\ln{|x+2|}+C$ vs $\frac{1}{2}x-\frac{3}{2}\ln{|2x+4|}+C$ Problem:
$$\int\frac{1}{2}-\frac{3}{2x+4}dx$$
Using two different methods I am getting two different answers and have trouble finding why.
Method 1:
$$\int\frac{1}{2}-\frac{3}{2x+4}dx$$
$$\int\... | $$\begin{align}\frac12 x + \frac 32 \ln |2x + 4| + C_1 &= \frac12x + \frac32\ln(2\cdot|x+2|) + C_1\\&=\frac12x + \frac32(\ln 2 + \ln|x+2|)+C_1\\&=\frac12x + \frac32 \ln|x+2| + (\frac 32 \ln 2 + C_1)\\&=\frac12x + \frac32\ln|x+2| + C_2\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Rate of change with $f(x)=4x^2-7$ on $[1,b]$ As part of a textbook exercise I am to find the rate of change of $f(x)=4x^2-7$ on inputs $[1,b]$.
The solution provided is $4(b+1)$ and I am unable to arrive at this solution.
Tried:
$f(x_2)=4b^2-7$
$f(x_1)=4(1^2)-7=4-7=-3$
If the rate of change is $\frac{f(x_2)-f(x_1)}{x_2... | A small sign-mistake! See the highlighted parts in red and blue:
Tried:
$f(x_2)=4b^2-7$
$\color{blue}{f(x_1)}=4(1^2)-7=4-7=\color{blue}{-3}$
If the rate of change is $\frac{f(x_2)-f(x_1)}{x_2-x_1}$ then: $\frac{(4b^2-7)\color{red}{-3}}{b-1}$ = $\frac{4b^2-10}{b-1}$
Which should be:
$$\frac{f(x_2)\color{red}{-}\color{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3322138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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How can we not use Muirhead's Inequality for proving the following inequality? There was a question in the problem set in my math team training homework:
Show that $∀a, b, c ∈ \mathbb{R}_{≥0}$ s.t. $a + b + c = 1, 7(ab + bc + ca) ≤ 2 + 9abc.$
I used Muirhead's inequality to do the question (you can try out yourself):... | Yes, your proof is correct. I have a similar proof where, for the last step, we need just the AM-GM inequality.
So it suffice to show that
$$2(a+b+c)^3 + 9abc \geq 7(a+b+c)(ab+bc+ca)$$
that is
$$2(a^3+b^3+c^3) \geq (a^2b +b^2c+c^2a)+ (ab^2 +bc^2+ca^2).$$
Now the inequality
$$a^3 +b^3 +c^3 \ge a^2b +b^2c+c^2a$$
follow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3326605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Solving the integral $\int\frac{1}{\sqrt{x^2+1}}\,dx$ I need to solve
$$\int\frac{1}{\sqrt{x^2+1}}$$
but I'm nowhere near the solution. I tried substituting
$u = x^2 + 1$
such that $dx = \frac{1}{2x}$
yielding
$$\int u^{-1/2} du = \frac{1}{2x}\frac{u^{1/2}}{1/2}=\frac{\sqrt{u}}{x} =\frac{\sqrt{x^2+1}}{x}$$
but this i... | Hint: Substituting $$x=\tan(t)$$ then we get $$x^2+1=\tan^2(t)+1=\frac{\sin^2(t)+\cos^2(t)}{\cos^2(t)}=\frac{1}{\cos^2(t)}$$
and $$dx=(\tan^2(t)+1)dt$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3326836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Finding the integral $\int \frac{dx}{(1-x)^2\sqrt{1-x^2}}$ One may take $x= \cos t$ and get
$$I=\int \frac{dx}{(1-x)^2\sqrt{1-x^2}}= -\frac{1}{4}\int \csc^4(t/2)~ dt=-\frac{1}{4} \int [\csc^2(t/2) +\csc^2(t/2) \cot^2(t/2)]~ dt.$$
$$\Rightarrow I=\frac{1}{2} \left [\cot (t/2)] +\frac{1}{3}\cot^3(t/2)\right]=\frac{(2-x)}... | Let us first find evaluate the indefinite integral $ \displaystyle \int \dfrac1{ (1-x)\sqrt{1-x^2} } \, dx $.
