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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2
votes
1answer
46 views

How to solve $\int \frac{\tan^{-1}x}{(1+x)^2}dx$?

I know how to solve the following integral $$\int \frac{\tan^{-1}x}{(1+x^2)}dx$$ . We have to substitute $\tan^{-1}x$ as $t$ and we will be done. After this one, I tried to find out $$\int ...
1
vote
2answers
38 views

Solving for $\theta$ in a circle

Let's say you have a pendulum hanging straight down and touching the ground at the lowest point. The pendulum has length $l$. If you pull the pendulum back so that the end is height $h$ above the ...
0
votes
0answers
30 views

Series representation for $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 \pi ^2 A^2+W^2}+6 \pi ^2 A^2+3 W^2}}{\sqrt{2}}$

My question is, is there a series representation or other function of $L$ and $A$ I can use when I solve the following equation for $W$? $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 ...
3
votes
1answer
29 views

first order linear PDE solving

$$\dfrac{\partial{\phi}}{\partial{i}}=0$$ $$\dfrac{\partial{\phi}}{\partial{v}}=E-v-i R_0$$ Where E,$R_0$ are constants. How do I solve these kind of PDE's.
-1
votes
0answers
45 views

Represent math problems as Markov chains [on hold]

The step by step that takes to solve a math problem (algebra, calculus, etc.) could be seen as a Markov chain? When solving a problem, the next math rule that you are going to apply only depends of ...
0
votes
3answers
47 views

Derivation of the “Combined Work Formula”

Before I get to my question, some background: Person $A$ can paint a fence at the rate $9 \frac{hour}{fence}$ (or equivalently $\frac{1}{9} \frac{fence}{hour}$) Person $B$ can paint a fence at the ...
0
votes
1answer
26 views

Finding distance using rates of change — best approach?

The question: A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 ...
48
votes
7answers
2k views

Problems that become easier in a more general form.

When solving a problem, we often look at some special cases first, and then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, ...
1
vote
3answers
43 views

Consider the following system of linear equations..

Story cut short, I have an exam in a weeks time and this is a question off a previous exam paper - I'm unsure as to how I should go about it as there are 4 variables with only three linear equations.. ...
7
votes
3answers
370 views

How to find natural solutions of an equation?

When I'm solving problems, I'm often confronted to solving equations, and when I'm solving equations, I'm often confronted to find the natural solutions of these equations. My actual personal ...
1
vote
0answers
25 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 ...
2
votes
1answer
59 views

Russian Old Merchant Problems

Anybody know where I can find more of these old merchant problems: Lui: Please tell us a little bit about your early education. Were you already interested in math- ematics as a child? ...
1
vote
1answer
20 views

Problem involving pseudomonotone mappings on Banach space

I have the following question regarding mappings on a Banach space $X$. If anyone has an idea or hint as to how to resolve this question it would appreciated. Let $X$ be a Banach space, $X^{*}$ its ...
0
votes
2answers
24 views

Die Probability Question + Basics of Conditional Probability

A die is rolled twice. What is the probability of observing: a) a four and a three P (obtaining a four and a three) or P(obtaining a three and a four) therefore P(obtaining a four)* P(obtaining a ...
0
votes
0answers
46 views

A simple repartition problem

Suppose the following conditions exist: A and B together own two values, X and Y; X > Y; A buys X for a value of X1; B buys Y for a value of Y1; X1 > Y1; X > X1; Y > Y1. We need to calculate how ...
4
votes
2answers
190 views

Solving awkward quadratic equation to obtain “nice” solution.

I would like to solve the following quadratic equation to get a "nice" analytic solution for $\rho$. $\rho^2(r\sin\theta-2nr^2)+\rho(2nr^3-2r^2\sin\theta-2\sin\theta+2nr)-2nr^2+3r\sin\theta=0$ where ...
4
votes
0answers
90 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 ...
2
votes
1answer
76 views

Exponential Diophantine: $2^{3x}+17=y^2$

Is there a way of solving the following equation, in integers $(x,y)$, by hand? : $2^{3x}+17=y^2$. You can also try: $2^{2x}+17=y^2$ or more generally $2^x+17=y^2$; each of these has at least 1 ...
0
votes
0answers
34 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
1
vote
1answer
136 views

Comparing $\pi^e$ and $e^\pi$

Comparing $\pi^{e}$ and $e^{\pi}$ I read the answer there but I didn't understand one thing. How I should know to put $\dfrac{π}e-1$ instead of $x$? If I had this question on a test, I had no idea ...
1
vote
2answers
23 views

Problem of bodies in motion in circles.

Consider two circles of radii $4\;cm$ and $8\;cm$, respectively, both circles have the same center $C$ and is two bodies $A$ and $B$, so that $A$ is smaller circumference of the trajectory at a ...
3
votes
0answers
59 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
2answers
108 views

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
2
votes
0answers
30 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 ...
1
vote
2answers
226 views

Can't solve this trignometric equation, why am I wrong?

There is this trig equation: $$ 5\tan x - 2\tan 2x = 0 \text{ for 0 < 0 < 360 } $$ So far I've gotten $$\tan x = \text{0, 180}$$ and all I have to solve now is $$\tan ^2x = 0.2$$ which gives ...
0
votes
1answer
43 views

Best approach or algorithm to solve equation with multiple variables?

