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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I have an equation like ax + by + cz = 0, where a, b, c are constants, I need to find x, y, z

I have an equation like $a_ry_1 + b_ry_2 + c_ry_3 = 0$, where $a_r, b_r, c_r$ are constants, I need to find $y_1, y_2, y_3$ where $\hat{y} = y_1 \hat{r} + y_2 \hat{t} + y_3 \hat{n}$ and $\hat{x} = a_r ...
IsTas's user avatar
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1 answer
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What is the average of $n(100-n)$ for $n \in {1,2,...,100}$

What is the average of $n(100-n)$ for $n \in {1,2,...,100}$ I saw this problem on the internet without a method of solving it without brute force or a computer. The answer is 1666.5. If you were to ...
Xerium's user avatar
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0 answers
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Probability of event happening after n times with chance 1 in x

Context: Hi, I started by calculating the chance of an event happening with a chance of 1 in x after n times. Next I wanted to flip this approach to be able to say how many n attempts are needed to ...
ALZlper's user avatar
1 vote
0 answers
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understanding over an example presented in Engel's book

Each of the numbers $a_{1},...,a_{n}$ is 1 or -1, and we have $$ S=a_1a_2a_3a_4+a_2a_3a_4a_5+...+a_na_1a_2a_3$$ Engel has written that if we replace any $a_i$ by $-a_i$ , then $S$ does not change $...
Alberto's user avatar
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1 answer
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Expected radius of throwing a dart at a dartboard

I am doing a problem that states: If you are throwing a dart at a circular board with radius $R$, what is the expected distance from the centre? If $x$ is the expected radius, then it would be the ...
Xerium's user avatar
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0 answers
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How to find good maths problems [closed]

Pls can you tell some good ways to find math problem and think deeply about simple things ... I want to think more but need some good problems to work on ... Not that collartz conjecture , or twine ...
MathSolver's user avatar
4 votes
1 answer
90 views

List all topologies of $X = \{1,2,3\}$ up to homeomorphism.

Problem 2-2 in Lee's Introduction to Topological Manifolds reads: Let $X = \{1, 2, 3\}$. Give a list of topologies on $X$ such that every topology on $X$ is homeomorphic to exactly one on your list. ...
ummg's user avatar
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61 views

Solving $\frac{1 - \cos(y/2)}{\pi d} \cdot 180c=y$ for $y$

I have been trying to solve a problem that came to me randomly about a week ago, and I am very close to solving it but have reached a roadblock when re-arranging the final equation to solve it. I do ...
Jim's user avatar
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2 votes
1 answer
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Find the potential $\Phi$ and the field $\mathbf{F} = \nabla \Phi$ for a two-dimensional dipole at the origin. [closed]

I'm struggling with one of the problems from mathematical analysis II course. The problem is next: Find the potential $\Phi$ and the field $\vec{\mathbf{F}} = \nabla \Phi$ for a two-dimensional ...
Nebula's user avatar
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I need help with this propositional logic problem

I study compound statements, and I encountered this problem in the book: The problem I tried a solution: Let p be proposition "The first door leads to freedom" and let q be proposition "...
Begginer2005's user avatar
3 votes
3 answers
155 views

Proving the convergence of a series with very little information

Let $m \in \mathbb{N}$ be a fixed natural number and $ (a_n)_{n \geq 1}$ be a sequence of positive real numbers such that $\forall n \geq 1\colon a_{n+1} \leq a_n - a_{mn}$. Prove that the series $\...
Shthephathord23's user avatar
2 votes
1 answer
47 views

Find the domain of a function for different values of $g$

Find the domain of: $$f(x) = \frac{1}{(g+1)x^2 + 2(g-1)x + g-3}$$ for the various values of $g\in \mathbb{R}$ I am trying to solve this. First of all, I take the $g$ value equal to $-1$ so the domain ...
Makarius's user avatar
1 vote
2 answers
54 views

Show that $\frac{1}{\sqrt{n}}\lVert x\rVert_1\le\lVert x\rVert_2\le \lVert x\rVert_1$

$\frac{1}{\sqrt{n}}\lVert x\rVert_1\le\lVert x\rVert_2\le \lVert x\rVert_1$ My idea is to show first that $\lVert \vec x\rVert_\infty \le \lVert \vec x\rVert_2 \le \sqrt{n}\lVert \vec x\rVert_\infty$, ...
Roma_Rayado's user avatar
1 vote
1 answer
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Solving a pretty difficult convergence for series

