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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2answers
261 views

Probability in a Dice Game (Zombie Dice)

In the game of Zombie Dice (Rules) there exist 13 dice: 6 Green - 3 Brains, 2 Footprints, 1 Shotgun 4 Yellow - 2 Brains, 2 Footprints, 2 Shotguns 3 Red - 1 Brain , 2 Footprints, 3 Shotguns A ...
9
votes
3answers
244 views

On solutions of an equation in $\mathbb{Z}_3$

For integer numbers $x_1, x_2, y_1, y_2, y_3$ suppose that $$ x_1 + x_2 \equiv y_1 + y_2 + y_3 \pmod 3. $$ For $k=0, 1, 2$ define $$ s_k = \Big| \{ y_i \,|\, y_i \equiv k \pmod 3 \} \Big| - \Big| ...
1
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1answer
63 views

solution verification

here is solution of my old question but i can't see it would someone explain to me the principal idea and what he wants to show Solution from ...
1
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0answers
57 views

How would I find this constant?

I have this equation, and I'm not sure how to solve for the constant $\nu$, since everything else is known: $$\begin{equation} a + \sqrt{a_i + 4 b_i \nu} + \sum^N_{j=1} (\sqrt{a_j + 4 b_j \nu}) ...
1
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2answers
45 views

Simplification ideas

Looking for a neat simplification idea to be able to solve for $x$ analytically in the expression below: $$S=k\tan x-Bk^2\frac{1}{\cos^2x}$$ where $\{S,k,B\}\neq0$ and $\in \mathbb{R}^+.$ Of ...
2
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8answers
130 views

Prove $4^k - 1$ is divisible by $3$ for $k = 1, 2, 3, \dots$

For example: $$\begin{align} 4^{1} - 1 \mod 3 &= \\ 4 -1 \mod 3 &= \\ 3 \mod 3 &= \\3*1 \mod 3 &=0 \\ \\ 4^{2} - 1 \mod 3 &= \\ 16 -1 \mod 3 &= \\ 15 \mod 3 &= \\3*5 ...
10
votes
2answers
334 views

Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components

Start with $n$ paiwise different integers $x_1,x_2,...,x_n,(n>2)$ and repeat the following step: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
0
votes
1answer
26 views

sides of a rectangle given a ratio and a surface

I am trying to find the sides of a rectangle given a ratio and a surface area. Here is where i am: Given the ratio formula where m:n height * (m / n) = width Given the surface is width * height = ...
1
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2answers
122 views

How to show that $3^x+4^x=5^x$ has only one solution? [duplicate]

How to show that $3^x+4^x=5^x$ has only one solution? Thanks in advice.
2
votes
2answers
100 views

How can I understand solving the equation?

$$\begin{align} &\left[(\sqrt[4]{p}-\sqrt[4]{q})^{-2} + (\sqrt[4]{p}+\sqrt[4]{q})^{-2}\right] : \frac{\sqrt{p} + \sqrt{q}}{p-q} \\ &= ...
0
votes
1answer
50 views

Find smallest $x$ such that $a^x \equiv b \bmod p$

Problem: How do we find smallest $x$ such that $a^x \equiv b \bmod p$, where $p$ is a prime and $1 \le b,a \le p$ and $a$, $b$, and $p$ are given and fixed. If there is no such $x$, how do we check ...
1
vote
1answer
58 views

Explain the result of this urn problem?

Suppose n balls are distributed in m urns. The probability that the first r urns receive k balls is $$\frac{\binom{n}{k}r^k(m-r)^{n-k}}{m^n}$$ I am most confused about the $r^k$ part. I know there ...
1
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1answer
33 views

Find the equation for the line that satisfies the following:

being parallel to the plane $P:x+2y-3z=1$ intersects orthogonally with the line $k:(x,y,z)=(1+2t,t,-1)$,$t\in R$. intersects with the x-axis in any point. I must be missing out on some information, ...
2
votes
2answers
26 views

Solve for a variable in the power when the base are two different values

I would like to solve for $C$ $$7^C \times 2^{n-C-1} \le \frac{2^n}{100}$$ Real questions. The different base is really throwing me off. I got up to $$7^C 2^{-C} \le \frac{1}{50} $$
0
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0answers
84 views

Polar coordinate for complicated curves

In general polar representation of a closed curve is done by coordinate $(\theta,r(\theta))$, $\theta\in (0,360)$. When working with real data, I got a closed curves whose graph looks like the below ...
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1answer
44 views

Systems of Modular Equations

Given the following systems of modular equations: $$ 4^{x}+x^{2}\equiv 1 (mod \: 6)$$ $$7x\equiv 3 (mod \: 9)$$ $$15x\equiv 10 (mod \: 25)$$ Which x solves the system ? It is possible to make ...
0
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2answers
46 views

Number of complex solutions

Given the following equation: $$ x^{259}=1 $$ $$ x^{413}=1 $$ How many complex solutions for x have? Thanks
1
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1answer
422 views

Maximum vertical distance between the line $y = x + 30$ and the parabola $y = x^2$ for $−5 ≤ x ≤ 6$

What is the maximum vertical distance between the line $y = x + 30$ and the parabola $y = x^2$ for $−5 ≤ x ≤ 6$? This is what I did but didn't work: Set $y_1=x+30$ and $y_2=x^2$, plugged ...
4
votes
1answer
282 views

Getting stuck on difficult problems.

