Questions tagged [problem-solving]

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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How many clients should I trust?

I hope this is the right place to put this problem. Otherwise, I beg you to teach me where to put it Supposing a laptop company organizes a month-event in which it shows and sells its new 15 laptops (...
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2 votes
1 answer
69 views

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Find the number of subsets $A$ of $S$ such that $x \in A$ and $2x \in S \implies 2x \in A$.

Let $S=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Find the number of subsets $A\subseteq S$ such that $x\in A$ and $2x\in S$ $\implies 2x\in A$. My Attempt I broke the problem into cases. I made pairs $(1,2),...
4 votes
3 answers
71 views

Counting ordered triples whose product is at most $n$

There is a classical question in discrete math such a kind that Let $a\times b \times c =24 $ then , how many possible $(a,b,c)$ triples are there where $a,b,c \in Z^+$ The answer is the following $...
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Maximum likelihood estimation for a binomial probability function

For a random variable with a binomial probability function $$f(x;p)=\binom{n}{x}p^x(1-p)^{n-x}$$ What is the maximum likelihood estimation of $p$ for a sample of size $1$ of this random variable? In ...
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DSolve producing an InverseFunction Inactive[Integrate] ProductLog as a solution.

When using DSolve I get the below output, a1, b, and c1 are real and positive. ...
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1 answer
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A system of equations resembling the Vandermonde matrix [closed]

I need help with this assignment. If we were to write it in matrix form, the right-hand side looks almost like a Vandermonde matrix. Any hint or solution? $$ {\begin{cases}{x_1 + ... + x_n = a}\\ {x_1^...
-1 votes
1 answer
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Number of imperfections in a fine copper wire [closed]

A previous investigation has shown that the number of imperfections in a fine copper wire averages $28$ imperfections per centimetre of length. What is the probability that there is a distance of $0....
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Solve a system of equations with n unknowns. [closed]

I need help with this assignment. If we were to write it in matrix form, the right-hand side looks almost like a Vandermonde matrix. Sorry for the misspelled dots. $$ {\begin{cases}{x_1 + ... + x_n = ...
-1 votes
1 answer
39 views

What fraction of triangle ABC is shaded? [closed]

I tried to find the area of this triangle but I don't think that's a right way to solve this problem. My question is : What is the ratio of area of Quadrilateral to the area of $\Delta\ ABC$ Please ...
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Solve a system of linear equations for single entry. [closed]

my problem is to solve a big sparse like system A*x = b for a single entry of x. Do you know any algebraic or numerical method that applies? Best regards and thanks in advance!
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0 answers
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Finding the distribution of an integer function of a uniform RV

Let X be a random variable uniformly distributed on [0,1] and let Y = [6X] + 1. Determine the distribution of Y and compute E(Y). My attempt at solving this: My professor advised us to always consider ...
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5 votes
2 answers
199 views

Collatz-like problem involving prime factors

Unfortunately I am not well-versed in LaTeX so I will try my best to keep this looking presentable. As an overview, I was investigating a variation of the Collatz conjecture: Define $f(1) = 1$ Then, ...
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-1 votes
1 answer
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Why is $\int_{\pi/2}^0 \cos x dx$ negative? [closed]

$$\int_{\pi/2}^0 \cos x dx = -1$$ But we know that integration is an area under the curve. $\cos x$ is positive between $0$ and $\pi/2$. So, the area should be positive. But why is the integration ...
0 votes
1 answer
50 views

Is it true that if you understand the content well enough, you won't need to know the solution (proofs)

I am currently self-studying an intermediate level martingales textbook that does not have solutions. I read somewhere that if you understand the content well enough, you won't need to verify, as you ...
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2 answers
35 views

Number of possible combinations in the game

The question is taken from my math book and is supposed to be solved using generating functions without any tools like a calculator, but I don't know how. Person A and person B is playing a game. In ...
1 vote
2 answers
96 views

Prove that $\det(M) = 0$ iff $f_i$ are linearly dependent, where $M_{ij}= \int f_i f_j dx$ and $f_i$ are real and continuous [duplicate]

