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
52 views

Show if $0 \le a <b$ implies $0 \le a^{\frac{1}{n}}<b^{\frac{1}{n}}$

Given that $0\le a<b$ show that $0\leq a^{1/n}<b^{1/n}$ Is this proof by induction? Show it's correct for $n=1$ Assume true for $n=k$, then $0\leq a^{1/k}<b^{1/k}$ holds for some $k$, ...
1
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2answers
171 views

How do you calculate P(A/B), when event B occurred after event A?

There's really only one question I can't begin to handle when it comes to probability, literally. It's not the only type of question I struggle with, though it's the type of question where I can't ...
1
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1answer
38 views

Isolate points of a metric space with some properties?

Suppose that all dense subspace of a metric space $(X,d)$ is open. Prove that the set of the isolate points of $X$ is dense in $X$. My Idea: all isolate points of $X$ are in any dense subspace, ...
0
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2answers
118 views

Is there a method or algorithm to solve “in what base is the equation true” questions?

I have been given some exercises in which I'm given some equation that doesn't hold in base ten, and I need to figure out in which base the equation does hold. For example: $\sqrt{41} = 5$ which I ...
1
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1answer
189 views

Does there exist continuously differentiable function $f:\mathbb{R}\longrightarrow\mathbb{R}$?

Does there exist continuously differentiable function $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that for all $x\in \mathbb{R},\,\,f(x)>0$ and, $f'(x)=(f\circ f)(x)$? I see this question in ...
2
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1answer
247 views

What is the answer of this problem solving question?

You need to order 4 plastic cups for each of the 800 runners. Plastic cups are sold in 2 different pack sizes and you must choose one type of pack only. A pack of 500 costs £12.50 or a pack of 800 ...
0
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1answer
184 views

Wrong ILP solution with LPSolve (simple example)

I added the following example into LPSolve and found a strange issue. I don't want S1 and S2 to overlap within certain margins. ...
0
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1answer
154 views

Math Exercises, Highschool Student

I really want to improve my problem solving skills.. However I have found that most of the exercise books (lots of problems) are above my curent level (e.g. Olympiad and College) I am currently in ...
0
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1answer
60 views

Finding subgroups of a group from specific order

Given the following group: $$ \left<\left\{ \begin{bmatrix}a & b \\0 & c \end{bmatrix} \mid a,b,c\in \Bbb Z_{5},a,c \neq 0 \right\} ,\:\: * \right> $$ where ∗ is multiplication. ...
1
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2answers
78 views

List of N items, randomly putting them in order, showing the procedure ends.

On a bookshelf, there are N tomes of the encyclopedia in random order. Every hour, a librarian takes a tome which is not in place and puts it in its place, and we must show this process will stop ...
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5answers
646 views

Examples of open problems solved through short proof

Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with ...
1
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0answers
48 views

Techniques for approximating a partial sum formula for any function.

There are several ways of computing the partial sum formulas of many summations, but is there a technique that can approximate a closed form for any summation? So far I found for $\sum_{x=0}^{n} \...
3
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4answers
50 views

Prove $F^2_{n+1} - F_nF_{n+2} = (-1)^n$

This is a question about Fibonacci sequences, a sequence in which the previous terms build up upon the current term (e.g. $F_1 + F_2 = F_3$ where $F_1 = F_2 = 1$). How would I go about proving $F^2_{n+...
0
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1answer
194 views

Finding the Compound Interest on 7500 Dollars at 4% per annum for 2 Years

Find compound interest on $\$7500$ at $4\%$ per annum for $2$ years, compounded annually. The choices are as follow: $\$512$, $\$552$, $\$612$, $\$622$. I tried to solve this problem by: C.I. $= ...
1
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1answer
53 views

Experiments with random selection: when to respect order?

I have a hard time imagining experiments of this nature without noting the arrangements (I draw that ball with my left hand, the other with my right. I draw that ball first, the other second) so I'm ...
0
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1answer
89 views

Help establishing restrictions for consistency on a linear system.

I'm having trouble wrapping my head around this problems, and others similar to it. I can typically solve systems of linear equations, but some give me trouble, especially dealing with unknown ...
0
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1answer
165 views

The Profit Gained by a Shopkeeper who Uses an 800gm Weight Rather than a Kilogram weight

A shopkeeper professes to sell his goods at $200 but uses a weight of 800gm instead of kilogram weight. Thus, he makes a profit of The choices are as follow: 20% 22% 25% None of these The ...
7
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2answers
84 views

For which $n, k$ is $S_{n,k}$ a basis? Fun algebra problem

Here it is a nice algebra problem I had some fun with Let $V$ be a vector space over $\mathbb R$ of finite dimension $\dim V = n$. Let $v = \{ v_1, \dots, v_n\}$ be a basis for $V$. Let $$S_{n,k} = \{...
1
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1answer
298 views

What's the Total Number of Candidates who Applied for the Exam?

In an examination, 5% of the applicants were found ineligible and 85% of the eligible candidates belonged to the general category. If 4275 eligible candidates belonged to other categories, then how ...
1
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1answer
59 views

Solving equation with exponentials

How to solve $ {z = x^y, x = y^z, y = z^x }$ for $ x, y $ and $ z? $ Is some sort of triple Lambert W to be introduced? Done so far: Taking logs, $$ \log z = y \log x , \log x = z \log y , \log ...
1
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1answer
59 views

How to best approach a problem of this kind (problem solving and simple linear equations)

This is a common type of problem that appear on algebra tests to test problem solving abilities (this is not from a real test or homework). I am curious on how to best approach these kind of problems ...
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3answers
246 views

Trouble with Vakil's FOAG exercise 11.3.C

I'm having trouble with the exercise in the title, even with part (a), which asks to prove that if $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least 1 and $H$ is a non-empty ...
1
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1answer
55 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
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1answer
50 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
107 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 'bridge-of-...
66
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23answers
10k 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, ...
0
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0answers
52 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
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1answer
50 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 the ...
1
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1answer
343 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 $X_1,X_2,\ldots,X_n$...
0
votes
1answer
41 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 ...
3
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1answer
55 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
57 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
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1answer
78 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
61 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 months,...
0
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1answer
134 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
105 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 $\frac{a}{b}...
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2answers
370 views

Prove Divisibility In Fibonacci Sequence Over A Prime Number

In The Fibonacci sequence which is defined as $$ F_n=F_{n-1}+F_{n-2}, $$ 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 ...
3
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2answers
111 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
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1answer
126 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 ...
6
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1answer
109 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
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1answer
82 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
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4answers
151 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
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1answer
167 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 $f(x)=0$...
1
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2answers
272 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
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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 $u_n=\sqrt{n}\prod_{k=1}^{n}\left(1-\frac{1}{2k}\right),...
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}) p_{i,...
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
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2answers
336 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: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...