Questions tagged [problem-solving]

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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2
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1answer
59 views

Evaluate in closed form: $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$

Evaluate in closed form: $$\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$$ where $|x|<1$ I am stuck on this problem. I tried decomposing the denominator into a geometric sum and tried to use ...
2
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1answer
36 views

A falling object does not keep accelerating indefinitely but, due to air resistance, reaches a terminal speed. What is the terminal speed?

Suppose that the speed of such an object, t seconds after the fall commences is vm/s where v= $$\frac{200}{3}(1-e^{-0.15t})$$ Find the speed of the object after five seconds. I have substituted t=5, ...
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1answer
33 views

A coin is flipped 15 times. How many possible outcomes contain exactly four tails? contain at least three heads? [closed]

A coin is flipped 15 times where each flip comes up either heads or tails. How many possible outcomes (a) contain exactly four tails?, (b) contain at least three heads? Hello everyone, I am currently ...
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0answers
39 views

How will you find out which box has the marbles of the Indian kind with only one weighing?

There are 8 boxes each containing 8 marbles. There are two kinds of marbles – Brazilian kind and Indian kind. Each marble of the Brazilian kind weighs 12 units and each marble of the Indian kind ...
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2answers
28 views

Find the number of elements in $s$ as per following criteria

Let $$s=\left\{\left(x_{1}, x_{2}, x_{3}\right) \mid 0 \leq x_{i} \leq 9 \text { and } x_{1}+x_{2}+x_{3} \text { is divisible by } 3\right\}$$ Then the number of elements in s is My approach With ...
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0answers
25 views

(a) Call a telephone number non-repetitious if no pair of adjacent digits are the same. [closed]

For the purpose of this problem, a telephone number is an arbitrary sequence of 7 decimal digits (Telephone numbers can start with a ’0’). (a) Call a telephone number non-repetitious if no pair of ...
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0answers
20 views

Short exact sequence proof [closed]

Short exact sequence and splits in some types of modules with proof the two proposition a. If the short exact sequence 0 →A →B→C →0 splits, then B~Imf ⊕ Img
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1answer
36 views

Cumulative Probabilities: What am I missing here?

Sorry if this question is a bit lower-level, yet complicated, but I feel like there is something wrong and I cannot put my finger on it. This scenario is adapted from something I read elsewhere on the ...
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0answers
11 views

Appropriate combination

How do you work out the appropriate combination? In a family, each of the six children has one packet of crisps on each weekday (Monday to Friday) for their packed lunch. The triplets have Cheese ...
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2answers
34 views

Divisibility Problem from an Olympiad Book

In a practice problem set of the Olumpiad book that I am solving, the following question has me stumped. $$\text{Prove that }3(7^{200}+7^{202}+7^{204})+7(3^{200}+3^{204}) -210 \text{ is divisible by ...
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1answer
27 views

Maps between Power Sets

Let $f:A \rightarrow B$ be a map and let $X \subseteq A$ and $Y \subseteq B$ Then by definition, we have the following two sets: The image of $X$, which is the set $f(X)=$ { $b\in B$ | $b = f(x)$ for ...
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1answer
69 views

Tournament plan

Last week, my math teachers and I were asked to solve a problem for a gym teacher. He has 12 sprinters and 3 lanes to run on. The problem is: How many runs do he has to do to guarentee that each ...
0
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1answer
31 views

Safe packing Constraint satisfaction problem - is it optimal?

