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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11
votes
0answers
390 views

Every 4-regular simple graph contains a 3-regular subgraph

The following result was conjectured by Berge & Sauer, and proved by Tashkinov [T]. Theorem A. Every 4-regular simple graph contains a 3-regular subgraph. A simple graph is the one with no ...
10
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0answers
305 views

Approximating $\pi$ by an expression of the form $\sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$

Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student: Find an approximation of $\pi$ formed of the digits $0$ through $9$, each used at ...
8
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1answer
321 views

A “Theorem Style” Problem Book in Differential Geometry

I am trying to teach myself differential geometry using Lee's Introduction to Smooth Manifolds. To test my understanding, and learn the subject better, I am looking for a good problem book in ...
7
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0answers
72 views

Given that $f(x) :=x^6-6x^5+ax^4+bx^3+cx^2+dx+1$ has its roots as all positive, find $a,b,c,d$

From Larson's "Problem Solving Through Problems" 7.2.10: Given that $f(x) :=x^6-6x^5+ax^4+bx^3+cx^2+dx+1$ has its roots as all positive, find $a,b,c,d$ Thus chapter is about (generalized) ...
6
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202 views

I am a physicist with a difficult equation (quadratic exponential) I am curious about. No luck with a lit review, sympy or Mathematica.

I have an equation I have been playing with (the variable is $x$ and the constants are positive and real): $$( r_1 - x )^2 \frac{ a r_3 e^{x / r_2} + b r_2 e^{x / r_3} }{\left( a e^{ x / r_2} +...
6
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0answers
269 views

Division of a square and value of a disk

I cam across this problem and I really don't know how to solve it. So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the ...
6
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0answers
658 views

A system of equations of Vietnamese Mathematical Olympiad 2013

This is a system of equation of Vietnamese Mathematical Olympiad 2013, the first day. Solve the system of equations $$\begin{cases} \sqrt{\sin^2 x + \dfrac{1}{\sin^2 x}} + \sqrt{\cos^2 y + \dfrac{1}{\...
5
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0answers
61 views

Minimizing a project costs through dynamic programming

I have a project and I want to minimize the costs. I am responsible for the inspection of 1000 miles of sewer grid in Canada. My goal is to provide time high quality inspection reports. I tried to ...
5
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1answer
130 views

A pre-calculus problem about a quadratic function

This is from a test (I'm not a high school student) given to rising high school juniors. The problem was designed to take less than 20 minutes, preferably 5-15. Judging from its source, this is not a ...
5
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0answers
226 views

Evaluating Coefficients for a Fourier Series when Exponential terms are present [Approach needed]

On the last step of solving a three-dimensional Laplace equation,($\nabla^2T=0$) with BC(s) as $T(0,y,z) = T(L,y,z) = T_a$, $T(x,0,z) = T(x,l,z) = T_a$, $\frac{\partial T(x,y,0)}{\partial z} = p_c\...
5
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2answers
75 views

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$ I don't know where to start as in is there a ...
5
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0answers
60 views

Network Connectivity Problem

Assume a network (a complete undirected graph) comprised of $N$ participants (vertices). Each person holds a different piece of information, unknown to all other members. The participants can ...
5
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1answer
119 views

Survival dynamics in zombie post-apocalypse

I'm new here. I have a world-building problem that I'm trying to translate into a simple mathematical model. I apologize in advance if I'm using the wrong terminology. A world (2d plane) is filled ...
5
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0answers
315 views

Good problem books at a relatively advanced level?

I have been searching for problem books on advanced topics. By advanced I am referring to the undergraduate level and above. I am looking for something analogous to the olympiad type problem books ...
5
votes
1answer
229 views

How to find a list of summands and factors adding up to a total?

