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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Confusion regarding pole of a complex function.

I am a graduate student.I am studying complex analysis.I encountered the following problem in a lecture: Find the residue of $f(z)=\frac{z-\sinh(z)}{z^2\sinh(z)} $ at $z=\pi i$. Now,this problem is ...
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Conceptual problems in measure theory.

I am studying measure theory as a graduate student. I have studied the theory up to integration. Now I want to test my concepts by doing some problems as the best way to learn a topic is to do ...
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2 answers
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Domain and range of $f(g(x))$ given only the graphs of $f(x)$ and $g(x)$? [closed]

I am a little confused on how to find the domain and the range of the function $f(g(x))$ when only given the two graphs. I kind of understand the solution, but I am struggling to use the domain of $g(...
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1 answer
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How to quickly find unique values of $\sin x$ given that $\sin (nx) = k$, where $-1\le k\le 1$ and $n\in\mathbb N$?

First question Find distinct values of $\sin x$ given that $\sin 5x =0.5$ my method There will be $5$ unique values of $\sin x$ working in first quadrant $5x = \pi/6 + 2\pi m$ for $m=0,\pm 1,\pm2,......
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Determine the location of a logistics center which is optimally close to its providers

Excuse me if my question is not worded perfectly in mathematical terms. I don't have a strong math background. So, here's the problem which has been brought up by a real-life situation: For simplicity,...
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1 vote
1 answer
23 views

Finding a vector maximising and minimising the inner product on two others vectors

I'm tying to solve a simple linear algebra problem. Let us assume that I have a vector space of dimension $R^N$. In that space I know two vectors $u$ and $v$. I want to find a vector $w$ that ...
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1 answer
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Evaluating $\int_0^\infty e^{-ax}\cos(bx)dx$ and $\int_0^\infty e^{-ax}\sin(bx)dx$ for $a>0$ using complex analysis.

I am a Mathematics student.Currently I am doing exercises of complex analysis from Stein Shakarchi's book.In the second chapter,there is a problem which is as follows: Evaluate $\int_0^\infty e^{-ax}\...
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Locus of a trajectory [closed]

If a stone is thrown at an angle of elevation of 45 degrees. What would be the locus of its trajectory? How would you sketch this? Problem The answer appears to be the arc of a circle, but shouldn't ...
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1 answer
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Is formula $\mathtt{(∃x)(∀y)ϕ(x,y)}$ provable or refutable from $\mathtt{T}$ (in a Sound & Complete Proof System using the Axioms of $\mathtt{T}$)?

Is the formula $\mathtt{(∃x)(∀y)ϕ(x, y)}$ provable or refutable from $\mathtt{T}$ (in a Sound and Complete Proof System using the Axioms of $\mathtt{T}$) where $$\mathtt{T} = \{\mathtt{(∀x)¬E(x, x), (∀...
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3 answers
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Calculating the probability of $P(A|B)$? [closed]

Our question was phrased to us as: Passing the emission check is mandatory for cars. A high emission detector has 1% false positives rate meaning that 1% of cars with low-emission wrongly turn the ...
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iven: $R_3=M_3 \oplus X$ and $R_4=M_4 \oplus X$ Can I select 2 different values for $M_3$ and $M_4$ such that $R_3$ and $R_4$ are different? [closed]

Given: $R_3=M_3 \oplus X$ and $R_4=M_4 \oplus X$ Can I select 2 different values for $M_3$ and $M_4$ such that $R_3$ and $R_4$ are different for sure? Please Note, I know nothing about the value of $X$...
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3 votes
2 answers
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(convex function) Let $f: [a,b] \to \mathbb R$ twice differentiable, and $f''(x) \ge 0$, $\forall x \in [a,b]$....

Let $f: [a,b] \to \mathbb R$ twice differentiable, and $f''(x) \ge 0$, $\forall x \in [a,b]$.Prove that $f(\frac{x_1 + x_2} {2}) \le \frac{1}{2}[f(x_1) + f(x_2)], \forall x_1, x_2 \in [a,b].$ My ...
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(extremum)Let $f: \mathbb R \to \mathbb R$ be a polynomial function ...