Let $P_n(x)=x^n-nx+1$ be a sequence of polynomials, where $P_n\colon[1, +\infty) \to \mathbb{R}$ and $n \ge 2$ a) Show that for each $n$, $P_n(x)=0$ has exactly one solution, and for each $n$ let $...
Shthephathord23's user avatar
3 votes
0 answers
166 views

Find the indeterminate values of $x_1$ and $y_1$ if $\vec x=(x_1,-2,1,-1)$ and $\vec y=(-2,y_2,-1,-2)$ and $\lVert \vec x\rVert=2\lVert \vec y\lVert$

Let $\vec x=(x_1,-2,1,-1)$ and $\vec y=(-2,y_2,-1,-2)$, wich satisfies $\lVert \vec x\rVert=2\lVert \vec y\lVert$. Find the indeterminate values of $x_1$ and $y_1$. So, assuming $\lVert _\dot{} \lVert ...
Roma_Rayado's user avatar
1 vote
1 answer
53 views

Seeking suggestions for a book with hard problems about surface and volume integrals

I am interested about the hard problem of surface and volume integral, so can anyone suggest me a book based on the problem on surface and volume integral (containing a lot of hard problem) for ...
Albert's user avatar
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1 answer
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Number of functions (mappings) between sets

Hello, I would like to ask if somebody has any site or materials that would explain these kinds of exercises. Examples: Let $A = \{1,2,3,4,5,6,7\}$ and $B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$. How ...
peterparker321's user avatar
3 votes
2 answers
334 views

When to give up on a problem (for maximal learning of a specific area)? [closed]

There are many things to admire about the so-called Moore method, in which all the theorems and aspects of a course (e.g. real analysis) become problems for you to solve on your own. But sometimes, ...
Chris Sanders's user avatar
6 votes
2 answers
104 views

Let $0<\alpha\leq\frac{\pi^2}{6}.\ $ Does $\exists\ A\subset\mathbb{N}$ and $f:A\to\{-1,1\}$ such that $\sum_{n\in A} \frac{f(n)}{n^2}=\alpha?$

If we let $0<\alpha\leq \frac{\pi^2}{6},\ $ then it is not always true that $\exists\ A\subset \mathbb{N}$ such that $\displaystyle\sum_{n\in A} \frac{1}{n^2} = \alpha.\ $ To see this, consider the ...
Adam Rubinson's user avatar
2 votes
1 answer
80 views

How do I know if a theorem has been proposed and proved [closed]

I'm trying to write a paper where the proof requires some theorems. I can prove these theorems myself, but I don't know if they have already been proved, and if so I only need to quote them. Is there ...
槿铃兔's user avatar
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1 answer
83 views

Question regarding monkeys and probabilities.

So this question is derived from problem 4 in chapter 2 of Kittel & Kroemer's Thermal Physics. To be upfront: I am confused about how to properly compute the probabilities from a canonical point ...
Gerald's user avatar
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2 answers
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Sample space with equally likely outcomes: question to a worked out example

Attached is an extract from the book A First Course in Probability by Sheldon Ross. I am struggling to understand the flow of his logic at the place highlighted in red. For example, let $n = 6$, $k = ...
Lina's user avatar
  • 1
24 votes
3 answers
763 views

What's the area of the triangle in this geometry problem? I think I can solve it, but it's way too convoluted...

I am trying to solve this geometry problem from an exam. The exam is supposed to be 3 hours long and this is supposed to be 1 out of 10 problems. So given that, the solution should be something quick, ...
zlaaemi's user avatar
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4 votes
2 answers
95 views

Probability of 3 Aces, Last Picked is an Ace

"Aaron picks an integer $k \in [1,52]$. Then, he draws the first $k$ cards from a standard, shuffled 52-card deck. Aaron wins a prize if the last card he draws is an ace and if there exists ...
LaTate's user avatar
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6 votes
1 answer
180 views

A literally challenging math book

The only way to learn mathematics is to do mathematics. - Paul Halmos Most books about uni-level mathematics follow a strict scheme of giving you the content and letting you practice with it with ...
Luca T. Castrillón's user avatar
2 votes
1 answer
57 views

Proving the Inequality $1-x-\lambda \leq \frac{\prod_{i=1}^n (1+\lambda_i)}{\prod_{i=1}^n (1+x_i)}$

I am trying to show the following inequality: Let $\lambda_i \in [0,1]$ for $i=1,...,n$, and let $x_1,...,x_n$ be non-negative real numbers. Define $\lambda:=\sum_{k=1}^n \lambda_k$ and similarly $x:=\...
Maxi's user avatar
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0 votes
2 answers
62 views

Faster solution to puzzle: fill table with results of multiplication signs

Fill + or - in the blank squares below so that: for each row, the sign in the rightmost (gray) column is the result of the multiplication of the signs in the row for each column, the sign in the ...
conierju's user avatar
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0 answers
25 views

Problem 3-40 in "Calculus on Manifolds" by Spivak. Show that we can write $g=g_n\circ\cdots\circ g_1$ if and only if $g'(x)$ is a diagonal matrix.