Problem: You need to pack several items into your shopping bag without squashing anything. The items are to be placed one on top of the other. Each item has a weight and a strength, defined as the ...
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0answers
26 views

Naive probability and sample spaces

I'll start with a couple of questions: Is there a systematic way to construct the sample space for a given problem in naive probability theory? Or is it more of an art than a science? Is there a ...
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1answer
38 views

Questions about the Strong Induction Principle and the Well Ordering Principle

The Strong Induction Principle (SIP) is presented in Hrbacek and Jech's Introduction to Set Theory (3 Ed) as follows: Let $P(x)$ be a property. Assume that, for all $n \in \mathbb{N}$, (*) if $P(k)$ ...
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Some questions on mathematical induction

I am studying Hrbacek and Jech's Introduction to Set Theory (3 Ed) and have a couple of questions about the induction principle. Specifically I would like to know if the following solutions are ...
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0answers
22 views

How to systematically study the Inventor's Paradox (the more ambitious plan may have more chance of success)

I first encountered this phenomenon in undergraduate mathematical induction, similar to these StackExchange posts Examples where it is easier to prove more than less or Illustrative examples of a ...
10
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6answers
442 views

In a row of $40$ kids, $22$ are sitting next to girls and $30$ are sitting next to boys. How many girls are there?

There are $40$ kids sitting in a row. Number of kids sitting next to girls is 22, Number of kids sitting next to boys is 30. How many girls are sitting in a row? This is a problem from my brother's ...
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3answers
46 views

Find a basis for S $\cap$ T. Also find its dimension. Conditions are as following. [closed]

Let $S=\left\{\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\in\Bbb{R}:a_{1}+a_{2}+a_{3}+a_{4}=0\right\}$ And $T=\left\{\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\in\Bbb{R}:a_{1}-a_{2}+a_{3}-a_{4}=0\right\}$ ...
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2answers
51 views

Can someone explain the limit $\lim _{n \rightarrow \infty} \left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\right)$?

$$\begin{aligned} &\text { Find the following limit: } \lim _{n \rightarrow \infty} \left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\right)\\ &\text { **In a ...
7
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1answer
88 views

Showing that the Diophantine equation $m(m-1)(m-2)(m-3) = 24(n^2 + 9)$ has no solutions

Consider the Diophantine equation $$m(m-1)(m-2)(m-3) = 24(n^2 + 9)\,.$$ Prove that there are no integer solutions. One way to show this has no integer solutions is by considering modulo $7$ (easy to ...
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1answer
63 views

Solving $x+x\ln(x)+\ln(x)=y$ for $x$

For $x,y\in\mathbb{R^+}$ , consider the equation: $x+x\ln(x)+\ln(x)=y$ with constant $y$, which is the same as $x+\ln(x^{x+1})=y$ How do I solve for $x$?
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2answers
91 views

Number of $3$-digit numbers with strictly increasing digits

A positive integer is called a rising number if its digits form a strictly increasing sequence. For example, 1457 is a rising number, 3438 is not a rising number, and neither is 2334. (a) How many ...
2
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1answer
35 views

Prove that every collection of partitions $T$, there exists $\inf{T}$ and $\sup{T}$

I am self-studying Hrbacek and Jech's Introduction to Set Theory (3rd edition), and I want to know if the following solution to problem 5.10 (c) is correct. Unfortunately the book contains no answers ...
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2answers
24 views

Determine how many integer ordered pairs $(x, y)$ satisfy the equation $\mid x + 2020\mid + \mid y + 505\mid = 4$.

I do not see how this is possible if all of it equals to a number higher than $4$. If $x$ or $y$ is negative would it make a difference? How can $x$ and $y$ adding two big numbers equal $4$. Is there ...
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1answer
35 views

What is the largest number k < 100,000 such that k has an odd number of factors? [duplicate]

so i do not know how to solve this problem and is confused on a method to find odd number of factors. How can I find the odd number of factors for a number that high?
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1answer
30 views

Find the value of $a + b$ if $ 2 ^{16} · 3^ 8 · 5^ 3 · 7^ 3 · 11^2 · 13^1 · 17^1 · 19^1 = a! /b!$ [closed]

I have tried looking for a pattern but cant seem to figure put a method.
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1answer
78 views

How to find roots of this equation?