Let $a < b$ be real numbers and $f_1,...,f_n:[a,b] \to \mathbb R$ continuous funtions. Define an $n$ by $n$ matrix $M = (m_{ij})_{i,j = 1,...,n}$ where $$m_{ij} = \int_a^bf_i(x)f_j(x)dx. $$ Prove ...
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-5 votes
1 answer
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Can Someone tell me where I went wrong in this proof

The question asked to find the solution to $ \sum_{k=1}^{n}(2k+1)\binom{n}{k}$. Here's what I did: By the binomial theorem, we know that $ \sum_{k=1}^{n}\binom{n}{k}x^k = (1+x)^n$. Set $x=x_1^2$. Then ...
-1 votes
1 answer
42 views

Whats the smallest possible value off p and q? [duplicate]

The question is taken from a problem-solving test: Both p and q are positive integers. What's the smallest possible value of the numbers such that: $\frac{p}{q}=0,126126\overline{126}$ I don't really ...
0 votes
1 answer
88 views

Prove that if f is polynomial function then $f+f'+f''+\cdots+f^{(n)}\geqslant0$ [duplicate]

Suppose $f$ is a polynomial function with degree $n$ and $f(x)\geqslant0$ (so $n$ must be even). Prove that $f + f' + f'' +\ldots+f^{(n)}\geqslant0$. Put $\;g(x)=f + f' + f'' +\ldots+f^{(n)}\;$ and $...
0 votes
0 answers
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Complex Equation solving for another variable

I have been trying to re-arrange an equation to solve for another variable and I have not done algebra since college and am getting stuck in a rabbit hole. If anyone can help me figure this one out ...
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1 vote
0 answers
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topology: prove that a subset D of a topological space M is dense iff for all x in M there exists a filter that contains D and converges to x

for the implication from left to right, my idea is that if D is dense, then it shares elements with any open of M (no intersection is null), the set of neighborhoods of x is a filter that converges to ...
2 votes
1 answer
33 views

Largest permutation of subset of $[n]$ with the property that every element in the right half is near to ($<n/2$ from) every element in the left half

Let $\ n\in\mathbb{N}\ $ and consider rearrangements/permutations, $\ \sigma_k\ (k\leq n)$ of subsets of $\ [n]:= \{ 1,2,\ldots, n\}\ $ with the property that no member of the right half of $\ \...
0 votes
0 answers
43 views

What branch of mathematics is this?

This is the question: How many real solutions is there to the following equation? $ x+x^2 +...+ x^{2012} = 0$ The answer is the 2 solutions: $x=0$ and $x=-1$ I solved the question by just factorizing ...
2 votes
0 answers
35 views

Please solve the heat equation satisfying the neumann condition of zero flow at the infinite boundary

The equation is \begin{equation} \left\{\begin{aligned} &\frac{\partial N(t,x)}{\partial t}=\frac{\partial ^2 N(t,x)}{\partial x^2}&x\in \Bbb R\\ &N(0,x)=N_0(x) &x\in \Bbb R\\ &\...
2 votes
2 answers
132 views

Prove this $\sum_{S \subseteq\{1, \ldots, n\}}(n-|S|) \pi(S)=(n+1) !\left(\frac{1}{2}+\frac {1}{3}+\ldots+\frac{1}{n+1}\right)$

Set the natural number $n$. For each $S \subseteq\{1, \ldots, n\}$ define $\pi(S)$ as the product of the members of $S$, with the agreement $\pi(\emptyset)=1$. Prove that $$ \sum_{S \subseteq\{1, \...
3 votes
1 answer
59 views

Problems in mathematical analysis exam

I want to ask some problems that are asked in exam (actually the example of exam that I'm preparing in this post it is Mathematica analysis) I want some suggestions to my answer (some are obtained by ...
-3 votes
1 answer
57 views

How would one solve this equation: $x = \cot(x/2)$? [closed]

I would like to solve the equation: $$x = \cot(x/2).$$ So far, I was able to find only an approximate solution by Taylor expanding $\cot(x/2)$ up to a 3rd term and solving for $x$. But I was hoping ...
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-1 votes
2 answers
35 views

how to prove that the topology generated by the left-closed intervals is finer than the usual topology

The idea is to prove that the open intervals (like $]a,b[$) are contained in the topology of the left-closed sets ($[a,b[$), but I cannot see a way of generating open sets from half-closed sets. (Same ...
1 vote
1 answer
45 views

How to check if a double integral is finite?

