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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0
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2answers
433 views

How to solve the irrational inequality?

Solve the inequality $$\dfrac{2x^4+2x^2}{\sqrt{x+1}}+(x+2)\sqrt{x+1}>x ^3 + 2x^2 + 5x.$$ I tried. By putting $t = \sqrt{x+1}$, we have $$2t^8-t^7-8t^6+t^5+15t^4-4t^3-11t^2+4t+4>0.$$ Using Maple, ...
0
votes
1answer
49 views

Graphing a function

A function is defined in a article equation 37, $$A(t)= 0.456+\frac{2.58}{(1+0.136 t)^{0.249}}$$ The simulation of the equation is: But I have tried like by this: ...
48
votes
5answers
2k views

Finding the value of $\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$

Is it possible to find the value of $$\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$$ Does it help if I set it equal to $x$? Or I mean what can I possibly do? ...
3
votes
2answers
141 views

Evaluating decay rate with trigonometric explanation

According to the equation 4, $$\phi(0,t)= \frac{A_0}{(1+\frac{2t^2}{R^4})^{3/4}}\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 t}{R^2}\right]\right)\tag{1}$$ what conditions makes, ...
14
votes
1answer
319 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 ...
3
votes
4answers
331 views

What is an effective means to get senior high school students to write their complete working out as part of their answer.

In Australia and in the International Baccalaureate (2 systems I have worked in), for better or worse, mathematics is assessed by criteria. This increases the importance of students to express their ...
10
votes
2answers
576 views

Is “problem solving” a subject to be taught?

Note: This question has been cross-posted to MathOverflow: see here. I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all ...
2
votes
1answer
165 views

Problem-solving

I just finished my second year as a mathematics student at university. At university, we learn about advanced mathematics and problems. However, I'm also interested in some problems that doesn't ...
0
votes
2answers
79 views

How to calculate the number of banner appearance based on monthly page views.

Hello fellow mathematicians. I have a website that gathers more then 44.000 page views per month. In my website I have 1 rotating place of 4 banner positions, each time it rotates 4 new banners will ...
1
vote
1answer
113 views

torus filling curve

I'm trying solve this problem but I didn't many ideas how to do it. So, if someone can give me a hint or the step of a solution I would greatly appreciate it. This is the problem: "Let ...
-2
votes
1answer
2k views

Solve: $T(n) = T(n-1) +(1/n)$ by iteration [duplicate]

Use iteration method to solve: $1.$ $T(n) = T(n-1) + \frac{1}{n},\,(T(0)=1)$ $ 2.$ $T(n) = 3T\left(\dfrac{n}{3}\right) +1,\,(T(3)=1)$
11
votes
0answers
272 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 ...
1
vote
2answers
838 views

Find the least next N-digit number with the same sum of digits.

Given a number of N-digits A, I want to find the next least N-digit number B having the same sum of digits as A, if such a number exists. The original number A can start with a 0. For ex: A-> 111 then ...
0
votes
2answers
74 views

Help with a Function

I'd like some help getting this question on its way, any help would be greatly appreciated. For the following function $$f(x) = \frac1{-(x-5)^2+7}\;,$$ determine: a) The equation of the ...
3
votes
1answer
168 views

Partition Proof

Let $\lambda$ be a partition of $N$ of rank $r$. How can I show that: $$\sum_wx^\lambda(w)=f^\lambda(-1)^{t(\lambda)}\prod^r_{i=1}(\lambda_i-1)!(\lambda'_i-1)!$$ where $w$ ranges over all ...
0
votes
1answer
253 views

Solving Systems of Equations Question

Saw this question and have been unsure of how to solve it properly. Any help would be appreciated! A pilot of a downed airplane fires the emergency flare into the sky. The path of the flare is ...
1
vote
3answers
160 views

Exponential equation with a negative exponent

From the first sight, this equation: $\exp(-2at)=-\exp(-2bt)$ has no solution. However, Worfram Mathematica clams, it exists. I am wondering, what is the most common to solve it: perhaps, Taylor ...
1
vote
2answers
502 views

How to find exponent coefficients in a sum of exponents?

