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
votes
1answer
47 views

Transformation of $\log(X)+\log(1-X)$, where $X$ is uniform.

I am trying to calculate the variance of: $$\log(X)+\log(1-X)$$ where $X \sim \mathrm{Unif}(0,1)$. So far, I have tried to use a random variable transformation, i.e. define $Y=\log(X)+\log(1-X)$, ...
0
votes
0answers
28 views

Weird behaviour of CLT's application to binomial.

I am carrying out the simulations of the following experiment for all $n$ in the set $\{1,2,3,...,100\}$. (0) Set $k=0$. (1) Generate $n$ $Bernoulli(0.9)$ trials. (2) Construct estimate $\hat\theta=...
1
vote
1answer
42 views

Solving $x^e =c$ in $\mathbb{F}_{p}$

Find all solutions to the equation $x^3=7$ in $\mathbb{F}_{13},\mathbb{F}_{19}$ and $\mathbb{F}_{35}$. In An Introduction to Mathematical Cryptography (Hoffstein et al), we have that proposition 3.2....
0
votes
1answer
60 views

Show that one equation equals another (trigonometry)

I'm studying for a test and when going through old exams I find this one which I'm not able to solve. Show that $$a^2 = (b−c)^2 + 4bc \sin^2 \left(\frac A2\right)$$ equals $$a^2 = b^2 + c^2 ...
0
votes
2answers
81 views

Complex proof - Not sure where to go from here. (homework)

Knowing $2\pi r =\dfrac{h}{m \left(\sqrt{\frac{e^2}{mr}}\right)}$, How do I prove $r = \dfrac{h^2}{((2\pi)^2m e^2)}$? I started by dividing both sides by $2\pi$ to get $r = \dfrac{h}{m\left(\sqrt{\...
0
votes
1answer
16 views

Given an integer $i\in\{1,\ldots,NM\}$, find its place in a matrix of size $N\times M$?

The integers $N$ and $M$ are positive. Given the matrix $\mathbf{A}=\left[\mathrm{a}_{nm}\right]$ defined as follows: $\mathrm{a}_{nm}=m+(n-1)M$ for all $m\in\{1,\ldots,M\}$ and $n\in\{1,\ldots,N\}$. ...
0
votes
2answers
37 views

How can I solve this nature log equation?

$ln(x+2)=e^{(x-4)}$ Is there any way to solve this equation without graphing or using GDC ? Thank you
0
votes
1answer
25 views

Recover Marginal Distribution subject to a Constraint

I want to identify the marginal of a normal distribution subject to a restriction. Take two normally distributed random variables $x,y$. Their pdfs are denoted by$\phi(x)$ and $\phi(y)$. The moments ...
0
votes
0answers
68 views

In a sweepstakes giveaway scenario, how does having 2 chances to win the same prize affect the overall odds?

In a sweepstakes giveaway scenario where total entries are expected to result in final odds of 1:93,150.685 for/against a single entrant (after adjusting for multiple entries) and can be won by either ...
0
votes
1answer
28 views

Finding the number of multiples [closed]

I have recently been doing problem solving in math, and I came across this problem: Determine the number of positive multiples of $6$ or $9$ or both, less than $1000$. I appreciate any help. Thanks!
4
votes
1answer
81 views

Finding the Determinant of a particular Matrix

I've come across the question of finding the determinant of the $(n\times n)$ matrix, given by $$A:= \begin{pmatrix} x & 1 & 1 & \dots & 1 \\ 1 & x & 1 & \dots & 1 \\ \...
4
votes
2answers
43 views

Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim t\...
-3
votes
1answer
61 views

Counting and Abstract Problem Solving [closed]

Suppose that you have a bucket holds fiv-sev c, and one holds tw-one c. How could you use them to measure out thre c of water?
1
vote
2answers
60 views

Solve equation of inverse functions

I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and $f^{-1}_2(x)$, by ...
0
votes
0answers
49 views

How do I derive the cubic formula? (without substitutions)

I've heard of a number of ways that people have derived a cubic formula (I've even heard of a number of different ways to show the formula itself too). What I want to know is how to derive it without ...
1
vote
2answers
21 views

Survival bias and probability

Imagine the following situation: A new virus is discovered that is believed to have infected 20% of the population. Anyone infected with the virus has a chance of 50% of dying in their sleep every ...
0
votes
1answer
27 views

How can we solve this system of linear inequalities?