I am having a problem with a question asked in GATE $2022$ exam.The question is as follows: Let $f(x,y)=\begin{cases} \frac{x^2-y^2}{(x^2+y^2)^2},\text{ if } (x,y)\neq (0,0)\\ 0,\text{ if } (x,y)=(0,...
0 votes
2 answers
60 views

show that $\mathbb{Q}$ is disconnected

I'm trying to show that $\mathbb{Q}$ is disconnected. I will use the two sets $O_1 = \{x\in\mathbb{Q}: x^2 <2\}$ and $O_2=\{x \in \mathbb{Q}: x^2 >2\}$. I think this is the right approach. ...
2 votes
0 answers
55 views

If the distance to previous terms in a bounded sequence is a large set, then is the set (of points of the sequence) itself dense?

Define the distance between a point $x$ and a finite set $X$ to be $ d(x,X) :=\ \displaystyle\min_{y\in X} \left\{\ d\left( x , y\right)\ \right\}.$ Let $\ (x_n)_n\subset [0,1]\ $ be a real sequence, ...
4 votes
2 answers
132 views

Does such an "extremely" equally distributed real, bounded sequence exist?

For each $\ n\in\mathbb{N},\ $ define the set $\ [n] := \{1,2,\ldots, n\}.\ $ Does there exist a real sequence $\ (a_n)_{n\in\mathbb{N}}\subset [0,1),\ $ such that for each $\ n\in\mathbb{N}\ $ there ...
-1 votes
1 answer
34 views

Solving a non-linear equation of complex numbers

I have an equation as $$y=\beta_0|x|^2x + \beta_1x,$$ where $x$ is a complex number. I know $\beta_0$, $\beta_1$ and $y$. How can I get $x$ and solve the equation? Many thanks.
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How to solve this very hard nonlinear ordinary differential equation: $4y^{3}+7x\sin(x)+4x^{4}-(16\cos(y)-7x-\frac{7}{4}y^{2})y'=-e^{y-x}+\sinh(y)$

$4\cdot y^{3}+7\cdot x\cdot\sin(x)+4\cdot x^{4}-\left(16\cdot\cos(y)-7\cdot x-\frac{7}{4}\cdot y^{2}\right)\cdot y'=-e^{y-x}+\sinh(y)$ I tried a lot to solve this differential equation. I think it ...
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1 vote
1 answer
59 views

$S_{n}$ is the sum of every third element in the $n$th row of the Pascal triangle, beginning on the left with the second element. Find $S_{100}$.

Problem: Let $S_{n}$ be the sum of every third element in the $n$th row of the Pascal triangle, beginning on the left with the second element. Find the value of $S_{100}$. My work: For brevity, I ...
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0 votes
0 answers
15 views

Question about statement of the exercise 8.3.1(d) from Zorich

In this exercise the author defines two paths $x_{1}$ and $x_{2}$ to be equivalent at the point $x_{0}\in \mathbb{R}^m$ if $x_{1}(0)=x_{2}(0)=x_{0}$ and $d(x_{1}(t),x_{2}(t))=o(t)$ as $t\to 0$ (I ...
3 votes
1 answer
71 views

Does there exist a set $A\subset\mathbb{N}$ such that, for each $d\in\mathbb{N}$ there is a unique $n\in\mathbb{N}$ such that $n\in A$ and $n+d\in A?$

Does there exist a set $A\subset \mathbb{N}$ such that, for each $d\in\mathbb{N},\ $ there is a unique $n\in\mathbb{N}$ such that $n\in A$ and $n+d \in A\ ?$ My first thought was that the set of ...
0 votes
0 answers
30 views

Solving numerically a system of transcendental equations

I have to solve the following system of equations $$ \phi = \int \mathcal{D}z \left[ \frac{\int^{1+\Delta/2}_{1-\Delta/2} dt \cosh(\beta~ J \sqrt{q} ~z ~t) \exp[f(t; \beta, J, \lambda_1, \lambda_2)] t^...
1 vote
2 answers
79 views

How to determine the continuity of bivariate functions?

