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.
4,606 questions
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Maximum number of non-crossing edges in a 7-node network, such that nodes can be three-colored without same-colored nodes being connected
I've been struggling through this and created a bunch of options where 12 work such as alternating colour 1 and 2 around the edges and having colour 3 in the middle, but I have a suspicion that there ...
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Perpendicular segments from special points in a rectangle with ratio $BC = 2AB$
Consider a rectangle $ABCD$ with $BC = 2AB$. Let $L$ be the midpoint of side $AD$. From $L$, draw a perpendicular to diagonal $AC$ that intersects:
$AC$ at point $K$
$BC$ at point $F$
Let $M$ be the ...
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Isosceles trapezoid and special triangles in a square with perpendicular construction
Let $ABCD$ be a square. Let $Dx$ be a ray from vertex $D$ that intersects side $BC$ internally at point $E$. Draw $BH$ perpendicular to ray $Dx$, where $BH$ intersects $Dx$ at point $F$ and intersects ...
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Investigating the topological properties of a quotient space of $\mathbb R^2$, given by identification of irrational lines through $0$.
My question is about a very erratic quotient space. I encountered this space in some topology exercise. The space $X$ is described in the following:
Let $\mathbb R^2=\{(x,y):x,y\in \mathbb R\} $ be ...
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Investigating Countability Axioms for the Space of Ordinals $[0,\Omega)$, where $\Omega$ is the first uncountable ordinal.
I wish to discuss about the following question from general topology, involving set of ordinals:
Problem:
Let $X=[0,\Omega)$ be the set of all ordinals strictly smaller than the first uncountable ...
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If $a, b, c, d, e, f > 0$ prove that $\frac{ab}{a+b} +\frac{cd}{c+d} + \frac{ef}{e+f} \leq \frac{(a+c+e)(b+d+f)}{a+b+c+d+e+f}$ [duplicate]
Here is the problem statement again:
If $a, b, c, d, e, f > 0$ prove that
$$\frac{ab}{a+b} +\frac{cd}{c+d} + \frac{ef}{e+f} \leq \frac{(a+c+e)(b+d+f)}{a+b+c+d+e+f}$$
The solution given in my book ...
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Length of $EF$ in isosceles trapezium.
Given $ABCD$ is a isosceles trapezium. $EF$ is parallel to $DC$ and $AB$. If $AB=25$ and $CD=20$, and $DF=\frac{3}{5}BD$, what the length of $EF$?
Let $O$ is intersection of $DB$ and $AC$.
I think to ...
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An equation for box switching problem
I've heard a problem somewhere about two persons carrying the box.
It goes like this:
Two people were carrying a box along a path. To make it easier, they agreed to switch who carried the box every <...
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Stable formula to minimize a metric with several input and output variables
I'm not sure how to formulate that in terms of maths.
I have four input variables whose value is arbitrary (among positive integers):
nOpposition
nGovernment
nCrossbenchers
nSpeaker
I have four ...
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Does Peano’s theorem apply when the initial condition lies on the boundary of the domain?
I am trying to do an exercise on Peano's Theorem. I can't solve this one because I don't know if it works in a closed set.
Prove that the Cauchy problem:
\begin{cases}
\dot{x}(t)=\dfrac{-t+\sqrt{...
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How to find existing solutions to problems
It is pretty often that I think of an interesting math problem, and similarly often that I get stuck. Historically I have found it difficult to locate solutions online, since I don't really know what ...
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Prove that $a=b=c=0$ given: $a^3 + b^3 + c^3 = 4abc$ and $ab+2bc+3ca=0$
I am trying to solve the following problem:
Let $a, b, c \in \mathbb{R}$ satisfy the equations
$a^3 + b^3 + c^3 = 4abc$ and $ab + 2bc + 3ca = 0$.
Prove that $a = b = c = 0$.
If I ignore the ...
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Computing rate statistics from time intervals
Imagine that you observe events occurring at random times. More precisely, the time intervals between events are IID and drawn from the distribution described by the p.d.f.
\begin{align}
\rho(t;\tau,...
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$X\sim N(\mu,\sigma),\ P(15<X<17) = 0.2$ and $ P(12<X<16) = 0.5. $ Find $\mu$ and $\sigma$ using inverse normal.
Suppose $X\sim N(\mu,\sigma),\ P(15<X<17) = 0.2$ and $ P(12<X<16) = 0.5. $
Is there a way using the inverse normal function on a calculator to find $\mu$ and $\sigma?$
There should be a ...
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**HINT** for exercise 4 (a), Chapter 5 in Evans PDE (2nd ed) [duplicate]
I am currently reading Partial Differential Equations (2nd Ed.) by L.C Evans. I finished chapter 5 and I am doing the exercises.
I am currently on exercise 4, which goes as follows:
Let $u\in W^{1,p}(...
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Equation concerning different bases and exponents [closed]
I have an equation that I've been trying to solve for fun algebraically after helping someone learn logarithm and exponent rules:
$$3^x+11^{2x^2+4x-7}=311$$
So far, no manipulation using logarithms or ...
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Inequality for $n=m+1$ case—my approach fails
I’m trying to prove the following inequality in the special case $n=m+1$.
Let two positive integers $n>m$, positive reals $x_1,\dots,x_n$ satisfy $x_1x_2\cdots x_n=1$. Then
$$\sum_{k=1}^n \frac1{...
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What is "wrong" with this attempt at finding curves orthogonal to $xy=C$? [duplicate]
I was tutoring in math and helping with an exercise:
Find all curves that are perpendicular to the given curve $xy = 1$ where $x$ and $y$ are reals.
This was my immediate attempt:
We want to find a ...
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How to prove $\int_{0}^{1}\frac{2x}{(1+x^2)\ln(\frac{1-x}{1+x})}dx=-\ln2$
So earlier today I accidentally typed $(1+x^2) $ instead of $(1+x)^2$ into Wolfram and thus came accross the following:
$$
\large\int_{0}^{1}\frac{2x}{(1+x^2)\ln(\frac{1-x}{1+x})}dx = -\ln2
$$
I have ...
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Coin Toss Game HT subsequence [duplicate]
Two players, A and B alternatively toss a fair coin (A tosses the coin first, then B tosses the coin, then A, then B and so on). The sequence of heads and tails is recorded. If there is a head ...
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Finding Determinant of $2 \times 2$ Matrix in an elegant way [closed]
Let $A$ be a square matrix of order 2 with $|A| \neq 0$ such that $\big|A + |A|\operatorname{adj}(A)\big| = 0$, where $\operatorname{adj}(A)$ is the adjoint of matrix $A$, then the value of $\big|A − |...
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Question about probability, which involves different events.
In a lottery with 1000 numbers, there is only one prize in each draw. Antonio buys 2 tickets for a single draw and Paulo buys 2 tickets, one for each of 2 draws. Determine which of the two players has ...
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Finding the value of $x_0$ for $f(x)=e^{\sin(x)} + \sin(x)$ where $f(x_0)=0$ [closed]
This question is from a high school calculus course.
We had to find all critical points on an interval $[0,x_0]$ such that $f(x_0)=0$
For the function $$f(x)=e^{\sin(x)} + \sin(x)$$
Differentiating $f$...
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Proving Jacobi identity for a specific tensor
Following is an exercise problem V.1.9 from Kovantsov's "Differential geometry, topology and tensor analysis : a collection of exercises" found on the page 130 in the 1982 Russian edition.
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How many members have been on council in its history?
A friend of mine on BlueSky asked this question and I was completely at a loss to solve it. I'm hoping someone can help me with it. Here's the question: A council of vampires is formed 900 years ago. ...
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