It is easy to determine a coefficient 'c' of exp(c*x), just log it and find slope. Or if it's exp(c1*x) + exp(c2*x) then after log from 0 to the right of left we would find 'c1' and 'c2'. But what ...
3
votes
2answers
228 views

choosing $5$ non consecutive books from a shelve of $12$

In how many ways can you pick five books from a shelve with twelve books, such that no two books you pick are consecutive? This is a problem that I have encountered in several different forms ...
1
vote
2answers
44 views

integration with substitution - why is this so?

I have this problem: $$\int_0^2 \mathrm{(x-1-e^{-\frac{1}{2}x})}\,\mathrm{d}x$$ what I tried: $t=-\dfrac{1}{2}x \Rightarrow \dfrac{dt}{dx} = \dfrac{1}{2} \Rightarrow dx = \dfrac{dt}{2}$ ...
2
votes
2answers
76 views

Age word problem

Adam is now one quarter of his father's age and in $5$ years time, his age will be one-third the age of his father. How old is Adam now? I have trouble with these kind of questions and I've spent ...
1
vote
2answers
60 views

Problem Solving using Algebra

If Peter is $7$ years older than Sharon and John is twice as old as Peter, work out how old Peter is if the average of their ages is $19$. Thanks! :)
2
votes
2answers
63 views

Represent RPG Increasing Formula

I am having trouble with the following math problem. There is a variable "xp" that starts at 0. At xp of 0, the level is 1. At xp 500, the level is 2; at xp 1250 the level is 3; at xp 2250 the level ...
14
votes
2answers
313 views

Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$: $$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$ Any clues?
3
votes
2answers
312 views

Proving there are no integer solutions for $3x^2=9+y^3$

Prove there are no $x,y\in\mathbb{Z}$ such that $3x^2=9+y^3$. Initial proof Let us assume there are $x,y\in\mathbb{Z}$ that satisfy the equation, which can be rewritten as $$3(x^2-3)=y^3.$$ So, ...
15
votes
4answers
299 views

Find $x,y,z \in \mathbb Q$ such that $x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z$

Find $x,y,z \in \mathbb Q$ such that: $$x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z$$ Here is my thinking: $$x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z\\ \implies \left ( x ...
2
votes
1answer
52 views

Number $e(n)$ of trees with $n+1$ unlabeled vertices $n$ labeled edges

How do I find the number $e(n)$ of trees with $n+1$unlabeled vertices $n$ labeled edges. We're suppose to give a simple bijective proof, I guess? Help appreciated!
1
vote
3answers
153 views

Distinguishable telephone poles being painted

Each of n (distinguishable) telephone poles is painted red, white, blue or yellow. An odd number are painted blue and an even number yellow. In how many ways can this be done? Can some give me a ...
2
votes
1answer
96 views

X,Y are independent RVs with known characteristic functions. Find P(X+Y=2).

X,Y are independent random variables with the following characteristic functions: $ \phi_X(\theta) = \frac{1}{4}e^{i\theta}+\frac{3}{4}e^{i2\theta} \\ \phi_Y(\theta) = ...
5
votes
3answers
2k views

An equation about a rectangle with given perimeter

I am doing a revision calculator paper and am stuck on an algebra question. There is a picture of a rectangle. One side is $x-2,$ another side is $2x +1.$ The question is. Setup and solve an ...
48
votes
3answers
1k views

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer multiplicity of such transforms. Is ...
9
votes
3answers
284 views

show that $a_{n}=\frac{1}{4}[(1+\sqrt{2})^{2n+1}+(1-\sqrt{2})^{2n+1}+2](n>1)$ not have square numbers

Show that the sequence $$a_{n}=\dfrac{1}{4}[(1+\sqrt{2})^{2n+1}+(1-\sqrt{2})^{2n+1}+2]\qquad (n>1)$$ doesn't contain a perfect square. I think this problem is interesting, and my idea: we ...
2
votes
1answer
195 views

Searching for the value of $p_5$

Reference post: click here Given, \begin{eqnarray} &&\Delta p_5-p_5+3S^2p_5 +\frac{SZ}{576\sqrt{\lambda}}(3Z-5S^3) \left(\frac{15g_5}{\lambda^2}+1\right)^2\nonumber\\ ...
1
vote
2answers
132 views

Not able to solve $({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$

I'm not able to solve $$({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$$ If you put values of $p$ (like $\frac{1}{2}$ or 2) back in the equation it doesn't satisfy! So please ...
1
vote
2answers
63 views

How can this be the correct derivative?