Consider a problem of the form $f(x,y) = \frac{g(x,y)}{h(x,y)}$ for $x \neq y$ and $f(x,y) = a$ for $x=y$ such that $a$ is a fixed real number and $(x,y) \in R^2$. And we need to show that this ...
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0 votes
0 answers
26 views

Probability 2 teams in one group

Hello guys i need help with this Consider the following (modified) sports draw: There are 30 teams divided evenly into 6 pots. Then 5 groups with 6 teams each are formed. Each group is randomly ...
-4 votes
1 answer
40 views

Is it possible to isolate this variable from this equation?

Can someone with strong knowledge of math answer me? I need to isolate $x$ from this equation: $a = \dfrac{b((1+x)^n - 1)}{ x(1 + x)^n}$
0 votes
2 answers
25 views

Find the maximum width and length of a swimming pool under these conditions

"A 1x1 metre tile costs £45.00. A garden owner has £900 (for tiles) and wants to build a new swimming pool with the base of the pool tiled, as well as the sides. The pool is one metre deep and is ...
1 vote
1 answer
71 views

Why this functional is continuous?

Our functional analysis instructor assigned us a problem which is as follows: Let $(a_n)$ be a sequence of real numbers such that for each real sequence $(x_n)\to 0$ the sum $\sum\limits_{n=1}^\infty ...
0 votes
0 answers
33 views

A problem based on uniform boundedness principle.

I am a graduate student.I am studying functional analysis.I am having problem with the following question in my assignment: Let $X,Y,Z$ be Banach spaces and $A_n$ be a sequence in $B(X,Y)$ and $A_nx\...
0 votes
1 answer
24 views

Find the operator norm of the following linear functional. [duplicate]

Let $C[-1,1]$ be the normed linear space of all the continuous functions from $[-1,1]$ to $\mathbb F=\mathbb{R\text{ or }C}$,with the sup-norm $\|\cdot\|_\infty$ and define $T:C[-1,1]\to \mathbb F$ ...
0 votes
1 answer
32 views

Understanding a function $A(W+r)^{b}$.

I am given the following function $$Q = A(W+r)^{a},$$ where $A$ and $r$ are constants. I am asked to do several things with this function: Find $\frac{\partial Q}{\partial W}$. Given constants $a$, $...
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1 vote
0 answers
38 views

An application of open mapping theorem in linear algebra.

Open mapping theorem,which is taught in functional analysis,states that if $X$ and $Y$ are two Banach spaces and $T:X\to Y$ be a surjective bounded linear operator ,then the map $T$ is open.Now I ...
-3 votes
2 answers
53 views

Solve the inequality $\lfloor x\rfloor=\lfloor 2x\rfloor$ [closed]

Solve the inequality $\lfloor x\rfloor=\lfloor 2x\rfloor$ https://i.stack.imgur.com/28r1D.jpg
0 votes
1 answer
25 views

Let $V$ be a $K$-vector space of dimension $n \ge1$ and let $B \subset V$ . At following statements are equivalent:

a) $B$ is a base of $V$. b) Each element of $V$ is uniquely written as a linear combination of elements of $B$. I'm a little confused, doesn't this equivalence come straight from the definition? My ...
0 votes
1 answer
39 views

How do we solve $xa^x = b$ for $x$?

Consider $x9^x = \frac{3}{2}$. I used wolfram alpha and found its solution to be $x = \frac{1}{2}$. But I don't know the method to get this solution. So, how do we go about solving this problem, ...

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