Use this tag when you want to determine the thinking that is needed to solve a certain type of problem, as opposed to looking for a specific answer to a question.

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2
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3answers
79 views

Finding roots of a quartic

How do I find the roots of the equation $$(x+3)^5-(x+1)^5=7$$ I tried opening it up, it turns into a ugly quartic which doesn't factor. I don't know what to do next. Please help me out.
2
votes
1answer
132 views

Fundamental Matrix

Determine $\phi(x,0)$ for $A(x)=\begin{pmatrix} -1 & \cos(x) \\ 0 & -1\end{pmatrix}$, where $\phi(x,0)t_{0}$ is a solution of $\frac{d}{dx}t(x)=A(x)t(x)$. I am not entirely sure as to ...
0
votes
1answer
26 views

Set of vectors to span.

Find a set of vectors that span the subspace $W$ of $V$: 1) $V = P^3(\mathbb{R})$ (polynomial degree 3) $W = \{p|p(1) = p(3) = 0\}$ 2) $V = \text{span}(\{\sin x, \cos x, \sin 2x, \cos 2x\})$ ...
2
votes
1answer
53 views

Possible values of $\gcd(a+b, a\times b)$

Main Question: Let $N \in \mathbb{N}$. What are the possible values of $\gcd(a+b, a\times b)$ given that $\gcd(a,b) = N$? Fact 0. If $\gcd(a,b) = N$, then $N \leq \gcd(a+b, a\times b) \leq ...
0
votes
1answer
30 views

Simple probabilistic expression

For the following expression: $$ \prod_{i=0}^{n-1} \frac{2n-i}{3n-i} $$ I'm trying to get a simple expression, unsuccessfully. Many thanks, Jonathan
0
votes
0answers
20 views

Efficient method to calculate passes (rises and sets) for satellites

There is a function describing the characterisic elevation of ISS seen from an observers horizon. Calculating of an elevation at one time is pretty expensive. So I wanna try to avoid naive iterating ...
0
votes
2answers
21 views

Number of unique Team parings given 10 players and 2 teams

I yammer a wee bit too much, feel free to skip to TLDR unless you want more background as to why I care about this problem. I was just thinking that it would be a fun to figure out the best 5 players ...
1
vote
1answer
82 views

Is there a solution to the equation x^x^x^x^x^x^… = 2?

I have been asked the following brainteaser, is there a solution to the equation: $$ x^{x^{x^{...}}} = 2$$ (x to the power of itself an infinite number of times) I am not sure about how to approach ...
2
votes
2answers
67 views

How to solve trigonometric equations with a domain involving negative values of $x$?

I don't seem to understand the concept of a negative domain when solving trigonometric equations on "another interval" For example: Solve $\cos x=-\sqrt{3}/2$ given that the domain is $-\pi \le ...
3
votes
1answer
23 views

$y = ln(p+qe^x)/x$, solve $x$

$y = \ln(p+qe^x)/x$ $p$ and $q$ are constants. Express $x$ in terms of $y$. I believe I have to use Lambert W function, but I'm stumped. Thinking help is needed. Thank you very much!
4
votes
5answers
71 views

If $9\sin\theta+40\cos\theta=41$ then prove that $41\cos\theta=40$.

I tried it this way: $$ 40\cosθ+9\sinθ=41 $$ $$ 9\sinθ=41-40\cos\theta $$ Squaring both the sides: $$81\sin^2\theta=1681+1600\cos^2\theta-2\cdot 40\cdot 41 \cos\theta$$ $$81-81 ...
0
votes
1answer
18 views

Mgf of double exponential RV

In class the other day we were talking about a double exponential RV $X$ with a pdf $f(x)=\frac{1}{2}e^{-|x|}$ for $-\infty<x<\infty$. The professor noted that the mgf was $M(t)=\frac{1}{1-t^2}$ ...
4
votes
1answer
48 views

Normal distribution - how to solve P(-b<X<b)=0.95

$X\sim N(2,3^2)$ How do you find $b$ where $P(-b<X<b)=0.95$ other than trial and error? You can't directly transform to $z$ because if you find an appropriate $z$, transforming back will give ...
0
votes
0answers
35 views

What's the solution set $S \subset \mathbb{R}^2$ of this equation?

I see that $(1,1)$, $(2,4)$ and $(4,2)$ are in $$S= \{(x,y) \in \mathbb{R}^2: \, x^y = y^x\}$$ My question is: The set $S$ contains many others elements? Thanks for any suggestions and helpful ...
22
votes
9answers
430 views

Examples where it is easier to prove more than less

Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ (e.g. since the induction hypothesis gives you more ...
1
vote
2answers
75 views

How to solve a algebraic equation?

My maths teacher gave me this equation and I really don't know how to solve this: $$\overline{abc}+\overline{ab}+\overline{bc}+\overline{ac}+a+b+c=29,$$ where $a$, $b$, $c$ are digits. I need to ...
0
votes
0answers
17 views

Do you have a specific method to solve logiqual sequences or do you rely on intuiton?