Let $f: \mathbb R \to \mathbb R$ be a polynomial function $f = a_0 + a_1x + … + a_n x^n$. Let $a_1 = a_2 = … = a_k = 0$, ($k$ less than n) and $a_{k+1} \ne 0$. The function $f$ has an extremum at the ...
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($C^1$ Function)False or true? (justify) a) Let $f: X \to \mathbb R$. If $f'(x) = 0, \forall x \in X$, then $f'$ is constant...

False or true? (justify) a) Let $f: X \to \mathbb R$. If $f'(x) = 0, \forall x \in X$, then $f(x)$ is constant. b) If $f$ is differentiable, then $f$ is of class $C^1$. In the case of true prove, in ...
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2 votes
1 answer
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Are there derivatives of the following functions or not? Justify your answer. In the case of a positive answer and calculate the derivatives.

Are there derivatives of the following functions or not? Justify your answer. In the case of a positive answer and calculate the derivatives. a) $f:(a,b) \to \mathbb R, f(x) = \frac{1}{x-a}$. ...
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2 answers
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Find a model and a non-model of a theory $T=\{\mathtt{(∀x)¬E(x, x),(∀x)(∀y)(E(x, y)→E(y, x)),(∀x)(∃y)ϕ(x, y)}\}$ over language $L$ where $L=(E)$

Find a model and a non-model of a theory $$\mathtt{T} = \{\mathtt{(∀x)¬E(x, x), (∀x)(∀y)(E(x, y) → E(y, x)), (∀x)(∃y)ϕ(x, y)}\}$$ over the language $\mathtt{L}$ where $\mathtt{L}=\mathtt{(E)}$. $(\...
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2 votes
1 answer
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Evaluating improper integrals with the help of contour integrals.

I am a graduate student.I have been studying complex analysis from Stein Shakarchi's book.In chapter $2$ ,there is an exercise which is as follows: As given in the hint,I assume that the countour is $...
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Solve $g_1(x) \mathcal{F}[e^{-c_1 f(x)}] = g_2(x)\mathcal{F}[e^{-c_2 f(x)}]$ for $f(x)$

Is there any way to approach such a problem? Let $f:\mathbb{R} \to [0,\infty]$ and $g_1, g_2:\mathbb{R} \to \mathbb{R}$ be smooth, compactly supported functions, and let $c \in [0,\infty]$. Denote the ...
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-2 votes
0 answers
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Feller formula for the probability of a pair of heads or tails coming out in succession.

Referring to https://en.m.wikipedia.org/wiki/Feller%27s_coin-tossing_constants In the example given, If we toss a fair coin ten times then the exact probability that no pair of heads come up in ...
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3 votes
2 answers
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(Expectation)Consider the joint density $f(x,y)=c(x−y)e^{−x}, 0 \le y \le x$.

Consider the joint density $f(x,y)=c(x−y)e^{−x}, 0 \le y \le x$. a) Determine the value of c. b) Calculate the marginal of Y. c) Calculate the expectation E(Y). a)$1=\int_0^x c(x−y)e^{−x} = \frac{...
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1 answer
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algorithms applied to n-tuples of numbers to end up with $(0,0,...,0)$

I came to the following problem in Problem Solving Strategies by Arthur Engel on page 18 : Start with a sequence $S=(a,b,c,d) $ of positive integers and find the derived sequence $ S_1=T(S)=(|a-b|,|b-...
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1 answer
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some questions about IMO 1986 problem 3

To each vertex of a pentagon, we assign an integer $x_i$ with sum $s=\sum{x_i > 0}.$ If $x, y,z $ are the numbers assigned to three successive vertices and if $ y < 0$ , then we replace $(x, y,...
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1 vote
1 answer
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2n ambassadors are invited to a banquet

Problem: $2n$ ambassadors are invited to a banquet. Every ambassador has at most $n−1$ enemies. Prove that the ambassadors can be seated around a round table, so that nobody sits next to an enemy. ...
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1 answer
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Finding number of ways to put four balls in k bins

I'm trying to calculate how many ways we can put four indistinguishable balls in n indistinguishable bins, given that the maximum each bin can hold is r balls.
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Engel's Extreme principle question.

