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
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0answers
276 views

Anecdote about mathematicians leaping to tops of problems and then building a staircase down?

I've run across this cute little story before, and now for the life of me I can't find it anywhere. It goes something like: Two people are looking out onto a mathematical landscape, and there are ...
10
votes
0answers
247 views

Pólya and Szegő, Part I, Ch. 4, 174.

The following is a problem proposed in Pólya and Szegő's book "Problems and Theorems in Analysis" Assume that $0<f(x)<x$ and $$f(x)=x-ax^k+bx^\ell+x^\ell \varepsilon(x),\,\;\;\;\lim_{x\to ...
7
votes
0answers
225 views

Differentiable functions satisfying $f'(f(x))=f(f'(x))$

I am wondering whether or not there is a reasonable characterization of differentiable functions $f: \mathbb{R}\to \mathbb{R}$ such that $f'(f(x))=f(f'(x))$ for each $x\in\mathbb{R}$. (Or, if you like ...
6
votes
0answers
2k views

Ways to Score 18 Points in Football Excluding 2-point conversion

The Chicago Bears score 18 points in a football game. In how many different ways can the Bears score these points? Points are scored as follows: a safety is 2 points, a field goal is 3 points, a ...
5
votes
0answers
353 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 + ...
4
votes
0answers
112 views

How far away is that cloud?

A few weeks ago I was on an airplane and to pass the time started thinking about this problem. Using the following information, I wanted to know how far away a cloud I could see was. Under some ...
4
votes
0answers
74 views

How to find $f$ and $g$ if $f\circ g$ and $g\circ f$ are given?

The question is: Let $f:\mathbb R\rightarrow \mathbb R$ and $g:\mathbb R\rightarrow \mathbb R$ be two functions such that $(f\circ g)(x)=4x^2+4x+1$ and $(g\circ f)(X)=x^2+2x+2$. Find $f(x)$ and ...
4
votes
0answers
101 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 ...
4
votes
0answers
169 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
votes
0answers
42 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
votes
0answers
76 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
votes
0answers
481 views

Conditional expectation of the product of two independent random variables

Suppose that $a$ and $b$ are independently distributed random variables, with means; $\mu_a$, $\mu_b$ and variances; $\sigma_{a}^2$, $\sigma_b^2$, respectively. Further, let $c=ab + e$, where $e$ is ...
3
votes
0answers
25 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 ...
3
votes
0answers
39 views

A question on combinations of a set of numbers

I have the set of the first $n$ primes $\{2,3,5,\ldots,p_n\}$. There are $n^n$ ways of selecting $n$ numbers from this set. Each combination has a number ($C_k$) associated with it and it is the ...
3
votes
0answers
54 views

Problem on the digits of $n!$

let $m$ be a natural number, is it always possible to find an $N\in \mathbb{N}$ such that $m$ or more "$0$" digits (excluding the terminal ones) appears amongs the decimal digits of $n!$ if $n\ge N$
3
votes
0answers
66 views

A challenging non homogenous fractional inequality.

The following problem is a challenging generalization of several difficult inequalities, where none of the usual methods used in inequalities seems to work. I would like to know if someone has a ...
3
votes
0answers
75 views

why does table look like this? - multiplicative table for Group

I have a Group $\mathbb{Z_2}[x]/q$ where $q(x)=x^3+x+1$ In my textbook, there a multiplication table: but I dont understand how the last $x+1$ comes. it should be $(x^2+x+1)*(x^2+x+1) \quad mod ...
3
votes
0answers
72 views

No idea how to solve this equation using two exponentials

The equation I have is: $$A = B ( \exp(C x) - \exp(-Dx) )$$ How do I solve for $x$ given $A$, $B$, $C$, $D$? I have no idea how. The only idea I have is that I could express these in terms of a ...
3
votes
0answers
38 views

How to find the point in a closed geometrical figure which maximizes the “direct-line-of-sight function”

To expand upon the title, and put it in clear terms, I phrase the problem thusly: Consider the interior of any continuous, closed, non-self-intersecting curve in the plane. (I'm not sure if I'm ...
3
votes
0answers
143 views

Difference between two sets of data points

I'm making a simple calibration of a z-stage, by measuring a number of points in one direction with a constant $\Delta$Z between each sample. Then I reverse the direction and measure the same number ...
3
votes
0answers
47 views

Dividing planes with lines and spheres

What is the greatest number of parts a plane can be divided into using $n$ infinite straight lines? What about $n$ circles? Can you generalise this into 3-dimensional space, planes and spheres? For ...
3
votes
0answers
176 views

Cluster point of $a_{n}:=n+(-1)^{n}n$

I am trying to find the cluster point of the sequence $a_{n}:=n+(-1)^nn$. Can you please check my solution? The subsequence diverges for increasing even $n$ since $2n$ grows infinitely. The ...
3
votes
0answers
75 views

Optimizations for Travelling Salesman Problem

I have to design a branch-and bound algorithm that solves the optimal tour of a graph on the cartesian plane every time. I have been given the hint that identifying hopeless branches earlier in the ...
3
votes
0answers
288 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 ...
2
votes
0answers
37 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 ...
2
votes
0answers
74 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) ...
2
votes
0answers
11 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 ...
2
votes
0answers
57 views

Least sum of power of distances

Let $n$ points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of distances $\|A_1X\|^q+ \|A_2X\|^q + ... +\|A_nX\|^q $ (where $q \in \mathbb{Q}$). Are there any general ...
2
votes
0answers
39 views

Trying to make a formula to find maximum driving time.

