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
21 views

Find the probability for … [duplicate]

Suppose we uniformly and randomly select permutations from the 20! Permutations of 1, 2, 3,..., 20. What is the probability that 2 appears at an earlier position than any other even number in ...
1
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1answer
37 views

Defining addition of vectors of different dimensions

While doing real data analysis I came up with a problem. I have given lots of efforts to solve it and could not succeed. Here is the problem: Suppose, we have a set of vectors ...
0
votes
1answer
40 views

how to reduce a fraction?

I solved expression and saw this solving, but I didn't see the way to reduce one. $$\begin{align}\frac{a+2\sqrt{ab}-3b}{ab(a - \sqrt{ab} - 3\sqrt{ab -3b})}=\frac{1}{ab}\end{align}$$ Can you show me ...
2
votes
1answer
37 views

Sequence pattern question

I have the following question. Let $S_1$ be the sequence of positive integers $1,2,3,4,5 , \ldots$ and define sequence $S_{n+1}$ in terms of $S_n$ by adding $1$ to the integers of $S_n$ which are ...
1
vote
2answers
41 views

Making up groups of Coins [duplicate]

In how many ways can a group of 100 coins be made up from 50,20,10,5,2 or 1 coin(s) respectively? An alternative way of phrasing this would be how many ways can a group of 100 coins be made from ...
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1answer
43 views

Polar coordinate system : Is radial coordinate is a function of angular coordinate?

In polar coordinate system: The polar coordinates $r$ is called the radial coordinate and $\theta$ is called the angular coordinate, often called the polar angle. I am confused when answering the ...
0
votes
1answer
47 views

How do you compare carsharing plans to calculate the cheapest?

Call hourly rate = HR. Assume that I can guess my monthly usage in hours, which I call $g$. Beware that the fixed fees are presented in different units of time, so first convert everything into ...
0
votes
1answer
72 views

Different ways to write $n$

What is a general formula $n(n)$ for this? We know that starting from below, we can see how many numbers a certain $n$ generates by counting the number of numbers contained in the column $n$ is in, ...
0
votes
1answer
67 views

Solution to $b\sin({\theta})\cos({\phi})+a\cos({\theta})\sin({\phi})=0$ for $\phi$

I'm looking for a solution to $b\sin({\theta})\cos({\phi})+a\cos({\theta})\sin({\phi})=0$ for the variable $\phi$. In the equation both $a$ and $b$ are real numbers; in particular, I have ...
5
votes
1answer
151 views

Prove Divisibility In Fibonacci Sequence Over A Prime Number

In The Fibonacci sequence which is defined as: Lets say we have the number $p$ which is an odd prime. Prove that: $F_{p-1} + F_{p+1} -1$ Is divisible by $p$. Prove that for any given $n$ real ...
3
votes
2answers
98 views

Is it possible to solve this equation with logarithms and exponents?

$$-\frac{1}{3}\log(4x-12)+6=\left(-\frac{1}{2}\right)^x $$ Out of all the logarithm laws I've learned (which is pretty limited), I have not found a way to solve for what x is yet. Can someone verify ...
12
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0answers
101 views

Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $[-1,1]$ for every natural number $m$?

Is $\{\sin n^m \mid n \in \mathbb{N}\}$ dense in $[-1,1]$ for every natural number $m$? Progress For $m=1$, I can prove this using the fact that $\sin$ is continuous and $a+b\pi$ is dense in the ...
5
votes
1answer
83 views

Is continuous $f$ constant if every point of $\mathbb{R}$ is local minimum of $f$?

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous. Is $f$ constant if every point of $\mathbb{R}$ is local minimum of $f$? What metric spaces we can use instead of $\mathbb{R}$? I guess we ...
1
vote
1answer
44 views

Prove no odd number can be abundant.

A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of $28$ would be $1 + 2 + 4 + 7 + 14 = 28$, which ...
1
vote
4answers
139 views

What are solutions to $2^x=x$?

Are there any solutions (real, complex , matrix etc.) to $2^x=x$? The best I can come up with is $\ln 2 = \frac{\ln x}{x}$ or $x^{\frac{1}{x}}=2$
4
votes
1answer
159 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
61 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
243 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
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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
302 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
92 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
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0answers
36 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
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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
55 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
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0answers
34 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
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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
68 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
121 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
27 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
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2answers
727 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
62 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
95 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
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
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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
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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
35 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. ...