The problem-solving tag has no usage guidance.

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### Competitive functions: uniqueness of solution [closed]

Let $f_i(x_1, x_2, ..., x_n)$ for $i=1,...,n$, be real-valued differentiable functions with the following properties:
1) $f_i(x_1, x_2, ..., x_n)=0$ if $x_i=0$.
2) $f_i(x_1, x_2, ..., x_n)=1$ if ...

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**2**answers

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### (Non)existence of mirrors with more than two foci

Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...

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### What can we say about variational energies here?

Let $V_{ij}^{lk}$ be any $nm \times nm$ real symmetric matrix, $\forall i,j,k,l$
\begin{equation}
V_{ij}^{kl}=V_{ji}^{lk}
\end{equation}
(So for the indices we have $1 \leq k,l \leq m$ and $1 \leq i,j ...

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**1**answer

231 views

### Strange problem about triplets of differential forms

Suppose we have the following map:
$$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$
...

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**2**answers

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### Why are there so few zero-dimensional polynomial system solvers and is this because there is no real market for them?

My questions involve the quotes below from wikipedia regarding solving polynomial systems, which given the size of the market for Big Data & Predictive Analysis applications I find puzzling:
...

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**1**answer

166 views

### Problem on the digits of $n!$

let $m$ be a natural number, does always exist a $N\in \mathbb{N}$ such that $m$ or more "$0$" digits (excluding the terminal ones) appears amongs the decimal digits of $n!$ if $n\ge N$?

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### Expected value of swaps

Suppose you have a list of non negative numbers of size N. Now you calculate the maximum element in the list by scanning the list linearly and constantly updating a variable which has initial value of ...

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### How to simplify conditional probability of union of several events

I have an output binary scalar, $y∈B=\{0,1\}$, and an input binary vector $x=[x_1, x_2,…x_M]$ where $x_i∈B=\{0,1\}$. I know that the output $P(y)=1$ depends entirely on the input x. Thus, I want to ...

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### A challenging non homogenous fractional inequality

I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in ...

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**1**answer

544 views

### Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...

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**4**answers

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### When did you “meet Polya”? [closed]

I guess most of us didn't meet Polya in person (this is the answer to the title)! Perhaps, it is much easier to guess that most of us have met one of his writings (or alike) on problem solving, and ...

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**3**answers

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### Is “problem solving” a subject to be taught?

I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...

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**2**answers

227 views

### Reducing system of polynomials with symbolic factors

Getting nowhere with maple using its triangularize and groebner decompositions for even moderate size systems with any symbolic factors. Any suggestions on how better to approach this would be ...

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**1**answer

734 views

### number of non-negative integer solutions for a set of equations [closed]

How to find the exact number of non-negative integer solutions of the following set of equations :
$$x_1 + x_2 + x_3 + x_4 + x_5 + x_6=6 $$
$$ 2x_1 + x_2 + x_3 = 4$$
$$ x_2 + 2x_4 + x_5 = 4$$
$$ x_3 ...

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**1**answer

188 views

### Problem solving equation of type x=a*b^x [closed]

I have the equation x = a*b^x and want to solve it for x. But every online solver I tried says that it is not possible.
But when I choose a==8 and b==0.5 there is a solution for x==2
Is it not ...

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**1**answer

163 views

### First moment of a function of a normally distributed random variable

I'm trying to find the first moment of the following function:
$f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the ...

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**1**answer

159 views

### Transformation problem involving 2 random variables

Any help in this problem?
Suppose U and V are independent random variables with density f(u) and g(v) respectively.
The domain of U is the interval (0, 1) and the domain of V is v > 0. After the ...

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**4**answers

2k views

### Help me find good math questions for my students

I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad.
Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...

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**3**answers

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### Truncated Exponential Series Modulo $p$: Deeper meaning for a Putnam Question.

Apparently B6 of the Putnam this year asked:
Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ of the numbers $\sum^{p-1}_{k=0} k! n^{k}$ are not ...

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### Contest problems with connections to deeper mathematics

I already posted this on math.stackexchange, but I'm also posting it here because I think that it might get more and better answers here! Hope this is okay.
We all know that problems from, for ...

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### Fast removal of weighted edges in a graph in a way such that all shortest paths are preserved

This problem is analogous to fast removal of the minimum number of edges in a weighted graph such that if the graph were to be drawn on paper with edge lengths linear in proportion to their weights, ...

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### Elementary mathematical books

I understand that this is a bit of offtopic but mathoverflow is my last resort, as google did not help.
I am about to publish an English translation of my Russian book for high school students. The ...

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**2**answers

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### Question about Banach's matchbox problem.

Hi,
I've been struggling with this for awhile ( http://en.wikipedia.org/wiki/Banach%27s_matchbox_problem)
and I put together this little bit of Python code
...

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7k views

### Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition
that is derived from your everyday experience of physical reality. Your
intuitions about how the spin of a ball affects ...

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**1**answer

649 views

### Math Puzzle: calculating the dimensions of variable rectangles in a fixed square

I've got the following problem,
I've got a fixed size square and within there are a fixed number of rectangles to be contained within it. I want the rectangles to cover the maximum amount of space ...

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**5**answers

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### List of recently solved mathematical problems

I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only ...

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**19**answers

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### Good books on problem solving / math olympiad

Hello,
I would want all book tips you could think of regarding Problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example would ...