Let $c_i$ be a given non-negative integer for all $i\in\{1,\ldots,n\}$. I would like to find the non-negative integers $a_i$ and $b_i$ for all $i\in\{1,\ldots,n\}$ such that: \begin{align} \begin{...
0
votes
0answers
23 views

Probabilities of each waitlist person

Coming from this, $10$ Applicants for a exclusive club membership. I found that you can use total probability to consider existing and old members leaving/returning as members in the club, ended up ...
0
votes
1answer
22 views

Mpg to l/100km conversion problem?

We have two vehicles A: old truck that does 17 mpg (13.84 l/100km) B: old car that does 47 mpg (5.00 l/100km) We are looking to replace one of these vehicles with a new one (of the same size)...
0
votes
1answer
17 views

Finding the rate at which the space diagonal of a cube is decreasing

This is the context of the question: I'm assuming by the space diagonal (although I'm not sure) to be the area of the right-angled traingle created by the diagonal. Let this space be $V$, then we ...
2
votes
1answer
73 views

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$?

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$? I calculated the integrating factor to be $e^x$. Then $e^x y'+ e^x y=e^x |x|$ hence $\frac {d(e^x y)}{dx}=e^x |x|$ hence $d(e^x y)=e^x|x|dx $ ...
3
votes
4answers
88 views

How to find unknowns $w_1,w_2,w_3$ that satisfy $t=w_1f_1 + w_2f_2 + w_3f_3$?

For any $i \in \{1,2,3\}$, let: $w_i \in [0,1]$ is an unknown number such that $\sum_{i \in \{1,2,3\}} w_i = 1$. $t$ is a known number in $[0,1]$. Suppose that $t = 0.8$. $f_i$ is also a known ...
0
votes
1answer
37 views

Let $S=[0,1) \cup [2,3]$ and $f:S \to \Bbb R$ be a strictly increasing map such that $f(S)$ is connected. Which of the following statements is true?

$f$ has exactly one discontinuity. $f$ has exactly two discontinuities. $f$ has infinitely many discontinuities. $f$ is continuous. I know theorems related to connectedness and continuity ...
3
votes
1answer
43 views

solve $54 x + 16 y = 2400$ for integer values of x,y

How to get integer values for x and y that satisfy: $$54 x + 16 y = 2400$$ Someone told me that I can do it using Euclid-Wallis algorithm, but I don't understand it so, if there isn't any else ...
0
votes
0answers
68 views

Reference request for a very particular problem solving skill

I want to start with an apology for a very verbose description of my question but if there is a way to cut it down, please let know and I will do so right away. I have been trying to get better at ...
0
votes
0answers
47 views

Books or website about solving IMO problems

Hey I want to solve IMO problems like the problem in the image below, but I cannot solve the problem or any of the problems in the IMO, so do you guys have some good website or books that teach how to ...
0
votes
0answers
26 views

Parameterization which is closed under addition

Suppose $\beta_1(t)$ and $\beta_2(t)$ are two parametric curves defined on $[0,1]$. Let $\beta_1^*(t)$ and $\beta_2^*(t)$ are two re-parametrized of the above curves. Now, I looking for a re-...
4
votes
5answers
82 views

Why is the solution to $\sqrt{6-5x}=x$ only $x=1$ and not $x=-6$? [duplicate]

I solved the equation $\sqrt{6-5x}=x$ as follows: $$(\sqrt{6-5x})^2=x^2$$ $$6-5x=x^2$$ $$0=x^2+5x-6=(x+6)(x-1)$$ $$x=-6 \quad \text{or} \quad x=1$$ If I plug in $x=-6$ into the original equation, I ...
0
votes
2answers
48 views

I need some help with Geometry. Is this a correct answer to this problem?

Good day, I have a question regarding geometry. I don't know whether my answer is correct because the answer in my book uses a totally different method for solving this particular problem. Here's ...
23
votes
3answers
2k views

Find a thousand natural numbers such that their sum equals their product

The question is to find a thousand natural numbers such that their sum equals their product. Here's my approach : I worked on this question for lesser cases : \begin{align*} &2 \times 2 = 2 + 2\\...
1
vote
0answers
53 views

Variation of the opaque forest problem (a.k.a farmyard problem)

I was wondering about the following variation of the opaque forest problem (see here and there for previous questions) : What is the least length set of segments that will intersect every straight ...
3
votes
2answers
41 views

Combinatorics Question with bridges and inability to cross over each other

Several small villages are situated on the banks of a straight river. On one side, there are $20$ villages in a row, and on the other there are $15$ villages in a row. I would like to build bridges, ...
0
votes
0answers
41 views

How can the lagrange multipliers in a simple constrained cost minimization problem be calculated? (for binding constraints)

Is there a simple algebric way to calculate the shadow prices (lambda) of the binding constraints given below? This is a cost minimization problem dependent on the generation output. The cost of ...
3
votes
1answer
90 views

Which is larger, $e^\pi$ or $\pi^e$? [duplicate]

I don't know how to approach this. I tried expanding $e^{\pi}$ using the power series but that was a dead end since I didn't know what to do with it. I tried estimating if $e \log({\pi})$ was ...
5
votes
1answer
74 views

What is the probability that the upturned faces of three fair dice are all of different numbers?

