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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1
vote
1answer
65 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
250 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
141 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
127 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
133 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: ...
5
votes
5answers
347 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
58 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
200 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
91 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 ...
1
vote
2answers
53 views

Show that one person must have x amount of dollars

A group of six friends discover they have a total of \$21.61 with them on a trip to the movies. Show that one or more of them must have at least \$3.61. How should I approach this problem? I can see ...
1
vote
2answers
73 views

Simple recursive equation sub-solution.

I have tried to solve a very simple recursive equation:))), but I don't know what's wrong with my brain but I got other solution when I partially solve the equation. Equation: $$T(n) = (n+2) + ...
3
votes
2answers
941 views

In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other?

In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other? I tried to draw diagrams onto a $8\times8$ square but I'm only getting $16$ ...
4
votes
6answers
148 views

Repeating Decimals [duplicate]

I'm just wondering how do we simplify repeating decimals into a fraction in general? Like, for example, $$0.5656\dots$$ $$0.12424\dots$$ $$4.23777\dots$$ Thanks!
3
votes
0answers
37 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 ...
-1
votes
1answer
71 views

Finding $5$ numbers with mode $3$, median $7$ and mean $6$ [closed]

I got this problem to solve in a non calculator paper test and didn't know how to solve it. The mode of five numbers is $3$. The median is $7$. The mean is $6$. Work out the $5$ numbers.
1
vote
0answers
41 views

Uniqueness of solution for boundary value problem

In the article : "Third order semipositone boundary value problems" They assume that $w(t)$ is nondecreasing and $w(t)>0$ on $(q,1]$ . and they found the Green's function ...
-2
votes
2answers
187 views

Three different even numbers add up to $90$. The second number is twice the first number and the first multiplied by the second is the third [closed]

Three different numbers add up to 90. All numbers are even. The second number is twice the first number. The first number multiplied by the second number is equal to the third number. Work out the ...
2
votes
2answers
686 views

Sum over cosines = dirac delta - how to get the coefficients?

Given this formula: $$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$ Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$? I googled and searched all kinds of ...
3
votes
3answers
263 views

Finding the number of ordered pairs that satisfies an equation

How many number of ordered pairs of positive integer $(a,d)$ satisfies: $$\frac{1}{\frac{1}{a+2d}-\frac{1}{a+3d}}-\frac{1}{\frac{1}{a}-\frac{1}{a+d}}=2012$$ This question is too complicated. Like I ...
-1
votes
4answers
172 views

How to solve the equation $-5(y+7)=25$? [closed]

I'm not sure how to solve with parenthesis
1
vote
6answers
295 views

Solve the equation $x-7=28$ [closed]

The question is $x-7=28$ But I'm not sure if when I subtract do I have to change the signs to negative?
3
votes
2answers
222 views

Irrational equation, How to solve it?

The equation $$\sqrt[3]{x^2-1} + x = \sqrt{x^3-2}$$ has a solution $x = 3.$ How to solve this eqution?
1
vote
2answers
56 views

I need help with this word problem.

A pet store owner wants to mix together an high quality dry cat food costing 1.10 per pound with a lower quality dry cat food costing 0.85 per pound. How many pounds of each should be mixed together ...
1
vote
1answer
32 views

Solving for $f(n+1)$ when $f(k)$ is known for $k=0,1,…,n$

I posted earlier about polynomials but this is different type of problem I think. I seem to have an answer but I mistrust it.... A polynomial $f(x)$ where deg[$f(x)$]$\le{n}$ satisfies $f(k)=2^k$ ...
32
votes
1answer
637 views

Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers

For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
0
votes
1answer
308 views

What branch of mathematics improves logical thinking? [closed]

So, that's the question. I dare to generalize it even wider: what branch of mathematics improves the general thinking ability, intilligence, the way the person thinks, and makes it more logical? I'm ...
3
votes
0answers
133 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 ...
20
votes
4answers
345 views

Ways to fill a $n\times n$ square with $1\times 1$ squares and $1\times 2$ rectangles

I came up with this question when I'm actually starring at the wall of my dorm hall. I'm not sure if I'm asking it correctly, but that's what I roughly have: So, how many ways (pattern) that there ...
4
votes
2answers
59 views

Find $x,y$ such that $x=4y$ and $1$-$9$ occur in $x$ or $y$ exactly once.

$x$ is a $5$-digits number, while $y$ is $4$-digits number. $x=4y$, and they used up all numbers from 1 to 9. Find $x,y$. Can someone give me some ideas please? Thank you.
2
votes
1answer
60 views

How to compute a product of logarithms?