$$\begin{array} {r c l} \displaystyle \int \dfrac1{ (1-x)\sqrt{1-x^2} } \, dx &=& \displaystyle \int \dfrac{1+x}{ (1-x^2)\sqrt{1-x^2} } \, dx \\ &=& \displaystyle \int \dfrac{1}{ (1-x^2)\sqrt{1-x^2} } \, dx ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3327801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Evaluate $\int_{0}^{2\pi} \frac{cos \theta}{2 + cos \theta} d\theta$ using the residue theorem My attempt to a solution for $I = \int_{0}^{2\pi} \frac{cos \theta}{2 + cos \theta} d\theta$ is as follows.
On the unit circle we have $z=e^{i\theta} \implies dz = izd\theta \iff d\theta = \frac{dz}{iz}$, and furthermore $co... | Your calculation for the residue at $z=-2+\sqrt3$ is wrong. In fact
$$ \text{Res}f(z)\bigg|_{z=-2 + \sqrt3}=\frac{z^2+1}{z(z+2+\sqrt3)}\bigg|_{z=-2 + \sqrt3}=-\frac{2}{\sqrt3}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3327944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1$
Show that $$\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1 .$$
I know that $\arctan 1 = \frac{\pi}{4}$ and that the sequence ... | Consider the function
$$\frac{x^{10}}{x^2+1} \equiv 1-x^2+x^4-x^6+x^8-\frac{1}{x^2+1}$$
The integral:
$$J = \int_0^1 \frac{x^{10}}{x^2+1} \text{d}x = \int_0^1 1-x^2+x^4-x^6+x^8-\frac{1}{x^2+1} \text{d}x$$
is clearly positive.
Since $1\leqslant x^2+1 \leqslant 2$, we have:
$$\frac{1}{2} \int_0^1 x^{10} \text{d}x < J < ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3328422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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How to draw graph of $|x+y-1| + |2x + y + 1|=1$?
How to draw graph of $|x+y-1| + |2x + y + 1|=1$ ?
My attempt
I can operate modulus in equations like $|x-1| + |2x + 1|=1$ but I don't know how to proceed in this question since it contains $2$ variable, $x$ and $y$.
| If you know already a bit of linear algebra you may proceed as follows:
You surely can draw $|u|+|v|=1$.
Now, consider the mapping
$$\begin{pmatrix}u \\ v\end{pmatrix} = \begin{pmatrix}1 & 1 \\ 2 & 1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}+\begin{pmatrix}-1 \\ 1\end{pmatrix}$$
The inverse mapping (just solve for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3329862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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$n$th derivative of $e^{ax}\sin(bx+c)$
*
*How can we substitute $r \cos \alpha$ and $r \sin \alpha$ for $a$ and $b$?
*How, on successive differentiation, is there another $r$ multiplied?
| First, note that the title in the excerpt is incorrect. It should be "$n$th derivative of $e^{ax}\sin(bx+c)$".
(1) Here we introducing new quantities $r$ and $\alpha$, so we may define them however we want. Essentially, this amounts to writing the pair $(a, b)$ in polar coordinates, as $(r, \alpha)_{\textrm{polar}}$.
