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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1answer
25 views

For $n \geq 2$, find $\theta_n, \theta_n > 1$ s.t. $-log(1-\frac{1}{n}) = \frac{1}{n} + \frac{\theta_n}{2n^2}$

For $n \geq 2$, show that $\exists$ a number $\theta_n, \theta_n > 1$ such that $-log(1-\frac{1}{n}) = \frac{1}{n} + \frac{\theta_n}{2n^2}$ lim$_{n\to \infty} \theta_n$ My attempt: I am not ...
0
votes
1answer
65 views

How can I solve this recurrence problem?

Given a function $$ f(n) = f(5n/13) + f(12n/13) + n \;\;\;\;∀n \geq 0 $$ I would like to find a function $g(n)$ such that $f ∈ Ө(g(n))$.
1
vote
1answer
41 views

Maths challenge problem: Why is the number of teams which require 4 substitutions 32?

I came across the following problem on a UKMT senior maths challenege: A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. There are four substitutes: 1 goalkeeper, 1 ...
2
votes
1answer
20 views

Let $n$ be a positive integer and $S$ the set of points $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers such that $x + y < n$.

Let $n$ be a positive integer and $S$ the set of points $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers such that $x + y < n$. The points of $S$ are colored in red and blue so ...
1
vote
0answers
17 views

A question on the representation of all integers in terms of the sum of other interger cubes [duplicate]

The question is from a book used for transition between high school mathematics and university mathematics, which states: Prove the following statement or give a counterexample $\forall n \in ...
-2
votes
1answer
45 views

Solve for b and d

Solve for b and d in the following equation. A triangle with sides $(a, a, b)$ has the same area and the same perimeter as a triangle with sides $(c, c, d)$ where $a, b, c$ and $d$ are positive ...
1
vote
1answer
18 views

More detailed explanation of how $2N_{h-2}$ becomes $2^{h/2}$?

I'm trying to learn the proof of the minimum number of nodes in an AVL tree of height h and I'm stumped on how $2N_{h-2}$ becomes $2^{h/2}$. I've read this [answer](How does $2N_{h-2}$ become ...
3
votes
5answers
97 views

Why count it this way?

This is a very very elementary problem solving technique I was taught some time back. I have been using it but now looking at it, I find it kinda strange why it should be this way. Typically, the ...
0
votes
1answer
379 views

Solve 4 A (L^(3/4)) - wL (((24 - L) w)^(-3/4)) = -(((24 - L) w)^(1/4)) for L? Using Mathematica

I'm trying to solve $$4 A (L^{3/4}) - wL (((24 - L) w)^{-3/4}) = -(((24 - L) w)^{1/4})$$ for $L$ using Mathematica, and it spits the following out: ...
4
votes
3answers
370 views

Students in a class, girls sitting with boys and boys sitting with girls

This is a very interesting word problem that I came across in an old textbook of mine. So I mused over this problem for a while and tried to look at the different ways to approach it but unfortunately ...
1
vote
1answer
19 views

Finding the smaller number of two given the ratio between sum, difference and product

How would you find the smaller of two numbers given the ratio between their sum, difference and product? I've been struggling with this one for a while. For example: the ratio between the sum, ...
-1
votes
2answers
94 views

Showing that $x^{11} \equiv 5 \pmod{47}$ has only solution $x \equiv 15$.

I don't understand the proof. Where did they get the first line from, i.e., $21 \times 11=1+5 \times 46$? Fermat's theorem in my view is $a^{46} \equiv 1 \pmod {47}$.
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votes
1answer
69 views

A question in interview for trinity college, Cambridge

Let $M$ be a large real number. Explain why there must be exactly one root $w$ of the equation $ Mx=e^x$ with $w>1$. Why is log $M$ a reasonable approximation to $w$? Write $w = \log M +y$. ...
1
vote
3answers
53 views

What is the number of mappings?

It is given that there are two sets of real numbers $A = \{a_1, a_2, ..., a_{100}\}$ and $B= \{b_1, b_2, ..., b_{50}\}.$ If there is a mapping $f$ from $A$ to $B$ such that every element in $B$ has an ...
1
vote
5answers
143 views

If $a+b+c+d=1$ then why is the maximum value of $(a+1)(b+1)(c+1)(d+1)$ is ${\left(\frac{5}{4}\right)}^4$?

What I know is that for equations of type $x+y=8$, $xy$ attains its maximum value when $x=y$ and this can be proved by either solving the quadratic equation with completing the squares or finding the ...
4
votes
4answers
87 views

$(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$?