I'm preparing a presentation on Logical Sequences. Here's one : $2, 6, 12, 20, 30,42, [?]$ The goal is to find the following number in the sequence. In this particular case, a possible answer is ...
0
votes
0answers
9 views

Find the area of the portion of a plane inside the cylinder

How can I calculate this? I think at some point I will need to use symmetry and change this to polar coordinates. In that case my radius is $\pi$, and $\theta=2\pi$ to 0. I can calculate 2 ...
0
votes
1answer
42 views

How to find expected value of a portion of the normal distribution?

$X\sim N(67,4)$ What's the expected value of the portion of the curve $(X>72)$? I tried to use the definition of expected value ($\int xf(x) \mathrm{d}x$), but my integral was far too complicated ...
0
votes
1answer
47 views

How to solve this integer equations?

Conditions $$\begin{array}{ll} 1. \quad&1\le i<j\le n\\ 2. &p=i\cdot n-n-\frac{i^2}2+j-\frac i2, 1 \le p\le\frac{n(n-1)}2 \end{array}$$ given $p$, is there a way to solve for $i, ...
1
vote
1answer
29 views

Represent probability with multiple distributions. Archer shooting bullseyes problem.

The goal is to come up with two ways to represent this probability: An archer shoots a bulls-eye with probability $0.4$. If the archer shoots ten arrows, what's the probability that at least 3 are ...
1
vote
2answers
67 views

Does almost every whole number integer contain any of the digits zero through nine?

For example, how many whole numbers contain an eight? Well, for whole numbers less than ten, it's just eight itself, so that's 10% and for whole numbers less than 100, there are 8, 18, 28, 38, 48, ...
2
votes
0answers
75 views

Isolating x and z in two equations.

I am working on a computer program and at some point I need to isolate an x and a z. I am basically trying to isolate x and z in these two equations: 1) $xn_{x} + yn_{y} + zn_{z} = n_{d}$ 2) ...
1
vote
1answer
25 views

Joint Probability Function

Two hats are drawn randomly w/o replacement from box containing $8$ black, $4$ red, and $2$ yellow hats. If $X$ denotes the number of black hats drawn and $Y$ the number of red hats drawn. What is the ...
1
vote
1answer
47 views

Tricky Substitution to get AM-GM inequality

So, I'm reading the literature to find different proofs of the AM-GM inequality, the following proof quite hit me, and I don't seem to understand at all. The proof is as follows: For any positive ...
1
vote
2answers
46 views

Probability of Game Series

A world series is a best of $7$ series between team $A$ and team $B.$ It takes $4$ wins to win the series. How many ways can a team win the World Series? I said: Suppose that a World Series is ...
0
votes
2answers
45 views

Palindromes less than a number

How many positive palindromes are less than $1,000,000,000$? I think one way to do this is to count palindromes with a fixed number of digits, and take the sum of these values from $1$ digit to ...
1
vote
1answer
26 views

How to express combined discrete-continuous RVs in one pdf?

Let's say we have a random variable $X$ that behaves in two different ways where $X\sim$Bernoulli(1/3) AND $X\sim U(0,1)$. $X$ follows the Bernoulli distribution 25% of the time and the uniform ...
0
votes
0answers
9 views

Are there tools for presentation and vizualization of deduction?

I read that Kalish and Montague introduced a natural deduction method (http://en.wikipedia.org/wiki/Donald_Kalish), which can be easily implemented in software. Any other tools who can help a logician ...
0
votes
0answers
21 views

Solve equation with simplex method

I have equation below and I'm newbie to this method. Can you help me with tutorial or maybe with steps to solve this equation? I know I can use simplex tables, but I don't know a good explanation of ...
0
votes
1answer
20 views

$5$ General Planes make how many CLOSED SPACES?

Actual problem is How many spaces $5$planes divide a space into? and by some analogy and proof, I found that $5$planes divides a space into $26$spaces. in fact, I considered first "How ...
1
vote
1answer
35 views

straightforward calculus problem

Find the arc length of the graph of $\displaystyle \large x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$. Hint: Use symmetry with respect to the line $y=x$. Let $y=x$ intersect at $a$. So, $\displaystyle ...
1
vote
2answers
62 views

Finding supremum in $S=\{q\in\Bbb Q:q<x\}$ [closed]

Let $x$ be in $\Bbb R$ and $S=\{q\in\Bbb Q:q<x\}$. Prove $x=\sup S$. Don't understand how to approach this or solve this.
2
votes
4answers
143 views

If $\omega + 1 = \omega$, find $\omega$ ($\omega \not= - \infty$ or $\infty$)

If $\omega + 1 = \omega$, find $\omega$ ($\omega \not= - \infty$ or $\infty$). It does not have to be a real number. My teacher gave us this question just to play around with, and my first ...
1
vote
1answer
28 views

Finding the square roots of a complex number.