M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Find value for $k$ such that $(x^2-k)$ is a factor for $f(x)=2x^4+(3k-4)x^3+(2k^2-5k-5)x^2+(2k^3-2k^2-3k-6)x+6$ Find value for $k$ such that $(x^2-k)$ is a factor for, $$f(x)=2x^4+(3k-4)x^3+(2k^2-5k-5)x^2+(2k^3-2k^2-3k-6)x+6$$
My Try
Since $x^2-k=0$ when we substitute $x=\pm k$ to $f(x)$ it should be equal to $0.$
But ... | Factor out $x^2-k$:
$$f(x)=2x^4+(3k-4)x^3+(2k^2-5k-5)x^2+(2k^3-2k^2-3k-6)x+6=\\
[2x^4-2x^2k+2x^2k-2k^2+\color{red}{2k^2}]+\\
[(3k-4)x^3-(3k-4)xk+\color{red}{(3k-4)xk}]+\\
[(2k^2-5k-5)x^2-(2k^2-5k-5)k+\color{red}{(2k^2-5k-5)k}]+\\
\color{red}{(2k^3-2k^2-3k-6)x+6}\Rightarrow \\
\begin{cases}2k^3-2k^2-3k-6+(3k-4)k=0\\
2k^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3332730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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How to equalize correctly? If i have this number:
$2 \sqrt{2-\sqrt{3}}$ and i want to find some $x,y$ nonzero real numbers such that $2\sqrt{2-\sqrt{3}} = \sqrt{x} + \sqrt{y}$
And for that, i do this:
$(2 \sqrt{2-\sqrt{3}})^2 = x + 2\sqrt{xy} + y$
$4(2-\sqrt{3})=(x+y)+2\sqrt{xy}$
$(8)+(-4\sqrt{3})=(x+y)+(2\sqrt{xy})$
... | I can write $2\sqrt{2-\sqrt{3}} = \sqrt{y}+\sqrt{x}$ as: $2\sqrt{(\frac{\sqrt3}{\sqrt2}-\frac{\sqrt2}{2})^2} = \sqrt{y}+\sqrt{x}=2(\frac{\sqrt3}{\sqrt2}-\frac{\sqrt2}{2})$ or $2\sqrt{(\frac{1}{\sqrt2}-\frac{\sqrt6}{2})^2} = \sqrt{y}+\sqrt{x}=2(\frac{1}{\sqrt2}-\frac{\sqrt6}{2})$. This can be easily obtained by the syst... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3334982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Proving that complex numbers form a regular pentagon Let there be five nonzero complex numbers having the same absolute value and such that zero is equal to their sum, which is equal to the sum of their squares. Prove that the affixes of these numbers in the complex plane form a regular pentagon.
I am familiar with the... | Let $|a| = R$. Then $\frac{R^2}{a} = \overline a$, and thus
$$ 0 = \overline a + \overline b + \overline c + \overline d + \overline e = \frac{R^2}{a} + \frac{R^2}{b} + \frac{R^2}{c} + \frac{R^2}{d} + \frac{R^2}{e} = \frac{R^2}{abcde}\left(bcde + acde + abde + abce + abcd\right).$$
Thus, $\sum_{sym}abcd = 0$. Similarly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3335956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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For which $a$, $b$, $c$ does this linear system have exactly one solution? $x+ay+a^2z=0$, $x+by+b^2z=0$, $x+cy+c^2z=0$
For which $a$, $b$, $c$ does this linear system have exactly one solution?
$$x + ay + a^2z = 0$$
$$x + by + b^2z = 0$$
$$x + cy + c^2z = 0$$
I started this problem by recognizing that if the RREF of ... | The other solution uses determinants which is going to be the standard route to solving this problem. You mentioned RREF which I don't think is impossible, but you should be careful with putting it in that form:
$$\begin{pmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{pmatrix}$$
subtracting row 1 from row 2 and row 1 from row 3:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3336600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate $\tan 195^{\circ}$ without using the calculator How to evaluate $\tan 195^{\circ}$ without using the calculator, and how to give the answer in the form $a+b \sqrt{3}$, where $a$ and $b$ are integers?
| Using the identity $\tan \left(180^\circ+\theta\right)=\tan\theta$ and $\tan \left(\alpha-\beta\right)=\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$,
$\quad\tan195^\circ \\=\tan 15^\circ\\=\tan \left(60^\circ-45^\circ\right)\\=\dfrac{\tan60^\circ-\tan45^\circ}{1+\tan60^\circ\tan45^\circ}\\=\dfrac{\sqrt{3}-1}{\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3337434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there a way to calculate the enclosed volume of two unit balls centered at $(0,0,0)$ and $(1,1,1)$ using a triple integral?
