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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4
votes
2answers
750 views

Using Sticks and Stones for Counting number of Ways

From the first twenty positive integers, how many ways can we select 6 integers so that no two integers from the six chosen ones are consecutive? I tried using sticks and stones, but my thought ...
3
votes
1answer
90 views

Find all positive solutions of the system of equations

Find all positive solutions of the system of equations $x_1+x_2=(x_3)^2$ , $x_2+x_3=(x_4)^2$ , $x_3+x_4=(x_5)^2$ , $x_4+x_5=(x_1)^2$ , $x_5+x_1=(x_2)^2$ What i have done : ...
1
vote
0answers
32 views

When setting up a probability problem, when is it appropriate to use conditioning?

I understand the principles of conditioning and its rules, but when do I decide if a problem will be easier using conditioning versus determining through other methods? I'm teaching myself probability ...
0
votes
1answer
298 views

Use of Delaunay Triangulation and Voronoi Diagram to find alpha shape using Edelsbrunner's algorithm

I am learning how to find the shape of a set of points in 2-D. I understand that Alpha Shape method is a good way to find the shape of a set of points. Alpha Shape was originally introduced by H. ...
1
vote
0answers
44 views

The Jugs of Water Problem - with constraints

Given three jugs containing any amount of water such that a1 <= a2 <= a3 and each jug is large enough to contain all the water, show that it's possible (or not) to empty one jug. Only ...
3
votes
2answers
385 views

Does an elegant solution exist for this trigonometric equation?

I'm trying to solve this: $\cos ^{-2}x + A\tan{x} = B$ Wolfram alpha spits out an incredibly long and convoluted solution for x. Is there no simple, straightforward analytical way to solve this?
0
votes
2answers
40 views

Is this proposition posible? [duplicate]

In a board, you have $13$ White round pieces, $15$ Black round pieces, and $17$ Red round pieces. In each round you can choose two different color pieces and change them with two other pieces of ...
0
votes
1answer
68 views

How do you figure out the formula to convert between units?

I know that to, for example, convert from Fahrenheit to Celsius you subtract 32 and then divide by 1.8. I'm interested in how this type of formula can be developed. So, given two different sets of ...
1
vote
1answer
44 views

Modular arithmetic and using in well-ordering principle

I need to prove the following, but I do not know how to go about it. If $$ (*)\:\:\: x^{3} - y^{3}= 3^{n} $$ Then $$ x \equiv 0 (mod 3) \:\: and \:\:\: y \equiv 0 (mod 3)$$ In addition, ...
0
votes
1answer
38 views

Solve equation with two unknowns (maybe modulo)

Given the following equation: $$ x^{2} - y^{2}=17, \quad 0\neq x,y\in \mathbb N$$ I know for example that one solution is $x=9$, $y=8$, but I do not know how to get it.
3
votes
1answer
921 views

Find a seven digit number which describes itself

Find a seven digit number which describes itself. The first digit is the number of zeros in the number. The second digit is the number of ones in the number, etc. For example, in the number 21200, ...
1
vote
1answer
110 views

Question about “linear programming problem” in reference to joint pmf

I'm working on a homework problem and I'm not totally sure what the question is asking... The question reads: "Consider the linear programming problem: maximize $Ax_1+Bx_2$ subject to $x_1+x_2\leq ...
2
votes
3answers
89 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
0answers
172 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 ...
2
votes
1answer
57 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
34 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
26 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
135 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
95 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
241 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
30 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
177 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
74 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
197 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 ...
28
votes
14answers
934 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 ...
2
votes
2answers
85 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
1answer
81 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
63 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
40 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
257 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
84 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
34 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
82 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
99 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
411 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
71 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
1answer
30 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
41 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 ...
2
votes
4answers
161 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
46 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
25 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
28 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
44 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
14 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
29 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
424 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 ...
1
vote
2answers
64 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
1answer
10 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
59 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
85 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 ...