The question given is Show that $(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$. What I tried is suppose $a=(y+z-x),\ b=(z+x-y)$ and $c=(x+y-z)$ and then noted that $a+b+c=x+y+z$. So the ...
11
votes
4answers
475 views

The best balance in studying Mathematics? [closed]

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 ...
0
votes
1answer
1k 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
votes
0answers
70 views

How to solve for X^2-2Yx+Y=0? [closed]

How can I solve for $x^2-2Yx+Y=0$? Note: Y is an exponentially distributed random variable with parameter lambda>0. The solution is the following: no real solution for $4Y^2-4Y<0$, so when ...
0
votes
0answers
21 views

clarity in the solution of the following problem

$$(D^2+D)y=x^2+2x+4$$ I found the solution as $$CF=C_{1}+e^{-x}C_{2}$$ and PI=$$\left(\frac{x^3}{3}\right)+4x$$ but the solution from my teacher is PI = $$\left(\frac{x^3}{3}\right)+4x+C3$$ Where ...
2
votes
0answers
25 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 ...
3
votes
2answers
163 views

Solving $2^x - 3^x + 6^x =0$.

Are there any known methods to solve $$2^x - 3^x + 6^x = 0,$$ where $x$ is either in closed form, perhaps in terms of special functions, or to give inequalities on the answers, where $x\in\mathbb{C}$ ...
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votes
0answers
21 views

Need Help Building An Equation to Find an Angle of Departure for Zeroing on a Rifle Scope

I asked this question yesterday, but the equation ended up not working. I believe I am using it correctly, and I have experimented countless times to no avail. Thus, I am here asking again being even ...
0
votes
1answer
24 views

What area can this question be categorized into?

In a game of 12 players that lasts for exactly 75 minutes there are 6 reserves who alternate equally with starting players. It means that all players, including reserves, are in the game for exactly ...
2
votes
1answer
32 views

Need Help Building An Equation to Find an Angle for Zeroing on a Rifle Scope

My name is Michael, and I am trying to create a small video game. I am only in high school, so my math skills lack which is why I am here to find help from nice people! I am trying to find an ...
1
vote
1answer
40 views

Determinant of a matrix with binomial coefficients.

Let $n \in\mathbb{N}$ and $A=(a_{ij})$ where \begin{equation}a_{ij}=\binom{i+j}{i}\end{equation} for $0\leq i,j \leq n$. Show that $A$ has an inverse and that every element of $A^{-1}$ is an integer. ...
0
votes
0answers
31 views

Is there a better way to determine the function in the integrand?

I need to find $U(z)$ given that $\Delta\ll 1$. $$\int_{-\Delta/2}^{\Delta/2} U(z) \, dz = C$$ $C$ and $\Delta$ are constants. Since $\Delta$ is small I am just using $$ U(z) = C / \Delta\,.$$ It ...
0
votes
3answers
338 views

simplification of an complex exponential equation

There are these steps in a solutions manual I do not follow. I struggle to find any good and problem specific information about this kind of math wizardry on my own. I don't really know what to google ...
0
votes
0answers
7 views

Correct distribution for cell visibility in 3D grid

I have 3D grid of cells. Each cell can be in two states: visible, not visible. The camera is positioned on the side and looks at the grid. Random variable X is defined as a number of visible cells in ...
0
votes
2answers
55 views

How to solve this integral and have arccos(…) as a result?

$$\int {\sqrt{\csc^{2}x -1}} \, d(\cos^2x)$$ I need to solve this integral in order to arrive to a solution that looks like $x= \arccos(...)$ The main substitution is already done, I don't know how ...
3
votes
5answers
15k views

Probability of winning a prize in a raffle

My work is having it's annual Christmas raffle today. 1600 tickets have been sold, and there are 40 prizes to win. I have bought ten tickets. What are the odds I will win a prize? While an initial ...
91
votes
8answers
3k views

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

Edit (June. 2015) This question has been moved to MathOverflow, where a recent write-up finds a similar approximation as leonbloy's post below; see here. Randomly break a stick in five places. ...
1
vote
2answers
434 views

Number of people having shaken hands an odd number of times

This is from a book called USSR Olympiad Problem book: Every living person has shaken hands with a certain number of other persons. Prove that a count of the number of people who have shaken hands ...
1
vote
1answer
354 views

How to calculate per unit costs for multiple items

I had a supplier give me a quote last week that seems very strange, can someone help me out? The quote is for IT hardware, but for simplicity (and anonymity) I'll use apples and oranges: ...
0
votes
1answer
51 views