I am reading "Calculus on Manifolds" by Michael Spivak. Problem 3-40. If $g:\mathbb{R}^n\to\mathbb{R}^n$ and $\det g'(x)\neq 0$, prove that in some open set containing $x$ we can write $g=T\...
佐武五郎's user avatar
1 vote
1 answer
70 views

Combinatorial Probability Question Similar to Sheldon Ross's A First Course in Probability 10th ed Chapter 2 Problem 7

I wrote a combinatorial probability question, and I am not sure if my solution is correct. It is just a slight rewording of a problem from Sheldon Ross's A First Course in Probability 10th ed. Chapter ...
Karl Chester Galapon's user avatar
0 votes
0 answers
45 views

$x$, $y$, $z$ positive and $\frac{y}{x-z} = \frac{x + y}{z} = \frac{x}{y}$, find numerical value of $\frac{x}{y}$

$x$, $y$, $z$ positive and $\frac{y}{x-z} = \frac{x + y}{z} = \frac{x}{y}$, find numerical value of $\frac{x}{y}$ The solution in the textbook seems straight forward. Because they are equal ...
Jordi's user avatar
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1 vote
1 answer
81 views

Prove $\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\ge 3$ for positive $x^2+y^2+z^2=x+y+z$.

I need your help to solve this Math Olympiad problem: Let $x$, $y$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = x + y + z$. Prove that : $$ \frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\...
Med ed's user avatar
  • 29
0 votes
1 answer
20 views

Solving for Device Position Using Geometric Methods with Distance Sensors

I am encountering a geometric problem in my project and would greatly appreciate your insights. The challenge involves using distance sensors to determine the position of a device within a room. I'm ...
CootMoon's user avatar
  • 103
0 votes
1 answer
90 views

$\frac{a}{\sqrt{2a^2+3bc}}+\frac{b}{\sqrt{2b^2+3ca}}+\frac{c}{\sqrt{2c^2+3ab}} \le \sqrt{ab+bc+ca}$

Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3$.Prove that $$\frac{a}{\sqrt{2a^2+3bc}}+\frac{b}{\sqrt{2b^2+3ca}}+\frac{c}{\sqrt{2c^2+3ab}} \le \sqrt{ab+bc+ca}.$$ By C-S we need proof $$4(...
Nguyen Anh Vu's user avatar
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0 answers
63 views

A proof of the fact that Heisenberg group is a closed subgroup of $GL(3,\mathbb R).

Let $\mathbb H$ be the set of all $3\times 3$ matrices of the form $\begin{pmatrix}1 & a & b\\0 & 1 & c\\0 & 0 &1\\ \end{pmatrix}$ where $a,b,c\in \mathbb R$ .I have to show ...
Kishalay Sarkar's user avatar
7 votes
2 answers
255 views

Inequality involving $f(x)=e^x-x^2/2$ with $f'(a)=f'(b)$ ($a\ne b$)

I'm currently working on a high-school level mathematical problem and have developed a solution approach, but I'm stuck at the final step. The problem is as follows: Consider the function $f\left(x\...
CaldariNavyFleet's user avatar
0 votes
0 answers
26 views

Normality of the derived family of a normal family.

I am trying to solve a problem from complex analysis which is concerning normal families.The problem is the following: Show that,if $\mathcal F\subset \mathcal H(\Omega)$ is a normal family of ...
Kishalay Sarkar's user avatar
2 votes
0 answers
63 views

Some maths journals for tough Olympiad and real analysis problems? [closed]

I wanted to know some maths journals and problems magazines like that in crux mathematicorum, American mathematical society, Romanian maths magazine etc. I want to know even more journals with ...
Soumyadip Das's user avatar
-1 votes
1 answer
74 views

How to solve this equation $8 \left(3^x+5^x+7^x\right)=5\cdot 2^x+2\cdot 4^x+17\cdot 6^x$? [closed]

