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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1answer
41 views

One hundred indistinguishable ants are dropped on a hoop of diameter 1

I have this question which I am not sure how to solve: One hundred indistinguishable ants are dropped on a hoop of diameter 1. Each ant is traveling either clockwise or counterclockwise with a ...
2
votes
1answer
39 views

Finding the best fit of 3 categories ( restaurants/meal/person analogy problem )

I have this problem that sounds tedious and long and I'm not sure if there exist an intuitive way to solve it. The problem is related to image recognition but I will try to give an analogy to it You ...
1
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1answer
105 views

Time & Distance : Pokemon Hunter and the Rogue Brook

I was working my way through some Puzzles in Discrete Maths by Rosen, when I came across the following question: A Pokemon Hunter is rowing upstream a brook As he passes under the ...
62
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23answers
9k views

An example of a problem which is difficult but is made easier when a diagram is drawn

I am writing a blog post related to problem solving and one of the main techniques used in problem solving is drawing a diagram. Essentially, I want to illustrate that some hard problems (for example, ...
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0answers
30 views

problem solving on work rate

please tell me this problem has insufficient data.. 5 men and 2 boys working together can do four times as much work as a man and a boy. working capacities of a woman and a boy are in the ratio ... ...
0
votes
1answer
24 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
51 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
40 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
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2answers
45 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 ...
0
votes
1answer
52 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
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1answer
51 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
76 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
76 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
178 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
99 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 ...
13
votes
1answer
113 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
85 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
47 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
142 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
164 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
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2answers
92 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
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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
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
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2answers
38 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
112 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
306 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
26 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
20 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
96 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
98 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
37 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
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
29 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
43 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
35 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
107 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
156 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
28 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
93 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
770 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
67 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
162 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
102 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
53 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
67 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? ...