Let $\Omega \subset \mathbb{R}^3$ be the subset $$\Omega = B((0,0,0),1) \cap B((1,1,1),1)$$ where $B(x,r)$ stands for the ball centered in $x$ with radius $r$.
I was looking for an easy way to... | Due to the perfect symmetry of the two-ball configuration around the axis (1,1,1), these three integrals can be carried out relatively straightforwardly.
Such symmetry is obvious for the simple volume integral,
$$I_1=\int_{\Omega} dV$$
For the other two integrals, we observe that
$$\int_{\Omega} xdV = \int_{\Omega} ydV... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3338959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Density of $ Y = X + \frac{1}{X}$ when $X\sim U(a,b)$ Let $ X $ be a continuous random variable uniformly distributed in $ \left[a, b\right] $, that is $X \sim U(a, b)$. Suppose $ 0 < a < b $.
We wish to find the density function of $$ Y = X + \frac{1}{X} $$ Therefore, we want $ f_Y(y) $.
It is obvious that for $$ y\in... | A possible path
Let $F_X(x)$ and $f_X(x)$ be the c.d.f and p.d.f. of $X\sim\mathcal U(a,b)$. Let also
$$g(x) = \frac1x + x,$$
$$\alpha(y) = \frac{y-\sqrt{y^2-4}}{2},$$
$$\beta(y) = \frac{y+\sqrt{y^2-4}}{2}.$$
Case 1 ($a\geq 1$)
$g(x)$ in monotonically increasing in $[a,b]$. Thus
\begin{eqnarray}
P(Y<y) &=& \begin{cases... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3340767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Knowing that $m$ and $n$ are positive integers satisfying $mn \mid m^2 + n^2 + m$, prove that $m$ is a square number.
Knowing that $m$ and $n$ are positive integers satisfying $$\large mn \mid m^2 + n^2 + m$$, prove that $m$ is a square number.
We have that $mn \mid m^2 + n^2 + m \implies mn \mid (m^2 + n^2 + m)(n + ... | Note $\ kmn = m^2+m+n^2\,\Rightarrow\, x^2+(1\!-\!kn)\,x + n^2\,$ has roots $\,x = m,m'\in \Bbb Z\,$ satisfying $ \underbrace{mm' = \color{#0a0}{n^2}}_{\textstyle\small p\mid m\,\Rightarrow\, p\mid\color{#c00} n}\!$ and $\: \underbrace{m+m' = kn-1}_{\textstyle\small p\mid m,m', \color{#c00}n\,\Rightarrow\,p\mid 1 }\,$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3342214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $ a + b + c = 90^{\circ}$, prove $ \tan(a) \cdot \tan(b) + \tan(b) \cdot \tan(c) + \tan(c) \cdot \tan(a) = 1 $ If $ a + b + c = 90^{\circ}$, prove $ \tan(a) \cdot \tan(b) + \tan(b) \cdot \tan(c) + \tan(c) \cdot \tan(a) = 1 $
Attempt:
Notice that $$ \tan(a+b+c) = \frac{\tan(a+b) + \tan(c)}{1 - \tan(a+b)\tan(c)} $$... | Since $\tan(a+b+c)$ is undefined, anything where you put it in an equation is suspect. But if you start with $\cot(a+b+c)$ then you can get
$1-\tan(a)\tan(b) - \tan(a) \tan(c) -\tan(b) \tan(c)$
in the numerator and show that it must be zero.
I don't see exactly how you intended to finish the proof that starts with $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3342989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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Volume of a solid created by an extended tetrahedron
Every edge of a tetrahedron with length $p$ is extended through the vertices by $p$.