I am neither a mathematician nor do I have an idea on how to write down my problem in accurate mathematic formulas. Please feel free to edit my question into shape and remove this paragraph. Also I am ...
5
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0answers
183 views

Different ways of operating an infinite continued fraction

Given the continued fraction below, $$ \cfrac{1}{\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}+\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}} $$ I wanted to know to which number it converged, so I ...
5
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0answers
313 views

Puzzle - zero knowledge proof

I am solving the following problem : I have edge-matching puzzles, where all pieces are squares and the grid has $n$*$n$ format. There is no global image to guide a puzzle solver. Despite the puzzles ...
4
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0answers
122 views

3D Homogenous Laplace equation with integral boundary conditions

I have the 3D heat equation (Laplace equation) $$\nabla^{(3)}T_s=0$$ where $\nabla^{(3)}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ ...
4
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2answers
145 views

Problem of rooms

A rectangle is divided into some smaller rectangles.Each two adjacent rectangles share a door which connects them.Prove that we can start from one of the small rectangles and pass them all without ...
4
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0answers
54 views

Help Solving Equation From A Paper

In the paper, entitled: A Closed Form Solution for the Pull-in Voltage of the Micro Bridge (Link to PDF: https://pdfs.semanticscholar.org/0d31/33707b1243f6b4e3344c4fa19b831b010b8b.pdf) ... the ...
4
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1answer
339 views

Chess tournament problem

$12$ chess players took part in a tournament. Each played against each other exactly once. After the tournament every chess player did $12$ lists of names. On the first list, the player only wrote ...
4
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0answers
139 views

Iterations $n, n^n, (n^n)^{(n^n)},…$

(Note: I'm reposting this, as I posted the original too late in the evening to gain anyone's notice.) A contest problem (#2 on the 2010 Virginia Tech Math Competition) proffers the solver the ...
4
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1answer
309 views

Math Olympiads: GCD of terms in a sequence equals GCD of terms in other sequence

Recently, someone asked for a proof of a problem from the Russian Mathematical Olympiad, 1995. Math Olympiads: GCD of terms in a sequence equals GCD of their indices. The problem was to show that if ...
4
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1answer
122 views

Is there a better way to solve this problem in Linear Algebra?

Well I have the following problem: Let $\alpha = \{v_1,v_2,v_3\}$ and $\beta=\{u_1,u_2,u_3\}$ be two bases of $\mathbb{R}^3$ such that $v_1=(1,0,1)$, $v_2=(1,1,0)$ and $v_3=(0,1,1)$. It's known that ...
4
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0answers
58 views

Solving a system of equations

I'm trying to prove the existence of a solution to the system of equations $$c_i = \gamma x_i + (1-\gamma) \frac{x_i^2}{\sum_{j=1}^\infty x_j}$$ for $i\in\{1,2,....\}$ where $\sum c_i=1$. I am also ...
4
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0answers
104 views

Arbitrary ratio sequences on a partition of $\mathbb{R}$ (Partition regularity of fixed ratio sequences)

Background: This question arose purely recreationally and doesn't really fit into any context that I know of. Let $A \sqcup B = \mathbb{R}$ be a partition of the ...
4
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0answers
952 views

How to deal with a mathematical problem if I don't know the answer?

The problem is I'm looking on shortest path between points problem and the intuition tells me that the shortest path between points happens when paths don't cross. It's a step one. Then for all sub-...
4
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1answer
56 views

Using induction to prove sum of certain number-theoretic function.

For any positive integer $ n > 1$, let $P(n)$ denote the largest prime not exceeding $n$. Let $N(n)$ denote the next prime larger than $P(n)$. (For example $P(10) = 7$ and $N(10) = 11$, while $P(11)...
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0answers
50 views

What is the probability of two even numbers given one is even?

Your friend has generated two random numbers from $1...10$, independently of each other. What is the probability that both numbers are even given the information that there is an even number among the ...
3
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0answers
50 views

About a problem of field extension in an algebra book by Fumiyuki Terada.

I am reading an algebra book by Fumiyuki Terada. There is the following problem in this book: $E_1, E_2, K$ are fields. $K$ is a subfield of $E_1$. $K$ is a subfield of $E_2$. $p, q$ are ...
3
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1answer
76 views

Understanding $\int_0^\pi{\frac{dx}{a^2\sin^2x+b^2\cos^2x}}$

$$I=\int_0^\pi{\frac{dx}{a^2\sin^2x+b^2\cos^2x}}$$ I am unable to understand why a substitution of mine in this problem gives a wrong answer. Here's what I did! $$I=\int_{\tan0}^{\tan\pi}{\frac{d(\tan ...
3
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0answers
90 views

Finding optimal solution to the utility function

I would like to find a solution to the following problem: I have the following utility function: \begin{equation} U(x,r)=\alpha\frac{x^{1-r}-1}{1-r}-(1-\alpha)(x-0.5)^2, \end{equation} subject ...
3
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0answers
155 views

Phragmén-Lindelöf Theorem

Let $f : G \to C$ be analytic and suppose that $G$ is bounded. Fix $z_0\in \partial G$ and suppose that $\limsup_{z→w} | f(z)| ≤ M$ for $w \in\partial G$, $w\neq z_0$. Show that if $\lim_{z→z_0} |z − ...
3
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0answers
79 views

How do you prevent being lead astray when you're working on a problem that takes months/years?

