# 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.

2,727 questions
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### Proving $\frac{\csc x + \cot x}{\tan x + \sin x} = \cot x\csc x$

I am currently working on understanding trig identities. A question has me stumped, and no matter how I look at it, it never leads to the proof. I believe I am making a mistake when dividing multiple ...
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### Petersen graph edge chromatic number

Hi I keep on getting 3 for the edge chromatic number of the Petersen graph. But the Petersen graph has edge chromatic number of 4 and I don’t know how to do that. Can someone please show this by ...
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### Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$ I don't know where to start as in is there a ...
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### Inscribe an equilateral triangle inside a triangle

Given a triangle ΔABC, how to draw all possible inscribed equilateral triangles with given side whose vertices lie on different sides of ΔABC?
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### How to make a kid understand geometry and help him solve problems?

I'm tutoring a 13 year-old boy, a middle school student. He has almost no problem with elementary algebra: he just applies the rules and everything falls into place. However he does struggle with ...
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### how to simplify and solve this equation :

Solve this equation for $x$: $$\left|x \sqrt{1-x^2} +x \right| = \sqrt{1+x^2}$$ I'm having a problem getting rid of the square root!
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### Is there anything that non-newtonian calculus can do, which newtonian calculus cannot?

As well, are there problems where non-newtonian calculus leads to a more elegant or simple solution than regular calculus?
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### 19 people on 8 benches with 1, 2 or 3 people. How many with only 2?

I found a math problem for 2nd graders in Norway. The article only presented drawing a table of everything and simply looking at it, instead of a mathematical approach, like a formula. I'm not sure ...
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### Solving $2y=\sqrt{3+\frac{1}{2y}}$

Any way to solve this irrational equation in $\mathbb{R}$? I think it has some artifice, but I do not see it $$2y=\sqrt{3+\frac{1}{2y}}$$
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### $m^{5}n+7$ and $n^{5}m+7$ are cubes of integer. [closed]

Find integer numbers $m$, $n$ such that $m^{5}n+7$ and $n^{5}m+7$ are cubes of integer. I guess the only pair is $(1;1)$, but I don't know what way we should follow to solve this problem. Help me ...
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### Given a vector between points $A$ and $B$, how determine the coordinates of point $C$ when $AC$ is collinear to $AB$ and we know the length of $AC$?

Say we have a vector $\vec{v}$ that defines the chemical bond between two atoms, and whose components are known $$\vec{v} = \begin{bmatrix}v_1 \\ v_2 \\ v_3 \end{bmatrix}$$ Lets define another ...
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### Every 4-regular simple graph contains a 3-regular subgraph

The following result was conjectured by Berge & Sauer, and proved by Tashkinov [T]. Theorem A. Every 4-regular simple graph contains a 3-regular subgraph. A simple graph is the one with no ...
### How to solve a $10$ degree Polynomial? (without using the normal tricks)
The problem specifically is a $x^{10}$ polynomial over $x^9$ polynomial that we have to graph. To graph, we need to find all the vertical intercepts so both polynomial need to be factored. Is there ...
### Express the equation $\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$ as a rational number. [closed]
$$\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$$ My first thought was to find the sum of this equation using induction to find a general solution that could bring this sum to a rational ...