Hello fellow mathematicians! I've been lately reading Problem Solving Strategies by Arthur Engel and in the Extreme principle section there's a problem that is a very well-known one, and I reckon a ...
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For Continuous $f: [1,4] \to \mathbb R$, $f(1)=f(4)$, Show $f(c) = f(c+ 1.5)$ for at least one c

For Continuous $f: [1,4] \to \mathbb R$, and given that $f(1)=f(4)$, Show there exists at least one $c \in [1,4]$ such that $f(c) = f(c+ 1.5)$. (this is not homework but I am practicing some questions ...
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2 answers
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Finding product of three numbers given respective products

While solving a problem about volumes of parallelepipeds, I came across three expressions involving the products of its respective sides: $a,b$ and $c$. The equations are the following: $ab = 60$ $ac =...
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5 votes
1 answer
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(Borel-Cantelli) If $X_i$ is a sequence of identical and independent variables. Using Borel-Cantelli, prove that...

If $X_i$ is a sequence of identical and independent variables. Using Borel-Cantelli, prove that $$E|X_1| \lt \infty \to P(|X_n| \gt n\ \text{infinite times}) = 0.$$ (uses that $E(x) = \sum_{n=0}^{\...
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0 votes
1 answer
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Which value a and e must have to meet the conditions? [closed]

I'm trying to implement an algorithm to solve conversion of base, and I'm stuck. I want to express value in this form: $ v = a×b^e$ . How can I solve this problem? If a $\epsilon \ I_{\ne 0}\ , e \ \...
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2 votes
2 answers
82 views

Finding the maximum $k$ such that $(7!)!$ is divisible by $(7!)^{k!}\cdot(6!)!$

If $(7!)!$ is divisible by $(7!)^{k!}\cdot(6!)!$, then what is the maximum value of $k$? At first glance I couldnt think of anything except Legendre's formula for calculating powers of a prime in a ...
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1 vote
1 answer
80 views

$\int_{-2}^xf(t)dt$ for $f(t) = \tiny\begin{cases} -1 \, &t<0 \\ 1 \, &t\ge 0 \end{cases} $, and its limit at $x=0$

Let $f: [-2,2] \to \mathbb R$, $$ f(t) = \begin{cases} -1 \, &t<0 \\ 1 \, &t\ge 0 \end{cases} $$ Define $g: [-2,2] \to \mathbb R$ as: $$g(x) = \int_{-2}^xf(t)dt$$ Plot $g(x)$ and find it'...
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Good intermediate level exams/ books.

I have recently finished my mathematics degree and am looking for exams/ puzzles/ books to continue with my mathematical journey. The reason I say intermediate is I do not wish for books at the ...
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6 votes
2 answers
71 views

Let $1=n_0<n_1<\ldots$ be an increasing sequence of positive integers. True/False: $\sum_{i=1}^\infty\frac{n_{i+1}-n_i}{n_{i+1}}$ diverges to $\infty$ [duplicate]

Let $1=n_0<n_1<n_2<\ldots$ be an increasing sequence of positive integers. Is it true that $\displaystyle\sum_{i=0}^{\infty} \frac{n_{i+1}-n_i}{n_{i+1}} $ diverges to $+\infty?$ For example, ...
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2 answers
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Integration of $\int_0^\frac{\pi}{4} (\sin^6 2x+\cos^6 2x) \cdot\ln (1+\tan x) dx$

I've found an integration problem from Molodova Matholympiad 2008. The problem is as follows. Find the Integration of $$\int\limits_0^\frac{\pi}{4} (\sin^6 2x+\cos^6 2x) \cdot \ln (1+\tan x)\ \mathrm ...
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6 votes
3 answers
300 views

Find $x,y,z$ satisfying $x(y+z-x)=68-2x^2$, $y(z+x-y)=102-2y^2$, $z(x+y-z)=119-2z^2$

Solve for $x,y,z$: $$x(y+z-x)=68-2x^2$$ $$y(z+x-y)=102-2y^2$$ $$z(x+y-z)=119-2z^2$$ After some manipulation, I obtain $$xy+xz=68-x^2$$ $$yz+xy=102-y^2$$ $$xz+yz=119-z^2$$ After combining equations, I ...
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0 votes
1 answer
52 views

How many equations are required to solve $n$ many variables?

