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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4
votes
4answers
81 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 ...
4
votes
3answers
334 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
3answers
51 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
6answers
131 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 ...
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 ...
-1
votes
0answers
63 views

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

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 ...
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
156 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 $x\in\mathbb{C}$ ...
-1
votes
0answers
20 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 ...
-1
votes
1answer
17 views

What is the answer to this problem solving question? [closed]

Saul plays a game where he scores 4 points for a hit and -6 for a miss. If he plays 20 round and scores 30 points. How many times did he miss?
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
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
54 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 ...
0
votes
1answer
50 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 ...
7
votes
2answers
161 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 ...
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
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 ...
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 ...
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 ...
2
votes
3answers
32 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 ...
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 ...
0
votes
2answers
23 views

System of equations that I'm having trouble with

$a/(x+y) - b/(x-y) = 1$ $b/(x+y) + a/(x-y) = (b^2-a^2)/2ab$ The answer to the values of $x$ and $y$ are given as $x=a-b, y=a+b$. How is that achieved?
1
vote
0answers
20 views

General and sufficient condition of independence

I'm having troubles with this proof: Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean and unit standard deviation. For $(a_0, a_1, ..., a_r)$ a sequence of $r$ real numbers ...
1
vote
1answer
65 views

$\mathbb{A}^2\setminus (0,0)$ is not affine

I want to prove that $X = \mathbb{A}^2\setminus (0,0)$ is not affine. My attempt: If $\Bbbk[X] = \Bbbk[x,y]$ then $X$ is not affine since $(x,y) \subset \Bbbk[x,y]$ is a proper ideal, but $V(x,y) \cap ...
4
votes
1answer
43 views

Inverse Fourier transform of $\frac{1-e^{-2\pi ift}}{2\pi if}$

I would like to calculate the inverse Fourier transform of the following $$H(f) = \frac{1-e^{-2\pi i f t}}{2\pi i f}$$ Can anyone tell me and explain to me how to do that? I don't want just an ...
2
votes
0answers
40 views

Reference request - Problem book by subject

I'm looking for good problem textbooks for self-study. I know only of two of this sort: "Introduction to Measure Theory" by Terry Tao, and "Problems in Algebraic Number Theory" by Esmonde and Murty. ...
1
vote
1answer
60 views

Strange sum of random variables

So guys, I'm having this hard proof to solve in probability. I don't really know how to tackle it! Hope that someone can help. Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean ...
3
votes
3answers
94 views

What is the value of $a^4+b^4+c^4$?

Consider $a,b,c$ such that $a+b+c =1, a^2+b^2+c^2=2$ and $a^3+b^3+c^3=3$. Find the value of $a^4+b^4+c^4$, if possible. Trial: I observe that \begin{align} a^4+b^4+c^4 ...
1
vote
1answer
12 views

Solving a matrix for color manipulation

I'm making an application that deals with color transforms. The idea is that if you give it an RGB color and apply a color matrix transform it outputs another color. In this case I'm giving the color ...
1
vote
2answers
48 views

Can I use eigenvalues to find the inverse of a vector?

I have two 1D matrices (say dimension 1xn) called A and B. Multiplying these: A . B = M. Where M is a scalar. Knowing B and M, can I find A? One cannot take the inverse of a vector, but is it ...
3
votes
4answers
385 views

SAT Maths Question About Fractions

Whilst revising, a problem caught my eye and I cannot seem to find an answer. I am usually bad at these types of questions. On a certain Russian-American committee, $\frac23$ of members are men, ...
1
vote
1answer
53 views

Kill the creeps with minimum cost

Oz plays popular ARTS Dota 2. Invoker is one of the favourite Oz's heroes. Oz's skills are not perfect yet, so he uses only two spells - SunStrike and Tornado. Each of these spells takes some mana ...
9
votes
2answers
154 views

Proving $\sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2}$

I've been going through some of my notes when I found the following inequality for $a,b,c>0$ and $abc=1$: $$ \begin{equation*} \sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2} ...
2
votes
5answers
219 views

Help With SAT Maths Problem (Percentages and Numbers)

I usually solve SAT questions easily and fast, but this one got me thinking for several minutes and I cannot seem to find an answer. Here it is: In 1995, Diana read $10$ English and $7$ French ...
16
votes
2answers
159 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 ...
0
votes
0answers
28 views

Movement of birds - Acceleration, Velocity, Time and Displacement. Needed for an assignment

Hi so there are a quandary of birds sitting on a tree.There are $3$ teams observing the movement of the birds. Team $1$ observes that on their first flight the birds move a short distance across a ...
4
votes
3answers
160 views

Numbers with 2015

I like to build math problems; to solve the one below I should first find a certain square and use it in my solution. I would want to know if anyone can solve this problem otherwise. Thanks. ...
0
votes
0answers
30 views

A little bit more difficult problem regarding rooted plane trees

A question regarding rooted plane trees bothers me. We know that the number of rooted plane trees with $n$ nodes equals to $n-{th}$ Catalan number, that is $|Tn| = Cn$. But what is this number if we ...
1
vote
3answers
46 views

Simplifying $\Big[\dfrac{5-\sqrt{a}}{5+\sqrt a}-\dfrac{\sqrt a+5}{\sqrt a-5}+2\Big]^{-2}$

Simplifying $$\Big[\dfrac{5-\sqrt{a}}{5+\sqrt a}-\dfrac{\sqrt a+5}{\sqrt a-5}+2\Big]^{-2}$$ When I try, the numerator cancels out to $0$, yet the answer sheet says $(25-a)^2/10000$. Where am I going ...
1
vote
1answer
21 views

Fitting the closest coefficients in a system of millions of simultaneous equations?

I don't really know the correct terminology to describe this, but let's say we have many values of $(x_n, y_n, z_n)$. Also let's say that our description of 'many' means that $i$ ranges from $1$ to ...
2
votes
4answers
74 views

Solving equations with $x^x$ on any given side [duplicate]

How would you solve such an equation if it's infeasible to just start trying different $x$ values? Example: $$x^x = 6.$$
1
vote
4answers
33 views

Inequality for sides and height of right angle triangle

Someone recently posed the question to me for the above, is c+h or a+b greater, without originally the x and y lengths. I used this method: (mainly pythagorus) $a^2+b^2=c^2=(x+y)^2=x^2+y^2+2xy$ ...
1
vote
1answer
38 views

Help Obtaining Numerical Approximation of Lambert W Solution

I am studying a particular generating function $$\frac{2e^x}{e^{2x}+1+2x}$$ and I thought I would try to solve the equation $$e^{2x}+1+2x=0$$ to determine for what value of $x$ if any the function ...