Now all 12 points create a new solid $J$ of which I seek the volume dependent on the volume of the tetrahedron in the centre.
With some help the solution becomes clear:
The whole ... | Here is an answer - complementing your 3d object to a cube with side - $3p/2^{0.5}$
Subtract 4 right angle pyramids with diagonal $p$ and 4 right angle pyramids with diagonal $2p$. The pyramids sides are, $p/2^{0.5}$ and $2^{0.5}p$.
The volume of the cube is given by:$(3p/2^{0.5})^3$
The volume of the pyramids - $4X(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3344522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Matrix diagonalization. Is $A = PDP^{-1} = P^{-1}DP$? Diagonalization of a square matrix $A$ consists in finding matrices $P$ and $D$ such that $A=PD P^{-1}$ where $P$ is a matrix composed of the eigenvectors of $A$, $D$ is the diagonal matrix constructed from the corresponding eigenvalues, and $P^{-1}$ is the matrix i... | No,
$$\begin{pmatrix}1&2\\1&0\end{pmatrix}
\begin{pmatrix}2&0\\0&1\end{pmatrix}
\begin{pmatrix}0&1\\\frac12&-\frac12\end{pmatrix}
=\begin{pmatrix}1&1\\0&2\end{pmatrix}$$
while
$$\begin{pmatrix}0&1\\\frac12&-\frac12\end{pmatrix}
\begin{pmatrix}2&0\\0&1\end{pmatrix}
\begin{pmatrix}1&2\\1&0\end{pmatrix}=\begin{pmatrix}1&0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3346006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Find $p,q$ such that $p+q-pq+1=0$ where $p,q$ are primes with $p
Find $p,q$ such that $p+q-pq+1=0$ where $p,q$ are primes with $p<q$.
My try:
If we take $p=2,q=3$ the equality is satisfied.
But how can I show that this is the only solution?
| $p + q-pq +1=0$ means $pq - q=p+1$ so $q=\frac {p+1}{p-1}$.
And $pq-p = q+1$ so $p =\frac {q+1}{p-1}$.
To solve $\frac {x+1}{x-1}\in \mathbb Z$, we can get $\gcd(x+1,x-1)= \gcd(x+1, (x+1)-(x-1)) = \gcd (x,2) = 1,2$ so $x-1=1,2$ (assuming $x-1 >0$) and $x+1=3,4$ and $x = 2,3$.
Alternatively we could have assigned $m = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3348395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
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Having trouble solving $\int x(2x+5)^8dx$. I am not sure how to solve $\int x(2x+5)^8dx$.
I have tried some different things, but nothing seems to give me the cancelling effect I need to solve the integral.
My Process:
Let $u=2x+5$
Then $du = 2dx$ and $dx = \frac{1}{2}du$
so $\int x u^8 \frac{1}{2} \;du = \frac{1}{2}\i... | Let $\displaystyle I=\int (2x+5)(2x+5)^8\,dx=\int (2x+5)^9\,dx=\frac{1}{20}(2x+5)^{10}$
And let $J=\displaystyle\int(2x+5)^8
\,dx=\frac{1}{18}(2x+5)^9$
I exclude constants until the end.
Then, $\displaystyle \int x(2x+5)^8\,dx=\frac{1}{2}(I-5J)+C=\frac{1}{40}(2x+5)^{10}-\frac{5}{36}(2x+5)^9+C$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3348933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Cubic diophantine equation with a prime $x^3 + y^3 + z^3 - 3xyz = p$. Question: Find all triple positive integers $(x, y, z)$ so that
$$x^3 + y^3 + z^3 - 3xyz = p,$$
where $p$ is a prime number greater than $3$.