Express $z=4\sqrt2(1+i)$ in modulus/argument form. Hence find the two square roots of $z$ and mark their representations on an Argand Diagram. So far I've worked out the mod/arg form of the ...
0
votes
0answers
27 views

how to solve these sort of problems

This problem was asked in Codeforces. This has been asked here too. The question is You have r red, g green and b blue balloons. To decorate a single table for the banquet you need exactly three ...
0
votes
0answers
18 views

Weighting with restrictions, but no clear objective function?

Here is the problem: I have 40 shares in an index and I want to weight them based on their market value, define the known value as $x_i$ In the traditional way, the weight of each share is ...
0
votes
1answer
26 views

Orthogonal parameterization

Consider the function $$f(a,b,c,d):=\frac{\left(a^*\right)^2b^2-\left(b^*\right)^2a^2+\left(c^*\right)^2d^2-\left(d^*\right)^2c^2}{a^*a+c^*c}$$ With complex parameters $a,b,c$ and $d$ Now find any ...
1
vote
1answer
30 views

How to set up problem involving Poisson RV

Consider an example where customers entering a store is a Poisson random variable with $\lambda=15$. How do you find the probability that 100 or fewer people will walk into the store in any five-day ...
2
votes
0answers
12 views

$\frac{dy}{dx}=\sum_{k=1}^{\infty}a_k(m-k)x^{m-k-1}$ or $\frac{dy}{dx}=\sum_{k=0}^{\infty}a_k(m-k)x^{m-k-1}$

If I have $y=\sum_{k=0}^{\infty}a_kx^{m-k}$ ,then is $\frac{dy}{dx}=\sum_{k=1}^{\infty}a_k(m-k)x^{m-k-1}$ correct because ..I'm confused whether $k$ should start from $0$ or from $1$. Please ...
0
votes
0answers
26 views

Help with a matrix problem

I'm stuck with the following matrix problem: Consider $A = $$\{ X \in \mathcal{M}_2(\mathbb{C})\ \mid X = \left( \begin{array}{ccc} a & 0 \\ 0 & b \end{array} \right); a, b \in \mathbb{C}; ...
0
votes
1answer
88 views

Finding Y coordinate of third triangle point when X coordinate and two other points are already known

Suppose you know the coordinates for points A and B of a triangle. We can refer to those coordinates as (Ay,Ax) and (By,Bx). Also, suppose you know the X coordinate for point C (Cx) but do not know ...
0
votes
0answers
29 views

Excercise: Find the volume of the parallelepiped

Find the volume $V$ of the parallelepiped whose four adjacent vertices are the points: $A = (−2, 1, 0)$, $B = (2, 3, 2)$, $C = (1, 4, −1)$, and $D = (3, 6, 1)$. I know how to find it with three ...
1
vote
2answers
49 views

Solving an unusual equation

I need to find a real number $n$ such that $n > 1$ and: $$ \sum_{k=1}^\infty \frac{2^k}{n^k} = \frac{n-1}{n} $$ Ideally, I'd find the minimum such $n$ (if more than one exists), but really, any ...
0
votes
0answers
49 views

Finding examples before solving

So I've been solving some contest problems,and most of them require a solution in order to be solved. For example $$S_n=\left\{{n\choose n},{2n\choose n},{3n\choose n},\ldots,{n^2\choose n} \right\}$$ ...
0
votes
1answer
8 views

Discrete algebra and exponents (See body text)

Let $a,b\in\mathbb{Z}^+$. If $a \equiv b\bmod 49$, and $\gcd(a,49) = 1$. How can I find any positive integer $n > 1$, so that $b^n\equiv a\bmod 49$? I'm completely stumped by this. I've been ...
2
votes
1answer
56 views

Special feature of the function f(z) = $|i + z|^2 + az + 3$

I have to solve following problem: Find all the values of a (a is a real number) that the function f : $f(z) = |i + z|^2 + az + 3$ (z is a complex number, i is an imaginary unit) has a following ...
2
votes
1answer
60 views

Problem about problem solving

I am having some problems on how to solve a problem.When I read a chapter on say group theory or real analysis,I feel that I have grabbed the concepts quite well,but when I start solving exercises ...
1
vote
1answer
10 views

Obtaining an expression between $s'(n,r)$ and $s(n,r)$

I've a doubt in this: We're given $[x]_n=(x)(x-1)\ldots (x-(n-1))$ and $[x]^n=(x)(x+1)\ldots (x+n-1)$ . Now as we can write : $[x]_n=(x)(x-1)\ldots (x-(n-1))=a_0+a_1x+a_2x^2\ldots ...
5
votes
1answer
77 views

No. of integral solutions of $x_1+x_2+x_3+x_4=20.$

I've to solve a no. of questions of this type but don't get how to do it: Determine the no. of integral solutions of $x_1+x_2+x_3+x_4=20.$ given the constraint that $$1\leq x_1\leq ...