# 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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### 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 ...
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### 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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### 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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### 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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### 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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### (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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### 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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### 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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### 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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### 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 ...
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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### 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 ...
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### 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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### 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 ...
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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### 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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### 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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### 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 ...
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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### 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 ...
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.
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 ...
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 $... 1answer 33 views ### Probability coin question [closed] Suppose each of three persons tosses a coin. If the outcome of one of the tosses differs from the other outcomes, then the game ends. If not, then the persons start over and retoss their coins. ... 0answers 57 views ### Find all functions$: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$and f is strictly increasing. I know that f is injective since it's strictly increasing and using ... 0answers 24 views ### There is a 4x4 checkerboard and we have 3 shapes to use to tile. right and isosceles triangle and a parallelogram. How many ways to be tiled? So I know that the right triangle is 2^16 because each box has 2 right triangle. I don't know about the parallelogram and the isosceles triangle. How would I also figure out about the mix of the three.... 2answers 74 views ### Show that$\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$I am having trouble finding the appropriate polynomial ... 2answers 87 views ### Prove that there exists a positive integer$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$modulo 24 and apply the Chinese Remainder ... 1answer 22 views ### A question on order-isomorphism with$\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$? ... 1answer 43 views ### How to computer$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 ... 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 =...
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 ...