A question from Hoffman's linear Algebra

the question is on Section 1.4 exercise 7, it says: find all solutions of $$2x_1 - 3x_2 - 7x_3 + 5x_4 + 2x_5 = -2$$ $$x_1 - 2x_2 - 4x_3 + 3x_4 + x_5 = -2$$ $$2x_1 - 4x_3 + 2x_4 + x_5 = 3$$ ...
0
votes
0answers
12 views

Steady state of advection diffusion

I am looking for the non trivial solution to the advection diffusion equation: \begin{equation} \frac{\partial}{\partial x}\left(D_x \frac{\partial c}{\partial x} - uc\right) ...
5
votes
3answers
168 views

How to solve this inequality, with the hypothesis more complicated than the conclusion?

Given $x,y,z \in \mathbb{R}$ and $x,y,z>2,$ I want to show that if, $$\frac{1}{x^2-4}+\frac{1}{y^2-4}+\frac{1}{z^2-4} = \frac{1}{7}$$ then, $$\frac{1}{x+2} + \frac{1}{y+2} + \frac{1}{z+2} \leq ...
1
vote
1answer
79 views

What is the relative strength of each of the players in this game?

This is a real life problem. A group of people meet once a week to play a game between two teams. Each round 2 people are randomly appointed captains. Each captain takes turns picking people to be on ...
7
votes
2answers
162 views

USSR Exam problem

I obtained this problem from here. A car starts from point $A$ towards $B$ at the same time as a motorcycle starts from $B$ to $A$ (but with a lesser speed). At the moment they meet, a second ...
16
votes
2answers
160 views

$xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$ implies $\int_0^1 f(x) \,dx\leq\frac{\pi}{4}$

I want to show that if $f\colon [0,1]\to\mathbb{R}$ is continuous and $xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$ then we have the following inequality: $$\int_0^1 f(x) \, dx\leq\frac{\pi}{4}.$$ The ...
188
votes
16answers
7k views

Optimizing response times of an ambulance corp: short-term versus average

Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. ...
1
vote
4answers
31 views

Simple mod problem

It’s kind of a silly question but I can't find a simple way for finding the value of variable $d$ . $(5*d) \mod 8 = 1$ I normally just do this recursively by saying $d=d+1$ until I get the right ...
0
votes
0answers
17 views

Weak Law of Large Numbers and Central Limiting Theorem problem

From past experience, a teacher knows that the result of an exam is a random variable, with average $75$ and standard deviation $8$. How many students must take the exam to guarantee, with a ...
1
vote
1answer
18 views

Matrix representation in exponential form

So having worked out beforehand that $Λ(v) = \begin{pmatrix} γ&0&\frac{-γv}{c}\\ 0&1&0\\\frac{-γv}{c}&0&γ\end{pmatrix}$ where $Λ(v) ∈ SO(2,1)$ is a matrix representation of a ...
0
votes
1answer
84 views

How do pupils solve 2nd degree equations in Germany? (different from Spain)

I'm from Spain and in Spain the undergraduate pupils learn to solve a 2nd degree (i.e. quadratic) equation using the formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ but years ago I had a colleague who did ...
5
votes
5answers
1k views

Dates with 8 consecutive digits

In many places, dates are written as DD/MM/YYYY. For example, the 25th of April 1736 is written as 25/04/1736. Dates such as this one that use 8 consecutive digits (not necessarily in order) will be ...
1
vote
2answers
63 views

Defining the $L^2$ norm of a vector valued function

I am considering a collection of function of the type, $ f:[0,2\pi]\rightarrow \mathbb{R^2}$. I want to define the $L^2$ norm of the function in that space. I am defining the a norm of ...
1
vote
2answers
30 views

Under what conditions would the function $\prod_{i=1}^{n}{\frac{r_i}{r_i - 1}}$ be decreasing with respect to $n$?

So I know that $$\frac{r}{r - 1}$$ is a decreasing function of $r$. My question is: Under what conditions would the following function be decreasing with respect to $n$? $$\displaystyle ...
1
vote
3answers
33 views

Problem leading simple equations

A sum of Rs. 8.85 is made up of 124 coins which are either 10 paisa coins or 5 paisa coins ; how many coins are there each Note : Rs. 1 = 100 paisas
0
votes
2answers
23 views

Problem leading equations

The question is : "A and B begin to play with 60$ each. If they play till A's money is double B's, what does A win?" Now i tried to solve it like they both have 60\$ each, then A got his money ...