I use Mathematica to solve this equation $$8 \left(3^x+5^x+7^x\right)=5\cdot 2^x+2\cdot 4^x+17\cdot 6^x$$ and get three solutions $x=0\lor x=1\lor x=2.$ I don't know how to solve by hand. How can I ...
John Paul Peter's user avatar
0 votes
1 answer
81 views

Direct integration and integration by parts doesn't yield the same result

I've encountered a discrepancy while calculating the same integral using two different methods: direct integration and integration by parts. The integral in question is: $$\int (b_1 x + b_2 x^2) x^{-2}...
t-rex's user avatar
  • 27
0 votes
0 answers
37 views

Question on Rudin's functional analysis Chapter 1, Exercise 21

The exercise asks to prove that for a given neighbourhood of $0$, $V$, in a topological vector space $X$, there exists a function $f:X\to\mathbb{R}$ such that $f(0)=0$ and $f(x)=1$, for any $x\notin V$...
math_cpt's user avatar
  • 184
1 vote
1 answer
158 views

Combinatorics question: teams vs missions

I know it's a combinatorics question, I'm just not sure how to approach it. Let's say we have $7$ types of agents: $A,B,C,D,E,F,G$. Agents work in teams of $4$. We send teams on missions, with each ...
Serge Shirokov's user avatar
2 votes
2 answers
108 views

How many numbers between 10 and 100 are divisible by 3 but not 2 nor 7?

My working: Let $|M_3|$ denote the number of integers between 10 and 100 that divides 3. So we have $$|M_3|= \lfloor \frac{100}{3} \rfloor-3=30$$, and similarly, $$|M_2|=\lfloor \frac{100}{2}\rfloor-...
Holland Davis's user avatar
0 votes
1 answer
37 views

Number of committees?

Q: "A committee composed of Morgan, Tyler, Max, and Leslie is to select a president and secretary. How many selections are there in which Tyler is president or not an officer?" I count as $1 ...
saner's user avatar
  • 519
3 votes
2 answers
327 views

Recommendations for a Comprehensive Problem Book(s) in Calculus $3$.

I am on the lookout for a comprehensive problem book specifically tailored to Multivariable Calculus and Vector Calculus (Calculus $3$), covering topics such as multiple integrals, partial ...
Mathematics enjoyer's user avatar
1 vote
2 answers
59 views

$X = [1_n \hspace{1mm} Z]$ with full column rank, then $(I_n - \frac{1}{n}J_n)Z$ is full-column rank

I'm trying to solve the following statement. Consider a $n \times p$ matrix $X = [1_n \hspace{1mm} Z]$, where $1_n = (1, \dots, 1) \in \mathbb R^n$ and $\text{rank}(X) = p$. Let $I_n$ be a $n \times ...
jason 1's user avatar
  • 695
2 votes
0 answers
75 views

Buffon's needle in one dimension

I want to solve Buffon's Needle problem but first I was trying to tackle a simpler case. So: consider an infinite line with points each $t$ units. Let's say that we have a "needle" of length ...
René Quijada's user avatar
1 vote
1 answer
82 views

Number of dinners

How many dinners consist of 2 optional appetisers, 3 main courses and 4 optional beverages? My result is $2\cdot 3\cdot4 + 2\cdot3 + 3\cdot4 + 3 = 45$. Where the first product $2\cdot3\cdot4$ ...
saner's user avatar
  • 519
0 votes
0 answers
131 views

$\int_D 1$, where $D:=\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 = r^2, (r, z) \in C\}$. (Problem 3-29 in "Calculus on Manifolds" by Michael Spivak)

The following problem is a problem in the section "FUBINI'S THEOREM". Problem 3-29. Use Fubini's theorem to derive an expression for the volume of a set of $\mathbb{R}^3$ obtained by ...
佐武五郎's user avatar
1 vote
1 answer
124 views

define $a@b=\frac{b^2+3a}{a+33b},$ calculate (1@2@...@100)×3303

Find the following question in a middle school math competition: define $a@b=\frac{b^2+3a}{a+33b}$, then what is $(1@2@3@\cdots@100)\times3303$? If we assume that $1@2@3=(1@2)@3$, the code below is ...
Lonitch's user avatar
  • 129
0 votes
0 answers
42 views

Linear Systems of Diff Eq - unsure how to solve

here is the problem I am trying to solve: "Consider a system of two well-stirred and interconnected tanks that are filled to capacity. Tank 1 holds 40 liters of water, initially containing 20 kg ...
cookiemonster's user avatar

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