First, a little background: I hope to go to graduate school in mathematics, but for financial reasons I will be unable to go back to school any sooner than the fall of 2016. However, since I feel ...
0
votes
1answer
34 views

Conditional expected value of a product of poisson processes

For $0<s_1<s_2<t$ evaluate conditional expected value $$E[N\left( s_1 \right) N\left( s_2 \right)|N\left(t\right)],$$ where $N\left( t\right)$ is Poisson process. Here is what I've got. By ...
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votes
4answers
98 views

How can I solve the system of equations?

How can I solve the system of equations? $$\begin{cases} x^2 y^2+12 x y^3-18 x y-18y^4-4 y^2+27=0,&\\ x^2 y^2-3 x y^3-3 x y+5 y^2=0. \end{cases}$$ I have not any idea to solve.
12
votes
2answers
962 views

Chess board combinatorics

STATEMENT: A dolphin is a special chess piece that can move one square up, OR one square right, OR one square diagonally down and to the left. Can a dolphin, starting at the bottom-left square of a ...
0
votes
0answers
313 views

Find the probability that event $A$ is right before $B$.

Problem: Let $S$ be a sample space of an experiment and $S = \left\lbrace A,B,C\right\rbrace $, where $P(A)=p$, $P(B)=q$, and $P(C)=r$. The experiment is repeated infinitely, and it is assumed that ...
0
votes
2answers
1k views

To find two sides of a triangle when it is circumscribed a circle

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the ...
1
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2answers
125 views

switch the colour until only one black square is left

Consider a standard chess board (8 × 8 squares). In each move, you pick one row or one column and switch the colours of all 8 squares (from black to white or from white to black). Is it possible to do ...
0
votes
3answers
165 views

Getting 90 degree coordinate of 2 coordinates that you know

I have 2 coordinates and I need to find the third with a 90 degree angle. How could I do this? ...
1
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1answer
111 views

Where is the fixed point? — Matlab is cluless too

Consider the differential equation $$\dot{k}(t)=f(k)-(r+t_1)k-f(1-k)+(r+t_2)(1-k)$$ where $k,t_1,t_1\in[0,1]$, $r\in\mathbb{R}$ and $f:[0,1]\to\mathbb{R}_+$. I'd like to solve for the fixed point ...
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votes
2answers
46 views

Quadratic Equations GRE Quants

It would be very useful if someone can give me an answer to this question with a proper explanation. One of the factors of the equation $x^2 +9x + c$ is $(x+11)$, where $c$ is a constant. Which of ...
2
votes
1answer
44 views

How can I solve an exponential equation of the following type?

I have an equation of the form $$ \frac{a^x}{d_1^x} + \frac{b^{x/2}}{d_2^x} = 1, $$ which I have already rewritten to $$ a^xd_2^x+d_1^xb^{x/2}-d_1^xd_2^x = 0. $$ However, I seem to be stuck here. ...
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votes
5answers
878 views

Give the number of solutions of $x+y+z = 30$, for $4 \leq x \leq 14$, $3 \leq y \leq 17$, $10 \leq z \leq 25$. [closed]

How would I find the number of solutions with both upper and lower bounds? Can anyone give a step by step way to solve this problem? This is question is in preparation for my discrete math final, so ...
3
votes
0answers
110 views

Shortlist of problems in linear algebra

A while ago I remember seeing a very nice shortlist of problems in linear algebra. It was a list of about 40-50 problems. The idea was that if you solve them, you learn linear algebra very well and ...
4
votes
2answers
139 views

Finding the convergent value of a recursion similar to Arithmetic-Geometric Mean recursion

The sequence is defined as follows : Start : $(x_0,y_0)$ with $ 0 < x_0 < y_0 $ Step : $x_{n+1} = \frac {x_n+y_n} {2}$ , $y_{n+1}= \sqrt{x_{n+1}y_n} $ Find $\lim_{n\to \infty}(x_n,y_n)$ . ...
4
votes
2answers
774 views

Using Sticks and Stones for Counting number of Ways

From the first twenty positive integers, how many ways can we select 6 integers so that no two integers from the six chosen ones are consecutive? I tried using sticks and stones, but my thought ...
3
votes
1answer
91 views

Find all positive solutions of the system of equations

Find all positive solutions of the system of equations $x_1+x_2=(x_3)^2$ , $x_2+x_3=(x_4)^2$ , $x_3+x_4=(x_5)^2$ , $x_4+x_5=(x_1)^2$ , $x_5+x_1=(x_2)^2$ What i have done : ...
1
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0answers
32 views

When setting up a probability problem, when is it appropriate to use conditioning?

