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
1answer
158 views

Additive function $f: \mathbb{Z}^\infty \rightarrow \mathbb{Z}$ is zero everywhere.

Let $f: \mathbb{Z}^\infty \rightarrow \mathbb{Z}$ be an additive function ($f(x+y)=f(x)+f(y)$ for every $x,y \in \mathbb{Z}^\infty$). In addition for every $x=(0,\dots, 0,1,0, \dots)$ we have ...
1
vote
2answers
57 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
242 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
vote
1answer
61 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
vote
0answers
56 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
vote
2answers
37 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
votes
8answers
103 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 ...
9
votes
2answers
301 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
0answers
24 views

Prove existence of (Nash) equilibrium

My question is about proving the existence of Nash equilibrium for a game involving two players. $x$ is player 1's strategy and $y$ is player 2's strategy; both strategies are continuous. For each ...
0
votes
1answer
18 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
vote
2answers
91 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
97 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
0answers
35 views

Match this urn problem to a distribution

An urn initially contains r red balls and b black balls. A holding area outside the urn initially contain no balls. Balls are randomly chosen from the urn and: the chosen ball and the balls in the ...
0
votes
1answer
47 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
53 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
vote
1answer
28 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
18 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
votes
0answers
33 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 ...
0
votes
1answer
31 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
votes
2answers
45 views

Number of complex solutions

Given the following equation: $$ x^{259}=1 $$ $$ x^{413}=1 $$ How many complex solutions for x have? Thanks
1
vote
1answer
66 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 ...
2
votes
1answer
114 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
25 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 ...
0
votes
4answers
90 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
722 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
59 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
89 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
vote
2answers
99 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
0answers
34 views

Is there an analytical solution to this system of equations?

$\alpha_i b_i (1-t_i) k_i^{b_i - 1} = \lambda \text{ and } \sum_i \alpha_i k_i =1$. $k_i$ are the variables--they should all take positive values. There may be an arbitrary number of them. This ...
0
votes
0answers
49 views

Minimize sum of squared error

I have an array of real numbers, I want to partition them into k sets. In each set, I calculate the sum of squared error. Then, I add up all the sum of squared error for all the set. I want to ...
0
votes
3answers
55 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? ...
0
votes
0answers
40 views

Partition partition with constraint of equal size

I see the problem here Polynomial complexity algorithm of partition problem with sets of equal size This is the well know partition problem but with constraint that the size of both sets must be ...
1
vote
1answer
58 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 ...
0
votes
2answers
34 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
39 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. ...
3
votes
1answer
53 views

How to deal with long and tedious logic problem? [closed]

I am always pretty bad at logic problems. Because most of the logics used aren't really logical (to me)So, as you might think, a long logic problem only adds to it already boring nature. The ...
0
votes
5answers
449 views

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

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
70 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 ...
3
votes
2answers
117 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
162 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
65 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
vote
0answers
26 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
107 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. ...
1
vote
0answers
22 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
381 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?
0
votes
2answers
38 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
38 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
39 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
28 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
172 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, ...