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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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
244 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
159 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
498 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
223 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
312 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
300 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
51 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
150 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
95 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
1k 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 ...
46
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
131 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
260 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
140 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
348 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
60 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
244 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
591 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
137 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
74 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 ...
0
votes
1answer
31 views

Creating simultaneous non-linear equations

Suppose, for each given $w_1 \in \mathbb{R}$, there exists unique $x_1,y_1,z_1 \in \mathbb{R}$ that satisfy the following system of equations: $F_1(w_1,x_1,y_1,z_1)=0\\ F_2(x_1,y_1,z_1)=0\\ ...
9
votes
2answers
171 views

Proving that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$

How do I prove that: $$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$$ Do I use induction?
14
votes
2answers
164 views

Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$

How do I simplify: $$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$ Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a ...
2
votes
1answer
71 views

Simplifying $\left|\left|\sqrt{-x^2}-1\right|-2\right|$

How do we simplify the expression $\left|\left|\sqrt{-x^2}-1\right|-2\right|$? This is very confusing. Do they cancel out and become just simply $\sqrt{-x^2}-1-2$?
1
vote
1answer
69 views

Power series of $f(x)=\sqrt{\frac{1+x}{1-x}}$

How do I find the power series form of $\,f(x)\,$: $$\displaystyle f(x)=\sqrt{\frac{1+x}{1-x}}$$ I tried to multiply the fraction by $\,\dfrac{1+x}{1+x}\,$ but it didn't help...
4
votes
7answers
328 views

Solving $x^{\log(x)}=\frac{x^3}{100}$

How do I find the solution to: $$x^{\log(x)}=\frac{x^3}{100}$$ So I multiplied 100 both sides getting: $$100x^{\log(x)}=x^3$$ Now what should I do?
5
votes
6answers
153 views

Simplify $\log_23\ \log_34\ \log_4 5\ \log_5 6\ \log_6 7\ \log_7 8$

How do I evaluate the product: $$\log_23\ \log_34\ \log_4 5\ \log_5 6\ \log_6 7\ \log_7 8$$ I know that $$\log_ba=\frac{\log\ a}{\log\ b}$$ How can I apply it? Thanks!
25
votes
12answers
2k views

Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$

Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)? $$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$ Using the calculator I can see that the first one ...
3
votes
2answers
130 views

Using the hypothesis $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ to prove something else

Assuming that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$$ Is it possible to use this fact to prove something like: ...
0
votes
1answer
185 views

Puzzle identification and solving algorithm

I am trying to solve 8x8 puzzle (total 64 buttons). Similar to LightsOut, but in this rules are different. Goal is turn ON every button. Example: ...
6
votes
5answers
396 views

For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .

If $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$.
2
votes
0answers
61 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}
3
votes
1answer
304 views

Problems on submanifolds

I am learning differential geometry and a basis of the theory of smooth manifolds but i'm feeling a lack of practice in solving problems on submanifolds in $\mathbb{R}^n$ (problems like 'prove that ...
2
votes
2answers
95 views

Simultaneously solving of equations

I am trying to refresh some math skills and I am struggling over the following problem. I tried to solve it with the help of a number of sources (i.e. http://www.idomaths.com/simeq.php), but I haven't ...