I understand the principles of conditioning and its rules, but when do I decide if a problem will be easier using conditioning versus determining through other methods? I'm teaching myself probability ...
0
votes
1answer
310 views

Use of Delaunay Triangulation and Voronoi Diagram to find alpha shape using Edelsbrunner's algorithm

I am learning how to find the shape of a set of points in 2-D. I understand that Alpha Shape method is a good way to find the shape of a set of points. Alpha Shape was originally introduced by H. ...
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0answers
45 views

The Jugs of Water Problem - with constraints

Given three jugs containing any amount of water such that a1 <= a2 <= a3 and each jug is large enough to contain all the water, show that it's possible (or not) to empty one jug. Only ...
3
votes
2answers
386 views

Does an elegant solution exist for this trigonometric equation?

I'm trying to solve this: $\cos ^{-2}x + A\tan{x} = B$ Wolfram alpha spits out an incredibly long and convoluted solution for x. Is there no simple, straightforward analytical way to solve this?
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2answers
40 views

Is this proposition posible? [duplicate]

In a board, you have $13$ White round pieces, $15$ Black round pieces, and $17$ Red round pieces. In each round you can choose two different color pieces and change them with two other pieces of ...
0
votes
1answer
71 views

How do you figure out the formula to convert between units?

I know that to, for example, convert from Fahrenheit to Celsius you subtract 32 and then divide by 1.8. I'm interested in how this type of formula can be developed. So, given two different sets of ...
1
vote
1answer
44 views

Modular arithmetic and using in well-ordering principle

I need to prove the following, but I do not know how to go about it. If $$ (*)\:\:\: x^{3} - y^{3}= 3^{n} $$ Then $$ x \equiv 0 (mod 3) \:\: and \:\:\: y \equiv 0 (mod 3)$$ In addition, ...
0
votes
1answer
38 views

Solve equation with two unknowns (maybe modulo)

Given the following equation: $$ x^{2} - y^{2}=17, \quad 0\neq x,y\in \mathbb N$$ I know for example that one solution is $x=9$, $y=8$, but I do not know how to get it.
3
votes
1answer
981 views

Find a seven digit number which describes itself

Find a seven digit number which describes itself. The first digit is the number of zeros in the number. The second digit is the number of ones in the number, etc. For example, in the number 21200, ...
1
vote
1answer
113 views

Question about “linear programming problem” in reference to joint pmf

I'm working on a homework problem and I'm not totally sure what the question is asking... The question reads: "Consider the linear programming problem: maximize $Ax_1+Bx_2$ subject to $x_1+x_2\leq ...
2
votes
3answers
89 views

Finding roots of a quartic

How do I find the roots of the equation $$(x+3)^5-(x+1)^5=7$$ I tried opening it up, it turns into a ugly quartic which doesn't factor. I don't know what to do next. Please help me out.
2
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0answers
175 views

Fundamental Matrix

Determine $\phi(x,0)$ for $A(x)=\begin{pmatrix} -1 & \cos(x) \\ 0 & -1\end{pmatrix}$, where $\phi(x,0)t_{0}$ is a solution of $\frac{d}{dx}t(x)=A(x)t(x)$. I am not entirely sure as to ...
2
votes
1answer
57 views

Possible values of $\gcd(a+b, a\times b)$

Main Question: Let $N \in \mathbb{N}$. What are the possible values of $\gcd(a+b, a\times b)$ given that $\gcd(a,b) = N$? Fact 0. If $\gcd(a,b) = N$, then $N \leq \gcd(a+b, a\times b) \leq ...
0
votes
1answer
34 views

Simple probabilistic expression

For the following expression: $$ \prod_{i=0}^{n-1} \frac{2n-i}{3n-i} $$ I'm trying to get a simple expression, unsuccessfully. Many thanks, Jonathan
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0answers
26 views

Efficient method to calculate passes (rises and sets) for satellites

There is a function describing the characterisic elevation of ISS seen from an observers horizon. Calculating of an elevation at one time is pretty expensive. So I wanna try to avoid naive iterating ...
0
votes
2answers
151 views

Number of unique Team parings given 10 players and 2 teams

I yammer a wee bit too much, feel free to skip to TLDR unless you want more background as to why I care about this problem. I was just thinking that it would be a fun to figure out the best 5 players ...