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
12 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
50 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
61 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
0answers
11 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 ...
2
votes
1answer
36 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
45 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
33 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
22 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 ...
2
votes
1answer
64 views

Proof that there are no solutions this equation. (3 variables, Square root and Sine) [on hold]

Hypothesis: There do not exist three different positive integers $a,b,c$ such that $$ -\sqrt{ab}\cdot \sin(p \cdot (a-b))+\sqrt{ac}\cdot \sin(p \cdot (a-c)) -\sqrt{bc}\cdot \sin(p \cdot (b-c)) =0 $$ ...
4
votes
5answers
64 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
35 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 ...
19
votes
2answers
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 + ...
1
vote
0answers
31 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
36 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
39 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
81 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 ...
-1
votes
0answers
16 views

Erlang and Poisson Distribution [closed]

Erlang distribution is a series of Exponential distributions random variables having same parameter(arrival rate per unit time), now instead of taking all series of exponential distributions in Erlang ...
5
votes
1answer
44 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 $6^3$ ...
1
vote
2answers
43 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
246 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
52 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
52 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
52 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
69 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
19 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 ...
0
votes
0answers
24 views

What is the probability of having the second child as a boy? [duplicate]

A couple had their first child, a boy, born on Wednesday. What is the probability that the second child is also a boy ? I thought it was a simple case where the probability is just 1/2 because there ...
0
votes
0answers
40 views

Show that the integers $a$ and $b$ can be chosen such that $ ha-kb=1$ holds for any given integers $h$ and $k$

During a longer calculation I encountered a problem where I need to show that one can pick two integers $a$ and $b$ such that $ha-kb=1$. Here $h$ and $k$ are two given integers. We have to assume ...
0
votes
0answers
12 views

How to determine the right initial and boundary conditions of the nonlinear PDE system

The nonlinear PDE system is from a research paper in 2000. The authors solved the system by using an ordinary differential equation integrator in FortranVariable-coefficient Ordinary Differential ...
1
vote
1answer
24 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
23 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
42 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
28 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 ...
0
votes
2answers
28 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, ...
-1
votes
1answer
45 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
41 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 ...
0
votes
2answers
49 views

How to find the maximum of this function $\dfrac{(1+x+y)^2}{(1+x)(1+y)}$?

The function with two variables is defined as follows: $$f(x,y)=\dfrac{(1+x+y)^2}{(1+x)(1+y)},$$ for all $0<x_{min}\leqslant x\leqslant x_{max}<\infty$ and for all $0<y_{min}\leqslant ...
0
votes
5answers
49 views

380 is what percent less than 600?

I'm New to percentages and this sum is confusing me a bit. If the question was " 380 is what percent of 600" , I would have converted it to an equation as follows.. 380 = ?% × 600 'n then I could ...
2
votes
0answers
26 views

expectation half-normal distribution or expectation Truncated Normal Distribution [duplicate]

I want to calculate integrals $$ \begin{split} \int_0^\infty x \exp\{ ax-b x^2\} dx &= \int_0^\infty x\exp\{-b(x^2-\frac{a}{b}x)\}dx\\ &= ...
1
vote
4answers
70 views

how to find $t$ from $2t^2-0.01t^4=100$?

how to find $t$, from $2t^2-0.01t^4=100$? I was guessing may be I can take $t^2$ common but if it is so so why cannot we take $t$ common in other cases? I mean, for example: $t^2+4t=-4$ why can we ...
0
votes
0answers
47 views

How to simplify a problem with two variables?

I am trying to solve this problem. Let $\Delta$ be a positive number. I would like to find the values of $x$ and $y$ such that: $$ \left(1+\dfrac{x}{1+y}\right)\cdot\left(1+\dfrac{y}{1+x}\right) ...
4
votes
4answers
310 views

Can you find the treasure??

My big bro gave this problem one week ago. I could not still solve it.Please HELP. STORY A man was just looking for items in his store room. Suddenly he found a map , which showed then it stated ...
13
votes
0answers
107 views

Every natural number in binary can be cut and added so that it is a power of $2$? [duplicate]

I was watching a google techtalk with Donald Knuth and he mentions for every binary number $\overline{a_1a_2a_3\dots a_n}$ there exists $c_1<c_2<\dots <c_r=n$ so that: ...
0
votes
2answers
20 views

Regarding a Markov chain example state space $\{0,1\}$

I have trouble formulating a question. The set up is $(X_n)$ is a Markov chain with the state space $\mathcal{S} = \{0,1\}$. We know $X_0 =1$ and $X_2=1$ and the transition probability matrix, $p$. ...
0
votes
1answer
18 views

Finding other point's values on a line knowing one point and distance between two others

So I have a normal line that has points $A, B, C$ and $D$ on it (same order). The distance between $A$ and $D$ is $392$. Point $B$ is equal to $293$. $$CD = 2AB = 4BC$$ Picture of the problem (drawn ...
2
votes
0answers
133 views

Prove that $(a-b)^n\mid (a^n-b^n) \iff n=1$ under given conditions

Suppose that $a,b,(a-b)$ are pairwise co-prime (i.e. $a\perp b\perp (a-b)\perp a$), and that $\frac{a}{2}<b<a$, where $a$ and $b$ are both positive integers greater than $2$. Let $n$ be odd. ...
0
votes
1answer
61 views

Paradox of Random Natural Numbers

I've got a question about a game taken from a book called Rachunek prawdopodobieństwa dla (prawie) każdego by Jacek Jakubowski and Rafał Sztencel. Adam and Bolek have a machine that generates a pair ...
0
votes
2answers
26 views

Cuboid room, hooks and strings proof

I'm trying to do the following problem: In a cuboid shaped room a hook is placed in the centre of each wall, the floor, and the ceiling. Every pair of hooks has either a piece of red or blue ...
0
votes
0answers
30 views

How to invert a transformation

I've come across a recursive equation involving vectors. You basically have one starting point $P = (x, y)$ and you transform it to another point $P'=(x', y')$ with the following equations $$ x' = x ...
2
votes
3answers
61 views

Where did two dollars go?? [duplicate]

One of my school friends gave me this sum.Its basically a story formated into a sum STORY There were 3 friends. They each gave 20 dollars to buy a radio. They bought the radio for 60 dollar. Later ...