Three fair dice are rolled ($6$ sides). What is the probability that the upturned faces of the three dice are all of different numbers? I got that the number of possible outcomes total is $6^3$ and ...
1
vote
2answers
46 views

How to solve this equation using logs

How do solve this equation for x using logarithms? $$4^x = 6^x-3$$ If it is not possible using logarithms, please provide another way. Thank you in advance
4
votes
2answers
262 views

Is there any easy way to solve two equations with three unknowns?

Is there a way to solve the below simultaneous equations? One possible solution is $a_1=20.0948$, $a_2=10.0948$, $a_3=6.3448$. The variables are actually dual variables of the binding constraints. ...
0
votes
2answers
59 views

Solving modulo equations with one variable

Given the following equation: $$10 = 4^x \pmod {18}$$ How can one know what are the correct values for $x$ ?
0
votes
1answer
55 views

Why do people say that some problem is hard when they do not actually prove it?

I have read many times in different papers something like the following (I do not remember the exact words though): "The problem is nonlinear non-convex programming problem which is hard to ...
1
vote
1answer
54 views

Launching a Plaintext Attack against Affine Cipher

Update 2 Being new to the world of Stack Exchange I did not realize that there exists a site solely devoted to cryptography. In light of this, I hope someone could help me migrate this question to ...
0
votes
0answers
90 views

Integration of the product of Hermite Polynomial and exponential function

how to proceed with these two integration.. $$\int^0_{−∞}e^{−ax2}H_{2k}(x)dx=?$$ $$\int^∞_{0}e^{−ax2}H_{2k}(x)dx=?$$ where $$H_n(x)$$ is the Hermite Polynomial (physicist's convention).
0
votes
1answer
34 views

PDF of negative $\cos(X)$

Let $Y = - \cos(X)$, then what will be the pdf? Please share if you have any idea. If $Y = \cos(X)$, where $X$ is uniformly distributed in the interval $(0, 2 \pi]$, then the pdf is given by $$...
0
votes
1answer
20 views

Find the lenght of a rectangle between two parabolas

I'm trying to find the length of $PQ$ but the best thing I have done so far is finding that the point $T$ is $(0,4)$, as well as finding the distance between the two turning points to be $6$. Can ...
1
vote
1answer
26 views

Technical meaning of two alike combinatorial problems

I am confused in how to interpret two alike combinatorial problems, because to me they both look the same. These are the problems: How many ways are there to put $24$ distinguishable flags on $18$ ...
1
vote
1answer
24 views

How to find a formula that is true for the given model in the First Order Logic?

I think I might get lost in the definitions. I am not sure if this is the right way to deal with models and formulas in the First Order Logic. I am not looking for the solution for this particular ...
0
votes
2answers
43 views

How many integers less than 2015 are multiples of 2 or 3 (or both)?

Here is what I did. To find all the multiples of 2 that is less than 2015 all we need to do is divide by 2. The same can be done for multiples of 3 that is less than 2015: 2015 / 2 = 1007 (...
0
votes
2answers
30 views

Basic probability exercise

Consider the problem of selecting two candidates from a group of five persons for a job. Knowing that the candidates differ in their degree of readiness (1 is the best prepared, 2 is less prepared ...
0
votes
2answers
33 views

Calculate number of sides of cylinder so each side is a certain width

I'm working on a video-game and as part of the level, I need to create one half of the room curved. For the cylinder, all sides should be of width 450cm, and the cylinder will have radius of 1475cm, ...
-2
votes
1answer
55 views

Balkan Olympiad in Mathematics 2001 [closed]

Where can I find the solutions of the problems from the Balkan Olympiad in Mathematics 2001, Belgrade?
0
votes
1answer
57 views

Finding the least number of dots to add into a 10x10 grid

I have a 10x10 grid where are some dots. What is the least number of dots that I need to add in order to have 3 dots in every row and column have odd number of dots in every row and column have ...