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
33 views

Solving special equation [on hold]

How can I find $x$ from the following function while we know that $a,b, c , d$ are constants? $$y= (a b x^{b-1}+ c d x^{d-1}) e^{-ax^{b} - c x^{d}}$$
3
votes
3answers
45 views

Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$?

It is the first time I met such a question: Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$? Intuitively I think $f(n)$ would gradually become larger as $n$ gets ...
0
votes
2answers
48 views

A tricky diophantine equation with factorials

I am being unable to solve this diophantine equation. Does anyone have any suggestions. Let $n$ and $m$ both be non-negative integers. Find all solutions to $$n(nm - 2)! = (n!)^m$$ How would one ...
2
votes
2answers
69 views

Show that $1+(x_1x_2…x_n)^{\frac{1}{n}} \leq [(1+x_1)(1+x_2)…(1+x_n)]^{\frac{1}{n}}$

Show that $1+(x_1x_2...x_n)^{\frac{1}{n}} \leq [(1+x_1)(1+x_2)...(1+x_n)]^{\frac{1}{n}}, \forall x_i \geq 0, i = 1,2,3...,n$ So, I have to make this function something like this: ...
8
votes
4answers
335 views

How To Develop A Higher Mathematical Aptitude? [closed]

First off I must say I'm pretty blown away by the vast majority of the people in this forum. I do aspire to reach the knowledge of mathematics as shown on the site, but honestly it's a little daunting ...
10
votes
4answers
169 views

A circle with $500$ points in its interior

Given any $1000$ points in the plane, show that there is a circle which contains exactly $500$ of the points in its interior, and none on its circumference. How do I approach this problem? I feel ...
12
votes
1answer
70 views

Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero coefficients than $f(x)$?

I saw this problem on a problem set and I have absolutely no idea how to proceed in a feasible way. Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero ...
3
votes
2answers
47 views

How To Tackle Trigonometric Proofs involving $4$th and $6$th powers?

How do I prove that $\cos^4A - \sin^4A+1=2\cos^2A$ $\cos^6A + \sin^6A =1-3\sin^2A\cdot\cos^2A$ I was going through a very old and very rich book of Plane Trigonometry to build a nice foundation for ...
1
vote
0answers
35 views

A harder long division puzzle than the first; what should “Algebra I” solution look like?

Here's another problem, significantly harder than the first, but still accessible to target audience. The statement of the problem (i.e., northwest corner only) comes from a PennyDell puzzle magazine: ...
6
votes
3answers
87 views

“Long-division puzzles” can help middle-grade-level students become actual problem solvers, but what should solution look like?

This is my first post. I hope it's acceptable. EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and ...
0
votes
0answers
62 views

Using Statistics to Detect Cheaters? [closed]

I'm working on an online open-source game and unfortunately, players are now cheating (they're using speedhack and other stuff). I want to be able to detect cheating using statistics from the central ...
-4
votes
2answers
59 views

Alternate solutions to seemingly simple problem [closed]

$$a+b=ab$$ Are there more than $1$ solutions to this problem? Can you prove it?
0
votes
0answers
42 views

Substitution method for solving recurrences

I am new to math.stackexchange and I need help understanding Substitution method for solving recurrence. This is the original problem T(n) = 2T(⌊n/2⌋) + n Our guess is T (n) = O(n lg n) so we need ...
1
vote
2answers
44 views

Application of Euler's theorem apart from finding last digits of huge numbers

I am looking for clever applications of Euler's Theorem. On browsing the internet, I see that nearly all the applications of the theorem asks for finding last few digits of a huge number. The only ...
1
vote
1answer
76 views

Calculating $a^{\sin x}=x^{\ln a}$ [closed]

$$a^{\sin x}=x^{\ln a}$$ Can you help me solve this equation for $x$ in terms of $a$?
4
votes
2answers
91 views

Is the following PDE boundary value problem well-posed?

My Question Is the following Poisson boundary value problem well-posed, as stated? If so, how could I go about solving it? If not, what would it need to be well-posed? Does it satisfy the ...
0
votes
0answers
28 views

Looking for problems which can be solved by the similar technique

While browsing on internet for different proofs of Fermat's theorem on sums of two squares, I came across Zagier's "one-sentence proof" which seems to be the most elegant and short proof. It invokes a ...
3
votes
1answer
36 views

Diophantus' Lifespan

Today I saw Diophantus' Epitaph. For those of you who don't know it and don't feel like googling: 'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God ...
-1
votes
1answer
40 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
76 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
44 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
24 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 ...
1
vote
1answer
20 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 ...
-2
votes
1answer
46 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
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, ...
3
votes
5answers
99 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 ...
-2
votes
1answer
78 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 $w = \log M +y$. ...
4
votes
4answers
91 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
410 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
57 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
147 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
26 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
26 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
190 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
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
25 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
34 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
42 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
11 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
56 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
53 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
13 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
172 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
171 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
32 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
89 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 ...