I have this equation and want to find its roots. $\left(a^2+1\right) \cosh (a (c -b))- \cosh (c a)=0 $. Any comment is welcome.
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2answers
62 views

Divisibility Number Theory problem, explanation needed

I can't understand the solution of the following problem: $x$,$y$,$z$ are pairwise distinct natural numbers show that $(x-y)^5$ + $(y-z)^5$ + $(z-x)^5$ is divisible by $5(x-y)(y-z)(z-x)$. No need to ...
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0answers
21 views

General strategies for solving functional equations.

I am very bad at solving functional equations, though I enjoy them a lot, so I'm looking for some strategies to solve functional equations. Like what are some approaches we would try to start solving ...
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2answers
91 views

Find all $(x,y)$ pairs : $x,y$ $\in \mathbb{Z}$ such that :- $x^4 - 4x^3 - 19x^2 + 46x = y^2 - 120.$

So here is the Question :- Find all $(x,y)$ pairs : $x,y$ $\in \mathbb{Z}$ such that :- $$x^4 - 4x^3 - 19x^2 + 46x = y^2 - 120.$$ What I tried :- I factored the LHS and got as :- $$x(x - 2)(x^2 - 2x - ...
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0answers
22 views

What is the average normal strain that is produced in post (1)?

P2.5 In Figure P2.5, rigid bar ABC is supported by a pin at B and by post (1) at A. However, there is a gap of A = 10 mm between the rigid bar at A and post (1). After load P is applied to the rigid ...
7
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3answers
99 views

Show that the polynomial $x^{8}-x^{7}+x^{2}-x+15$ has no real root.

Show that the polynomial $$x^{8}-x^{7}+x^{2}-x+15$$ has no real root. As I have learnt from my previous post, using Descarte's Sign rule I am getting $4$ positive and $0$ negative roots. So no of ...
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4answers
78 views

Find all real $x$ such that :- $\frac{4x^2}{1 - \sqrt{1 + 2x^2}} < 2x + 9$ [closed]

Find all real $x$ such that :- $\frac{4x^2}{1 - \sqrt{(1 + 2x^2)}} < 2x + 9$ How can I solve this ? . I tried by solving for x by normal algebra , but it didn't succeed as I thought . Do we have ...
1
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1answer
31 views

Why does $\sum_{i=n}^{2n-1}\binom{i-1}{n-1}2^{1-i}$ computes the probability of $n$ head or tails

$\sum_{i=n}^{2n-1}\binom{i-1}{n-1}2^{1-i}$ For $i = n,n+1,\ldots, 2n - 1$, the sum above computes $P(E_i)$, the probability that i tosses of a fair coin are required before obtaining $n$ heads or $n$ ...
0
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1answer
37 views

Solving an algebraic equation with fractions

How do I rework the following equation to solve for P when I know the other variables? (Apologies, I tried to Google but just couldn't get the right search terms.) T = ( 1/AP - 1/P ) * Q I got this ...
0
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2answers
48 views

Solving Inequality from the proof of First Order ODE.

Let $x'=f(x)$ $x(0)=x_0$ assuming that $|f(x)-f(x)|\le L|x-y|$. We have $x(t)=x_0+\displaystyle\int_0^tf(k)dk$. Let's say $$x_{n+1}=x_0+\int_0^tf(x_n(k))dk$$ and $$|x_{n+1}-x_n(t)|\le L\int_0^t|x_{n}(...
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3answers
52 views

I have this identity that I'd like to prove. $\sum_{k=0}^{n}\left(\frac{n-2k}{n}\binom{n}{k}\right)^2=\frac{2}{n}\binom{2n-2}{n-1}$

I have this identity that I'd like to prove. $$\displaystyle{\sum_{k=0}^{n}\bigg(\dfrac{n-2k}{n}\binom{n}{k}}\bigg)^2=\dfrac{2}{n}\binom{2n-2}{n-1}$$ Here's what I have done so far: (using a binomial ...
0
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1answer
22 views

we say a subset of positive integers is anti-closed, can you partition +integer into a finite # of subsets that are all anticlosed? [closed]