I have tried the following: The equation factors as
$$(x + y + z) (x^2 + y^2 + z^2-xy-yz-zx) = p.$$
Since ... | Result: If $p>3$ is a prime number and $x$, $y$ and $z$ are positive integers such that
$$x^3 + y^3 + z^3 - 3xyz = p,$$
then if $p\equiv1\pmod{3}$ we have, after permuting $x$, $y$ and $z$, that
$$(x,y,z)=\left(\tfrac{p-1}{3},\tfrac{p-1}{3},\tfrac{p+2}{3}\right),$$
and if $p\equiv2\pmod{3}$ we have, after permuting $x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3351039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Are there integer solutions such that the sum of the roots of x and y equal the root of a prime p? How could one prove whether or not integer pairs (x, y) exist such that $\sqrt{x} + \sqrt{y} = \sqrt{p}$, where p is prime? `
| Using elementary methods.
We'd have
$$x + 2\sqrt{xy} + y = p$$
and hence
\begin{align}
x + y - p = -2\sqrt{xy}
&\implies
x^2+y^2+p^2+2xy-2xp-2yp = 4xy
\\&\iff
(x^2-2xy+y^2) + p^2= 2p(x+y)
\\&\iff
(x-y)^2 +p^2 = 2p(x+y)\tag{1}
\\&\iff
(x-y)^2 +2p(x-y) +p^2 = 2p(x+y) + 2p(x-y)\tag{2}
\\&\iff
(x-y+p)^2 = 4px.
\end{align}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3353686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Expression of $x^n+\frac1{x^n}$ by $x+\frac1{x}$ where $n$ is a positive odd number. There was a problem in a book:
Denote that $y=x+\dfrac{1}{x}$, express $x^7+\dfrac{1}{x^7}$ using $y$.
It's not a hard question, but I find a special sequence:
$x+\dfrac{1}{x}=y\\x^3+\dfrac{1}{x^3}=\left(x+\dfrac{1}{x}\right)^3-3\lef... | Let $s_1:=x+\dfrac1x$. Then
$$x=\frac{s_1\pm\sqrt{s_1^2-4}}2$$ and
$$s_n=\left(\frac{s_1+\sqrt{s_1^2-4}}2\right)^n+\left(\frac{s_1-\sqrt{s_1^2-4}}2\right)^n.$$
Then by the Binomial development, after cancellation of the odd terms
$$s_n=\frac1{2^{n-1}}\sum_{2k=0}^n\binom n{2k} s_1^{n-2k}\left(s_1^2-4\right)^{k}.$$
Then,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3353877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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Formulas for Sequences Removing Multiples of 2, 3, and 5 First off, I am a programmer so please excuse if some of the terms I use are not the correct mathematical terms. I was working on devising a function to improve one of my prime number generation algorithms. With this in mind, I first set out to find the formulas ... | If this is for a programming task, you don't want a fancy formula. That will only slow you down. Instead, for the first table, just cycle through the values $6n+1,6n+5$ for $n=0,1,2,\ldots$ And for the second table, cycle through the values $30n+1,30n+7,30n+11,$ etc. (there are eight of them).
Updated to add:
Perhaps t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3355398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Find the determinant whose result is $(x-n)^{n+1}$
Find the determinant
$$
\left|\begin{array}{cccccc}{x} & {1} & {} & {} & {} & {} \\ {-n} & {x-2} & {2} & {} & {} & {} \\ {} & {-(n-1)} & {x-4} & {\ddots} & {} & {} \\ {} & {} & {\ddots} & {\ddots} & {n-1} & {} \\ {} & {} & {} & {-2} & {x-2 n+2} & {n} \\ {} & {} & {}... | Here's a possible method:
From up to down, add the row above to each row:
$$
D_n=\left|\begin{array}{cccccc}{x} & {1} & {} & {} & {} & {} \\ {x-n} & {x-1} & {2} & {} & {} & {} \\ {x-n} & {x-n} & {x-2} & {\ddots} & {} & {} \\ {} & {} & {\ddots} & {\ddots} & {n-1} & {} \\ {} & {} & {} & {x-n} & {x-(n-1)} & {n} \\ {x-n} &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3355740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
If $a$, $b$, $c$ are three positive integers such that $a^3+b^3=c^3$ then one of the integer is divisible by $7$ Let on contrary that none of the $a$, $b$, $c$ is divisible by $7$. Then either $a^3\equiv b^3\pmod{7}$ or $b^3\equiv c^3\pmod{7}$ or $c^3\equiv a^3\pmod{7}$.