I am trying to figure out how to make a formula (that will eventually be used in excel.) to figure out, how much driving time could be done in a block of time. In this case, 24 hours. And theses are ...
2
votes
0answers
34 views

Fermat pseudo primes

Is it possible for a number of the form $2^p-1$ with $p\in \mathbb{P}$ (the primes) to satisfy $3^{2^p-2}\equiv 1\pmod {2^p-1}$ and not be a prime? In other words, can a Mersenne number be a Fermat ...
2
votes
0answers
72 views

Is this graph problem already solved

I would like to solve the following: Let $G=(V,E)$ be a directed graph such as $\forall (x,y) \in E, x \neq y$. Find any (all would be even better) graphs $S$ such that: $S \subset V$ $\#\{(x,y) ...
2
votes
0answers
122 views

Can you explain the solution of this geometric problem

A year ago IBM research posted an interesting geometrical problem: A gardener plants a tree on every integer lattice point, except the origin, inside a circle with a radius of $9801$. The trees ...
2
votes
0answers
74 views

Analytical methods to solve a cubic equation

I'm having trouble solving this cubic equation: $x^3 - 7x^2 -27x+64=0$ I was able to prove that it has no rational roots, and by graphing the cubic function, I found that its 3 roots are all real, ...
2
votes
0answers
59 views

Non-linear system of 4 unknowns

What are the non-zero solutions in $x,y,z$ and $t$ of the following system of equations \begin{cases} (1+ax+bz)(1-x)=1\\ (1+cy+bt)(1-y)=1 \\ (1+dx+bt)(1-z)=1\\ (1+fy+bz)(1-t)=1 \end{cases}
2
votes
0answers
195 views

Sought-Given-Solution-Answers

I am a German student attending lectures delivered in English at the KTH Stockholm. There I am supposed to solve a problem sheet, using a "Sought-Given-Solution-Answer approach". I am not really into ...
2
votes
0answers
103 views

Feasibility of a cryptography transformation

This is a follow-up of the question: Transformation We are given $$g^{1/(x+m)},$$ (it is not possible to find $\frac{1}{x+m}$ due to the Discrete log problem), can we find a $k$ such that ...
1
vote
0answers
40 views

Finding the convergent value

The sequence is defined as follows : Start : $(x_0,y_0)$ with $ 0 < x_0 < y_0 $ Step : $x_{n+1} = \frac {x_n+y_n} {2}$ , $y_{n+1}= \sqrt{x_{n+1}y_n} $ Find $\lim_{n\to \infty}(x_n,y_n)$ . ...
1
vote
0answers
20 views

In spherical co-ordinates, 3 unknowns, possible to solve for 1?

So I am trying to get either $\theta$ or $\phi$ in terms of $\theta_1$ and $\phi_1$, where a,b,d and e are constants but c is unknown too and below is as far as I have gotten, I keep going in circles ...
1
vote
0answers
19 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 ...
1
vote
0answers
15 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 ...
1
vote
0answers
37 views

no. of regions a plane is divided into by $n$ lines in general position

My notes state the Counting process for knowing no. of regions a plane is divided into by $n$ lines in general position := Let $h_1(n)=$ No. of parts a line is divided by $n$ distinct ...
1
vote
0answers
25 views

Solving N for HN=0, Given H is a special type of skew symmetric (n x n, n is a odd number) matrix.

Solving $N\ \mathrm{for}\ H \times N =0$, given $H$ is a special type of skew symmetric matrix $(n \times n, n\ \mathrm{is\ an\ odd\ number}\ n=2k+1)$, 0 on diagonal and 1, -1 in off-diagonal ...
1
vote
0answers
24 views

Mathematical Rube Goldberg problem

Is there a book or website that has mathematical rube goldberg-style puzzles? In other words, puzzles that require you to compute something, then compute something based on that, and then iterate for ...
1
vote
0answers
58 views

Finishing a problem using equalities

This is my problem: Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$\frac{a^{n+2}}{a^n + (n-1)\,b^n} + \frac{b^{n+2}}{b^n + (n-1)\,c^n} + \frac{c^{n+2}}{c^n + ...
1
vote
0answers
32 views

How to solve this Ricatti-like ODE

I have been trying to solve the following ODE \begin{equation*} \dfrac{d\pi}{dx}x=c_1+\pi(x) c_2 + \pi(x)^2(c_3-x), \end{equation*} where, for every $i=1,2,3$, $c_i$ is a constant real value. ...
1
vote
0answers
35 views

Slicing through a cuboid containing spheres, how many are exposed to the surface and what is their combined volume

So I place spheres of radius chosen at random from a normal distribution of known mean and standard deviation in a cub or cuboid at random (not overlapping) until a known density of the entire cube is ...
1
vote
0answers
36 views

How many lines needed to not lose in tetris game?

Suppose we play a tetris game with tetris be given randomly. Is there exists a number of lines that we can play infinitely, i.e. do not lose the game?
1
vote
0answers
34 views

Trying to use the deformation theorem to solve integral

I have this integral: $$\int_{|z|=2}\frac{\cosh z}{(z+1)^3(z-1)}dz$$ Both singularities $z=1,z=-1$ are inside the circle. I have already solve this using partial fractions, and I don't have much ...
1
vote
0answers
46 views

bounding the sum of squares of lengths of a quadrilateral inscribed in a unit square

Consider this nice little problem: if $ABCD$ is a quadrilateral inscribed in a unit square, then $$2\leq AB^2+BC^2+CD^2+DA^2\leq4$$ (Evidently this is problem 1 on paper 1 of the 1989 Irish ...
1
vote
0answers
56 views

A problem related to Vectors.

A few days ago I posted an answer to a question on Phys.SE. The question is: Three particles $A,B$ and $C$ are at the vertices of an equilateral trinagle $ABC$. Each of the particle moves with ...