For any two unique elements in positive integer if a,b is in a subset then a+b is not.
1
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1answer
48 views

The sign of $-4\eta ^2\cosh\beta\cosh(\beta\eta)-4\eta\sinh\beta\sinh(\beta\eta)+2B^2\cosh(2\beta\eta)+2B^2+4$

Can I figure out when the sign of this expression is positive and when it is negative? $$-4 \eta ^2 \cosh (\beta ) \cosh (\beta \eta )-4 \eta \sinh (\beta ) \sinh (\beta \eta )+2 B^2 \cosh (2 \beta ...
1
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1answer
43 views

Is there any $C^\infty$ monotonically non-decreasing function $f$ which satisfies the conditions below?

As stated in the above title, is there any $C^\infty$ monotonically non-decreasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f((-\infty, -2]) = \{-1\}, f([2, \infty)) = \{1\}$ and $...
0
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1answer
33 views

Probability coin question [closed]

Suppose each of three persons tosses a coin. If the outcome of one of the tosses differs from the other outcomes, then the game ends. If not, then the persons start over and retoss their coins. ...
0
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0answers
57 views

Find all functions $:f:\mathbb N\rightarrow \mathbb N$ such that $f(f(n))=3n$. [duplicate]

Find all functions $:f:\mathbb N\rightarrow \mathbb N$ such that $f(f(n))=3n$ for all $n\in\mathbb N$ and f is strictly increasing. I know that f is injective since it's strictly increasing and using ...
0
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0answers
24 views

There is a 4x4 checkerboard and we have 3 shapes to use to tile. right and isosceles triangle and a parallelogram. How many ways to be tiled?

So I know that the right triangle is 2^16 because each box has 2 right triangle. I don't know about the parallelogram and the isosceles triangle. How would I also figure out about the mix of the three....
0
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2answers
74 views

Show that $\binom{n}{1}-3\binom{n}{3}+3^2\binom{n}{5}\cdots=0$

Show that if $n\equiv 0\pmod 6$ (although the statement holds true for $n\equiv 0\pmod 3$) $\binom{n}{1}-3\binom{n}{3}+3^2\binom{n}{5}\cdots=0$ I am having trouble finding the appropriate polynomial ...
2
votes
2answers
87 views

Prove that there exists a positive integer $k$ such that $k2^n + 1$ is composite for every positive integer $n$.

Prove that there exists a positive integer $k$ such that $k2^n + 1$ is composite for every positive integer $n$. (Hint: Consider the congruence class of $n$ modulo 24 and apply the Chinese Remainder ...
0
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1answer
22 views

A question on order-isomorphism with $\mathbb N$.

Let $A$ be a countable subset of $\mathbb R$ which is well ordered with respect to usual ordering $\leq$ of $\mathbb R$.Then does $A$ have an order preserving bijection with a subset of $\mathbb N$? ...
1
vote
1answer
43 views

How to computer $f(\frac{1}{2})$ given $f(f(x)) = x^2 + \frac{1}{4}$?

I have observed that $f(f(\frac{1}{2})) = \frac{1}{2}$ and $f(f(f(x))) = f(x^2 + \frac{1}{4})$, and when $x = \frac{1}{2}$, we have $f(\frac{1}{2}^2 + \frac{1}{4}) = f(\frac{1}{2})$. But I don't know ...
1
vote
3answers
65 views

Modulus operation to find unknown

If the $5$ digit number $538xy$ is divisible by $3,7$ and $11,$ find $x$ and $y$ . How to solve this problem with the help of modulus operator ? I was checking the divisibility for 11, 3: $5-3+8-x+y =...
2
votes
1answer
31 views

A Weird Modular Arithmetic Question

I saw this question somewhere and I was wondering if there's a nice closed form answer to it. It just seems like a troll question to me. $2016^{2016} + 2018^{2016} (\bmod{2017}^{2016})$ I proceeded ...

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