Now how to go further?
| There's 36 combinations of $a$ and $b$ in the integers mod 7, where they're not already $0$. Just try checking them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3356927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
A box contains 13 four-sided dice, 36 six-sided dice, and 8 eight-sided dice. A box contains 13 four-sided dice, 36 six-sided dice, and 8 eight-sided dice. One die is chosen at random and then rolled. The probability that the die rolled was a six-sided die, given that the outcome was 5 is / where and are relatively p... | There are 57 dice. If one is chosen at random (with uniform probability) then the probability of choosing a six-sided die and rolling a five is:
$\frac{36}{57} \frac{1}{6} = \frac{6}{57}$.
The probability of rolling a 5 in general is:
$\frac{13}{57} \frac{0}{4} + \frac{36}{57} \frac{1}{6} + \frac{8}{57} \frac{1}{8} = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3361656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the probability of being dealt a hand where all $5$ cards are different denominations? In a game of poker you are dealt $5$ cards at random from a standard deck of $52$. A standard deck of $52$ contains $13$ different denominations of cards (Ace, 2, 3, ..., 10, jack, queen, king), each in four different suits
(... |
What is the probability of being dealt a hand where all five cards are of different denominations?
There are $\binom{52}{5}$ ways to select a five-card hand.
There are $\binom{13}{5}$ ways to select five different denominations. For each such denomination, there are $\binom{4}{1}$ ways to choose the suit of the card... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3367555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
If $n$ numbers are generated, what is the probability that the product of all those numbers is a multiple of 10? A computer generates random numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ (each has equal probability). If $n$ numbers are generated (with replacement), what is the probability that the product of all those n... | To begin, I am making the assumption (that most people would do) that each of the digits is selected in sequence uniformly and independently at random. This makes each of the $9^n$ possible sequences of choices equally likely to occur. This is in direct contrast to only considering the number of each digit selected w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3368272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Factorizing $x^5+1$ as a product of linear and quadratic polynomials. I am encountering some trouble with this question:
Factorize $$x^5+1$$ as a product of real linear and quadratic polynomials.
I know that if we subtract 1 from $x^5+1$, we get that $x^5 = -1$, but I am unsure where to go from here.
Can anyone help wi... | First you seem to know that $(x+1)|(x^5+1)$. Carry out this division with polynomial long division to get
$x^5+1=(x+1)(x^4-x^3+x^2-x+1)$
The remaining fourth degree polynomial is symmetric or palindromic, its coefficients $1,-1,1,-1,1$ read the same forward and backwards. When this happens there will be quadratic fac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3371200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 1
} |
Given $0 < x \le 1 \le y$. Calculate the minimum value of $\frac{x}{y + 1} + \frac{y}{x + 1} + \frac{x^2 + 3y^2 + 2}{6xy(xy + 1)}$.
Given $0 < x \le 1 \le y$. Calculate the minimum value of $$\large \dfrac{x}{y + 1} + \dfrac{y}{x + 1} + \dfrac{x^2 + 3y^2 + 2}{6xy \cdot (xy + 1)}$$
We have that $$\frac{x^2 + 3y^2 + 2}... | For $x=y=1$ we'll get a value $\frac{3}{2}$.
We'll prove that it's a minimal value.
Indeed, let $y=1+a$.
Thus, $a\geq0$ and we need to prove that:
$$6x^2a^4+(3+9x+21x^2-9x^3)a^3+3(4+9x+3x^2-10x^3+2x^4)a^2+$$
$$+(1-x)(17+37x+14x^2-12x^3)a+2(x+1)^2(1-x)(5-3x)\geq0,$$ which is obvious.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3375653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.