I've been reading through Stewart's Calculus textbook, and came across the following problem fairly early on - What is $$\prod_{i = 2}^{31} \log_i (i + 1)\;?$$ I did some searching, and found ...
0
votes
1answer
64 views

Showing that $B_0=1$ and that $\sum^n_{k=0}{{n+1}\choose k}B_k =0$

Let $h(x)=x$ and $f(x)=e^x-1$ for $x\in \mathbb{R}$. It's known that $\frac{h(x)}{f(x)}$ has a Taylor series in $0$, which is written as follow: $$\sum^\infty_{n=0}\frac{B_n}{n!}x^n$$ So, how do I ...
2
votes
1answer
62 views

Showing that $f$ is analytic in $\mathbb{R}$

Okay, let $y\in \mathbb{R}\;$ be fixed and consider the function $\;f(x)=e^{x+y}$. How do I show that $\,f\,$ is analytic in $\,\mathbb{R}\,$ and use this fact and the Taylor series in $\,x=0\,$ of ...
87
votes
8answers
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (March. 2014) This question has been moved to mathoverflow; see here. Randomly break a stick in five places. Question: What is the probability that the resulting six pieces can form a ...
10
votes
4answers
852 views

Show that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.

The following problem was on a math competition that I participated in at my school about a month ago: Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions. I will outline ...
2
votes
1answer
628 views

Python numerical solution for a nonlinear second order ODE with two boundary conditions

I want to solve numerical the next equation, in Python $$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$ it is a nonlinear second order $ODE$ with two $B.C$. ...
0
votes
2answers
69 views

system of equations with three equations.

We have to find all real solutions to this system of equations: $$x=\frac{4z^2}{1+4z^2},y=\frac{4x^2}{1+4x^2},z=\frac{4y^2}{1+4y^2}$$
8
votes
2answers
297 views

Solving matrix equations of the form $X = AXA^T + C$

I'm trying to solve this matrix equation: $$X = AXA^T + C$$ In particular, $$ X = \begin{bmatrix} 1.5 & 1 \\ -0.7 & 0 \end{bmatrix} X \begin{bmatrix} 1.5 & -0.7 \\ 1 & 0 ...
5
votes
1answer
103 views

Chessboard and Catalan numbers [duplicate]

Let's consider labeling the square of a $2\times n$ chessboard from $1$ to $2n$ such that the numbers increase from left-to-right and bottom-to-top. Prove that the number of such labelings equals ...
9
votes
3answers
302 views

Solve the functional equation $f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)$ with $f : [0,\infty) \to \mathbb R$ continuous

Solve the functional equation $$f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)\qquad \forall x\geq 0$$ with $f : [0,\infty) \to \mathbb R$ continuous. I can't manage to get this one ...
0
votes
3answers
78 views

How to find remain factor of this trigonometic equation?

The equation $$3\sin^2 x - 3\cos x -6\sin x + 2\sin 2x + 3=0$$ has a solution $x = 0$. That is mean it has a factor $\cos x - 1$. I tried write the given equation has the form $$(\cos x - 1)P(x)=0.$$ ...
4
votes
3answers
73 views

Oceans and volume [closed]

What would be the math formulas to calculate the increase in the height of the ocean if one drop of water were released into it. Assuming that everything is static. How would you solve that ...
2
votes
0answers
173 views

Personal project help needed [closed]

This is the first time i have posted on this site. If the people here judge post as brutal as the programming forums. I apologize ahead of time my math background isn't deep enough to make a thesis ...
0
votes
2answers
46 views

How to find initial function of a function

I am stuck in this problem. How do I find the initial function of a given function? I am learning integral and there in the formula $S=F(b)-F(a)$ is $F$ the initial function of $f$. For example, ...
0
votes
0answers
25 views

Permutations of 1 to 9 and subsequences [duplicate]

Arrange the numbers $1,2,...,9$ in such an order that no four of them appear (adjacently or otherwise ie as a subsequence) in ascending or descending order. Show that there is no arrangement of the ...
2
votes
5answers
688 views

Different ordered triples $(a,b,c)$ of non-negative integers

How many different ordered triples $(a,b,c)$ of non-negative integers are there such that $a+b+c=50$? I tried to list the possibilities but the list is way too long, I know how to find the ordered ...
1
vote
1answer
1k views

Solving the inverse of cos^2

The following equation provides the inclination ($i$) of a galaxy, using the ratio of its two axes: $$ \cos^2 i = {(b/a)^2 − (b/a)^2_{eos} \over 1 − (b/a)^2_{eos}} $$ All I need however is to ...
1
vote
1answer
41 views

What are some ways to find the minimum of an expression?

I don't get how people solve these types of questions. If you have a single variable expression then I guess you could differentiate it, but how would you find the extreme values of an expression ...
2
votes
1answer
41 views

Solve the equation by putting new variable

I want to solve the equation $$x^2+(2 x+3) \sqrt{3 x^2+6 x+2}=6 x+5.$$ I tried. Put $t = \sqrt{3 x^2+6 x+2}.$ We have $$x^2+(2 x+3)t - 6x - 5 = 0.$$ Discriminant of this equation (with unknown $x$) ...
1
vote
2answers
77 views

How to solve this system of equations?

How to solve this system of equations? $$\begin{cases} 1+\sqrt{2 x+y+1}=4 (2 x+y)^2+\sqrt{6 x+3 y},\\ (x+1) \sqrt{2 x^2-x+4}+8 x^2+4 x y=4. \end{cases}$$