I am learning statistics and come across this calculation for Maximum-Likelihood estimator for the Binomial distribution. I don't understand the step from second to third row where they took the ...
7
votes
0answers
262 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 ...
-4
votes
1answer
142 views

How to find $p$ when $ ({\frac{1}{2}})^p + ({\frac{1}{4}})^p + ({\frac{1}{8}})^p - 1 = 0. $ [duplicate]

Kindly mention solution-techniques along with solution
1
vote
1answer
34 views

Is this the best solution (or a correct one) to this recurrence relation.

The $r_k$'s are terms of a sequence of positive integers satisfying $\sup r_k=\infty$. I am looking for a solution $F_k$ to $$F_{k}\ge 2r_{k+1} F_{k+1}-F_{k+1}\ (k\ge1);\ F_1=1 \ (*)$$ (I want ...
0
votes
3answers
87 views

$\log\left(\binom{n}{x} \pi^x (1-\pi)^{n-x}\right)=x\log \pi + (n-x)\log(1-\pi)\;\;?$

$$\log\left(\binom{n}{x} \pi^x (1-\pi)^{n-x}\right)=x \log \pi + (n-x)\log(1-\pi)$$ this is what i have. i dont understand how $\binom{n}{x}$ disappears, but the rest is fine. I tried this, but it ...
3
votes
1answer
354 views

Evaluating the time average over energy

For more info see the article equations 37 Edit: The $\varepsilon ^3 $ has vanished due to time average. But how to get the 4th order? Let us define some function for scalar field $$\phi= ...
0
votes
1answer
61 views

Confusion regarding probability of microbe producing everlasting colony.

My question is about the given solution to problem 4 in Newman's book 'A Problem Seminar'. Note that the book is available online at Springer. Problem 4 A microbe either splits into two perfect ...
5
votes
2answers
246 views

Let $k \geq 3$; prove $2^k$ can be written as $(2m+1)^2+7(2n+1)^2$

Prove: If $k \geq 3$, then $2^k$ can be written as $(2m+1)^2+7(2n+1)^2$, where $k, m, n \in \mathbb{N}$.
1
vote
1answer
47 views

Solving variables for the equilibrium distribution

This question is related to the equilibrium probability distribution of a markov chain. I have: $ \vec {π} * \vec {P} = \vec {π}$ $ [π_0,π_1,π_2,π_3] $ [ $\begin{matrix} 1/3 & 2/3 & 0 & ...
3
votes
6answers
599 views

Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$

Where do I start to solve a equation for x like the one below? $$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$ After squaring it, it's too complicated; but there's nothing to factor or to ...
5
votes
1answer
139 views

Maximizing an unusual function (Putnam 1996)

“Fish," he said, "I love you and respect you very much. But I will kill you dead before this day ends.” -- Ernest Hemingway, The Old Man and the Sea I have, with varying degrees of concentration, ...
0
votes
1answer
51 views

About continuous functions and aritmethic progression

I've try solve this question, but I haven't sucess... The problem is the following: A continuous functions $f:[a,b]\rightarrow \mathbb{R}$ assume positive and negative values in its domain, show ...
1
vote
2answers
80 views

Solving a system of equation:

Solve for $x,y$: \begin{align} x^3 + y^3&=2\\ x^2 +x + 9y - 3y^2&=8 \\ \end{align} I can find $x=y=1$ by guessing. Please help me solve it without using computer. Thanks Edited, sorry, I ...
3
votes
0answers
76 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 ...
0
votes
1answer
45 views

Frequency determination from Dimension analysis

the time averaged total energy, $\bar E$, has the following $\varepsilon$ expansion in $D$ dimension: \begin{equation} \bar{E}=\varepsilon^{2-D}\frac{E_0}{2\lambda}+ \varepsilon^{4-D}E_1 ...
6
votes
1answer
137 views

Why substitution method does not work for $\int (x-\frac{1}{2x} )^2\, \mathrm dx$?

Why $$\int \ \left(x-\frac{1}{2x} \right)^2 \, \mathrm dx$$ is easy to integrate once $$\left(x-\frac{1}{2x} \right)^2$$ is expanded, but impossible using substitution method? (tried 5 different subs ...