I have an equation : $A^6x_1 + A^5x_2 + A^4x_3 + A^3x_4 + A^2x_5 + A^1x_6 + x_7 = B$ What can be the best algorithm/approach I can use to crack this? $A$ and $B$ are constants. $x_1,x_2...x_7$ are ...
1
vote
1answer
95 views

If one plays $132$ games in $77$ days, there is a span of consecutive days with exactly $21$ games

This is a high school contest question. Simple answers are required A chess player has $77$ days to prepare for a tournament. During this time he wants to have a match everyday and to have $132$ ...
1
vote
2answers
35 views

(What is the formula to find) What is the probability that the sum of the numbers on the tickets chosen is at least 7?

Senario: Box A contains four equal-sized tickets, numbered 1, 2, 3 ,4 Box B contains three tickets of the same size, numbered 4, 5, 6 An experiment consists of selecting one ticket from the box A ...
1
vote
2answers
49 views

How can I solve this puzzle using equations?

There's a hall with 100 seats. I want to fill up these seats with men, women and children; they're going to purchase seat positions. The cost per seat for men is 5 USD, women 1 USD, and children 0.05 ...
0
votes
1answer
22 views

How to find plane that's equidistant from the origin

Objective: Give the equation of a plane that crosses the axes at points equidistant from the origin. How do I make sure that the points $A(1,2,-2)$, $B(-5,1,1)$, $C(4,-3,1)$ are equidistant from the ...
9
votes
3answers
255 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for me. It ...
0
votes
1answer
27 views

How to know when a line is parallel to the xz-plane

What are some features of the equations of a line that is parallel to the xz plane, but does not lie on the plane, and is not parallel to any of the axes? So far all I got: -dot product of plane's ...
4
votes
2answers
110 views

Difficult generating function

Define a beautiful number to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise ...
3
votes
1answer
87 views

Last 7 digits of 7th powers

Alice and Bob play the following game. They alternately select distinct nonzero digits from $1$ to $9$, until they have chosen seven such digits. Consider the resulting seven-digit number by joining ...
2
votes
1answer
80 views

How to see 4th dimension? [closed]

Anyone has tricks or methods? Great help, thank you. P.S.: Abstractly of course, not actually seeing 4D, which I don't think is possible.
3
votes
1answer
55 views

Re professional mathematicians working on several problems at once. Source needed.

Recently I read a quote from a working mathematician where he pointed out that professionals have to get used to carrying around several unsolved problems at once. Can anyone help me with the source ...
4
votes
4answers
170 views

Inequalities, when does the sign change here?

I have encountered a problem with inequalities. I have been looking at examples provided by two websites which 'solve' inequalities, however when I try using my own method, the extremely simple ...
6
votes
2answers
147 views

The best balance in studying Mathematics?

I'm a student studying Mathematics at a university level. I've completed Single Variable Calculus, Differential Equations, Multivariable Caculus, Real/Complex Analysis, and Linear Algebra and I've ...
6
votes
1answer
112 views

Coming up with short “magical” proofs

I was reading the solution to this problem: Prove that $f(n) = 2n$ is the only non-constant solution to $2f (m^2 + n^2 ) = (f (m))^2 + (f (n))^2 .$ The solution used these identities, pulled out of ...
10
votes
1answer
114 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
0
votes
1answer
50 views

Basic Mixture question [on hold]

Let's say that I have 50 ounces of 100% sugar water solution. If I add 50 more ounces of pure water to that solution. What percentage of sugar water solution would I have left?
1
vote
1answer
72 views

Books in mathematics based on problem solving

Hi I'm looking for mathematical books that try to teach mathematical concepts by motivating and solving real life problems. For example a book in topology that is motivated by problems of planetary ...
0
votes
3answers
14 views

How to find 2 data point in Bezier satisfy the condition the Chord Length Method?

Suppose bezier curve have 4 control point $P0$, $P1$, $P2$, $P3$. How to find 2 data point $D1$, $D2$ satisfy the condition the Chord Length Method? The Chord Length Method : ...
2
votes
3answers
101 views

Trouble with inequalities

I'm a 9th grade student, going into 10th grade. Math has always been a subject I enjoyed and excelled in. I'm writing a schoolboard-wide math contest next year in mid-February I believe. To prepare ...
40
votes
15answers
13k views

Interview riddle

On the Mathematics chat we were recently talking about the following problem @Chris'ssis had to solve during an interview : $$3\times 4=8$$ $$4\times 5=50$$ $$5\times 6=30$$ $$6\times 7=49$$ ...
5
votes
2answers
54 views

Putting objects in a line.

I'm working on a project outside of school, and I've run into a bit a problem. I thought, maybe there are some problem solvers on the internet who would enjoy this. I have 8 balls, 3 red cubes, and ...
1
vote
2answers
83 views

How to solve for $x$ in the function $y=x\ln(x)$?

I have $y=x\ln(x)$ and I need to solve for x. How am I supposed to do that? Because I get $y=x\ln(x)$ $y=\ln(x^x)$ $e^y=x^x$ and I am stack here... Can somebody help me? EDIT In my case the ...
0
votes
1answer
62 views

Finding the range from standard deviation and Gaussian Curve

The figure above shows a normal distribution with mean m and standard deviation d, including approximate percents of the distribution corresponding to the six regions shown. Suppose the heights of a ...
1
vote
3answers
28 views

Iterative Equation Problem with Constraints

I was given a programming riddle recently. It was eluded to that there is an equation, but I was told that finding the equation was not the goal, and that writing a simple program was the goal. I ...
3
votes
2answers
57 views

Sum of square roots of integers

Let $x, y$ be integers and consider the equation $$\sqrt{x}+\sqrt{y}= 8 \sqrt{31}.$$ It is claimed that this implies $\sqrt{x}= a\sqrt{31}$ and $\sqrt{y}=b \sqrt{31}$ for $a,b$ integers. While this ...