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

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 ...
10
votes
0answers
310 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
103 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 numbers $0$ through $9$, each used ...
5
votes
0answers
50 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
votes
0answers
112 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
votes
0answers
473 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
54 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 ...
4
votes
0answers
140 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
94 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
207 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
45 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
81 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
428 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 ...
4
votes
0answers
586 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
49 views

If $(x^2+y^2+z^2)=2(x+z-1)$, then show that $x^3+y^3+z^3$ is constant and find its numeric value.

I am trying to solve this question, If $(x^2+y^2+z^2)=2(x+z-1)$, then show that $x^3+y^3+z^3$ is constant and find its numeric value. I've tried this, $$x^2-2x + z^2-2z + 2 + y^2 = 0$$ $$ ...
3
votes
0answers
53 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
votes
0answers
83 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 ...
3
votes
0answers
37 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
44 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
57 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
81 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
86 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
77 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
44 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
190 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
51 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
191 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
83 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 ...
2
votes
0answers
23 views

Find the angle between asymptotes

Sketch the locus of the centres of circles which touch two fixed and unequal circles. Find the angle between the asymptotes How shall I find the locus when the size of the circles are not ...
2
votes
0answers
37 views

Help with Definition of Limits (Finding a delta given epsilon)

The problem says: Find a $\delta$ such that $|f(x)-l| < \epsilon$ for all x satisfying $0 < |x-a| < \delta$ when $f(x) = x^4; l = a^4$. What I did so far was $|x^4-a^4| < \epsilon$ so ...
2
votes
0answers
58 views

Is it possible to bruteforce a differential equation

Is there any method to solve differential equations which involves just a number of basic functions combined into various permutations (with various factors) which are then fed into the differential ...
2
votes
0answers
27 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 ...
2
votes
0answers
46 views

Reference request - Problem book by subject

I'm looking for good problem textbooks for self-study. I know only of two of this sort: "Introduction to Measure Theory" by Terry Tao, and "Problems in Algebraic Number Theory" by Esmonde and Murty. ...
2
votes
0answers
14 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 ...
2
votes
0answers
48 views

Is the matrix with these coefficients invertible?

Let $0 \leq x_{i-1} < x_i < x_{i+1} \leq 1$. Let $p, q$ be functions that depend on that such that $p$ is positive and $q$ is non-negative. Let $c_i = a_{i+1,i} = a_{i,i+1}$. Let all other ...
2
votes
0answers
44 views

On the second part of solution of a question due to Erdos

Problem. Let $a_1<a_2<\dotsb<a_n\le 2n$ be a sequence of positive integers. Then $$ \min [a_i,a_j]\le 6\left(\Big[\frac n2\Big]+1\right), $$ where $[a_i,a_j]$ denotes the least ...
2
votes
0answers
41 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 ...
2
votes
0answers
78 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 ...
2
votes
0answers
36 views

Get function definition from an equation

My question: I have to find a function $g$ fulfilling the equation $$2\frac{t_k \cdot t_0 - 1}{t_k-t_{-1}} = g(t_k) + g(t_{k+1}) + t_{k+1}\cdot g(t_k)g(t_{k+1})$$ Whereby $t_{n+1}=t_n + h$ with $t_0, ...
2
votes
0answers
146 views

Fundamental Matrix

Determine $\phi(x,0)$ for $A(x)=\begin{pmatrix} -1 & \cos(x) \\ 0 & -1\end{pmatrix}$, where $\phi(x,0)t_{0}$ is a solution of $\frac{d}{dx}t(x)=A(x)t(x)$. I am not entirely sure as to ...
2
votes
0answers
78 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
13 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
75 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 ...
2
votes
0answers
48 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
44 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
73 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
154 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
88 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
63 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}