If we have consistent system of linear equations, each of which has variables $x_1, x_2, \ldots, x_n$, then we will require having $n$ many equations in order to find the one-true single solutions set ...
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2 votes
1 answer
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Given irrational numbers $0<\gamma_1<\gamma_2<1,$ real numbers $0\leq a<b\leq 1,$ does $\exists n$ such that $\{n\gamma_1\}, \{n\gamma_2\}\in(a,b)?$

For this question, $\{x\}$ means the fractional part of the real number $x.$ Given irrational numbers $0<\gamma_1<\gamma_2<1,\ $ real numbers $a,b$ with $0\leq a<b\leq 1,$ does there exist ...
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-2 votes
0 answers
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Typical problems about the Fundamentel Group

more than a specific question, I want to ask you about what do you think are the most typical problems or generic introductory problems about the fundamental group.
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How to evaluate the following contour integral?

I have recently studied Cauchy's integral formula which states that if $f:\Omega\to \mathbb C$ be a holomorphic function and $C$ be a positively oriented simply closed curve whose interior is also ...
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0 answers
31 views

Solution of multi variables dependent first order differential equation?

I am solving first order differential equation: $$(x_0x_1)' =-8 \dfrac{x_0'}{R},$$ where $x_0=f_0(z)$, $x_1=f_1(z)$, $R=r_1-z(r_1-1)=f_2(z)$ (they are all dependent on $z$, but that is not the same ...
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1 vote
0 answers
60 views

What kind of problem is the Potter Kata? Is there a general solution?

I recently completed the Book Store exercise on Exercism. It is a copy of the Potter Kata from Coding Dojo. The basic premise is that a bookstore wants to promote sales of a particular book series. ...
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0 answers
37 views

A math logic problem [duplicate]

Say there are 12 people on an island, 11 weigh the same. One is either lighter or heavier. There's a beam balance to measure the weights. Is it possible to find the heavier OR lighter person in only 3 ...
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Using the Bolzano–Cauchy Theorem, prove that the equation $x^2+x−\cos x=0$ has at least one solution in the interval $[0,1]$.

Using the Bolzano–Cauchy Theorem, prove that the equation $x^2+x−\cos x=0$ has at least one solution in the interval [0,1]. Let $f : [a,b] \to \mathbb R$ be a continuous function if the signs of $f(a)...
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What does it mean that a function $f:\mathbb R \to \mathbb R$ is continuous on $c \in \mathbb R$?

a) What does it mean that a function $f:\mathbb R \to \mathbb R$ is continuous on $c \in \mathbb R$? b) Show that the function $f: \mathbb R \to \mathbb R$ given by $$ f(x) = \begin{cases} 3, & \...
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1 vote
1 answer
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Let $p : \mathbb R \to \mathbb R$ be a non-constant polynomial, that is, for all...

Let $p : \mathbb R \to \mathbb R$ be a non-constant polynomial, that is, for all $x \in \mathbb R$;$p(x) =a_0 + a_1x +... + a_nx^ n$, with $a_n \neq 0$ and $n \ge 1$. Prove that if $n$ is even then $...
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0 votes
0 answers
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Consider the following typographical error in the limit definition:...

Consider the following typographical error in the limit definition: $$\forall \epsilon \gt 0, \exists \delta \gt 0:x \in X, 0 \lt \lvert x-a \rvert \lt \epsilon\Rightarrow \lvert f(x) – L \rvert \lt \...
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0 votes
0 answers
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Are there any tricks to solving this sum?

Suppose that I have a generic, real function $f(x)$ where $x$ is a positive integer. Suppose further that $$\sum_{i=1}^{x} f(i)=B(x),$$ $$\sum_{i=1}^{x} \Phi_i=A,$$ for some $\Phi_i>0$, where $A>...
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0 votes
1 answer
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Why is this recursively-defined permutation $\sigma:\mathbb{N}\to\mathbb{N}$ symmetrical in the sense that $\sigma(\sigma(n))=n\ $?

Define a permutation $\sigma:\mathbb{N}\to\mathbb{N}$ recursively by the following rule: $\sigma(1)=1$ and $\sigma(n)$ is the least $x\ $ in $\ \mathbb{N}\setminus\{\sigma(k): k\leq n-1\}$ such that $...
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0 votes
0 answers
13 views

The following table represents the velocity of a body as a function of time....

The following table represents the velocity of a body as a function of time. \begin{array}{c|lcr} t(s) & 0 & 10 & 18 & 23 & 29 \\ \hline v(m/s)& 22 & 26 & 37 & ...
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0 votes
3 answers
76 views

Coin Flipping with Bias

This is a question on basic probability, and I wonder if my approach is correct. Moreover, if my answer is correct, I wonder if there is another way to do it without using the law of total probability ...
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