Usually, when you're a student, the textbook/teacher has organized the structure of your education such that you are led on the right path. You are basically "taken by hand" to make sure that after a ...
3
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0answers
103 views

A problem on floor function

Find $m,n\in \mathbb{Z}$ s.t. $$ \sum\limits_{k = 0}^{mn - 1} {\left( { - 1} \right)^{\left\lfloor {\frac{k} {m}} \right\rfloor + \left\lfloor {\frac{k} {n}} \right\rfloor } } = 0 $$ See here for ...
3
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0answers
64 views

Derivatives-Piecewise Function problem

I was doing exercises from a book when I tried this particular one: Find $f ^\prime (0)$ knowing that $g(0)=g^\prime (0)=0$ and $g^{\prime\prime}(0)=2$. $$f(x)= \begin{cases} g(x)/x & \text{if $...
3
votes
2answers
103 views

2 queen, $5\times 5$ chess board problem

Prove the following: In $5\times 5$ chess board the least amount of queens you need in order to threaten on each square is 3. (Square threat: the queens threatens on each square in the diagonals,row ...
3
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0answers
315 views

Easier version of “Problem Solving Strategies” by Engel?

I am preparing to take the Putnam next year. Currently I own the book in the title, but the material might be a little too difficult for me. I have gone through the first eight examples in the first ...
3
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0answers
262 views

What are some useful problem solving strategies for real analysis?

In this blog, Professor Tao exhibited some problem solving strategies that can help students in their study of (mostly) measure theory and some are intended for analysis in general. I'd love to see ...
3
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0answers
245 views

Does this equation have no solutions?

The question is this : The source from where I got this question was devoid of any answers to it, so I came here, this is how I proceeded : LHS : $((((({(x)^x})^{2x})^{3x})^{....x^2})^2 = (((((x)^{...
3
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0answers
138 views

What are the essential tools and proof techniques for beginning smooth manifolds and differential topology?

I am an undergraduate currently taking a first course in smooth manifolds. I feel that I understand the material intuitively. But, I'm having trouble turning my intuition into proofs. I was hoping ...
3
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0answers
52 views

What could be examples at calculus or introductory analysis level for the idea contained in the statement by David Hilbert?

I read the following quote in the book "As opposed to abstraction the art of doing mathematics consists in finding special cases which contain all the germs of generality. --David Hilbert", however ...
3
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0answers
72 views

Green's Theorem with respect to a given polar region.

Using Green's Theorem, compute the counterclockwise circulation $I$ of $\vec{F}=\langle-\sqrt{x^2+y^2},\sqrt{x^2+y^2}\rangle$ around the region defined by the polar coordinate inequalities $7 \leq ...
3
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0answers
247 views

Problem of $\pi$- and $\lambda$-systems

I have some trouble with a theoretic-like exercise about measure theory, and $\pi$- and $\lambda$-systems, and I would like to have some help. The problem is stated in the book Mathematical Statistics,...
3
votes
0answers
59 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are $\lambda_1\...
3
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0answers
42 views

Is there a test for tractability of nonlinear differential equations?

After lengthy attempts at tackling the problem one might say that coming up with a closed form solution for a nonlinear differential equation is not possible - that the problem is intractable. But is ...
3
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0answers
122 views

Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...
3
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0answers
63 views

Please check my problem solved. The task was to calculate $M^{100}$, where M is a $3\times 3$ matrix

Again, o points for this problem. And there's a small mis-type in the beginning where t1=t2=t3=t=1, it's actually -1
3
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0answers
489 views

How to learn problem solving strategy for Measure Theory?

I have taken both graduate level Algebra and Measure theory courses but found the latter much more difficult for me. I have put a lot effort on learning it by reading a few reference books and ...
3
votes
0answers
189 views

Shortlist of problems in linear algebra

A while ago I remember seeing a very nice shortlist of problems in linear algebra. It was a list of about 40-50 problems. The idea was that if you solve them, you learn linear algebra very well and ...